canonically equivalent to $\mathcal{F}ol(X/B)$. This follows simply from the observation that the $\infty$-category of graded mixed $\mathbf{DR}(A/B)$-cdga's can be also expressed as the comma $\infty$-category of graded mixed cdga's under $\mathbf{DR}(A/B)$. By gluing we also deduce that $\mathcal{F}ol(X/Y)$ is equivalent to $\mathcal{F}ol(X/B)$ in the sense of Definition 2.1.3.3 for any derived stack $X$ with a map to $Y = \mathbf{Spec} B$.

**EXAMPLE 2.1.5.4. Closed 1-forms.** We recall from [PTVV13] the notion of closed $p$-forms on a derived stack $X$ (relative to $k$). This notion can be expressed easily in terms of graded mixed complexes. For this, we consider the graded mixed complex $k(p)[p-n]$, which is $k$ sitting in pure weight $p$ and cohomological deree $n-p$ (and both the cohomological differential as well as the mixed structure being zero). The space of closed $p$-forms on $X$ of degree $n$ is defined to be the mapping space of graded mixed complexes

$$ \mathcal{A}^{p,cl}(X/k) := \mathbf{Map}_{\epsilon-dg_k^{gr}}(k(p)[p-n], \mathbf{DR}(X/k)). $$

We can use here the equivalence between graded mixed complexes and complete filtered complexes in order to express the space of closed $p$-forms in a more traditional manner in terms of truncated de Rham complex $F^{-p}\hat{C}_{DR}(X/k) := F^{-p}|\mathbf{DR}(X/k)|^t$ (see Definition 1.4.2.1).

Let now $\omega : k(1)[1-n] \to \mathbf{DR}(X/k)$ be a closed 1-form on $X$ of degree $n$. As $\mathbf{DR}(X/k)$ is the graded mixed cdga of global sections of $\mathbf{DR}_{X/k}$ (see theorem 2.1.3.8), the form $\omega$ can also be considered as a morphism of stacks of graded mixed cdgas on $\mathbf{dAff}/X$

$$ \mathrm{Sym}_k(k(1)[1-n]) \to \mathbf{DR}_{X/k}. $$

We set

$$ \mathbf{DR}_{\omega} := \mathbf{DR}_{X/k} \otimes_{\mathrm{Sym}_k(k(1)[1-n])} k, $$

where we use the canonical augmentation morphism $\mathrm{Sym}_k(k(1)[1-n])k \to k$. Note that, as a stack of graded cdga's, $\mathbf{DR}_{\omega}$ is of the form $\mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\omega}[1]^{\mathrm{big}})$ where $\mathbb{L}_{\omega}[1]^{\mathrm{big}}$ is the $\mathcal{O}_X$-dg-module whose values on $\mathbf{Spec} A \to X$ is the cone of the morphism $A[-n] \to \mathbb{L}_{A/k}$ given by the restriction of $\omega$ as a 1-form of cohomological degree $n$ on $\mathbf{Spec} A$. It is clear from this description that $\mathbf{DR}_{\omega}$ defines an object in $\mathcal{F}ol(X/k)$, denoted by $\mathcal{F}_{\omega}$, and called the derived foliation associated to $\omega$. Note that the cotangent complex of $\mathcal{F}_{\omega}$ is by construction $(\mathbb{L}_{\omega}[1]^{\mathrm{big}})^{qcoh}$ and thus sits in an exact triangle of quasi-coherent complexes

$$ \mathcal{O}_X[-n] \to \mathbb{L}_{X/k} \to \mathbb{L}_{\mathcal{F}_{\omega}/k} $$

where the first morphism is defined by $\omega$. In particular, when $X$ is a smooth scheme over $k$ and $n=0$, a closed 1-form (of degree 0) on $X$ simply is a section $\omega$ of $\Omega_{X/k}^1$ such that $dR(\omega) = 0$. The derived foliation associated to $\omega$ has a cotangent complex given explicitly by a length two complex $\mathcal{O}_X \to \Omega_{X/k}^1$, where $\Omega_{X/k}^1$ sits in degree 0. We see in particular that the derived foliation $\mathcal{F}_{\omega}$ is smooth in the sense of Definition 2.1.4.5 if and only if $\omega$ is nowhere vanishing, or in other words if $\omega$ identifies $\mathcal{O}_X$ with a subbundle of $\Omega_{X/k}^1$. When $X$ is moreover proper over $k$, Hodge theory implies that any global section $\omega \in \Gamma(X, \Omega_{X/k}^1)$ is closed for the de Rham differential, and thus defines a derived foliation $\mathcal{F}_{\omega}$. This provides many examples of derived foliations on smooth proper schemes over $k$.
**EXAMPLE 2.1.5.5. Vector fields.** Let $X = \mathbf{Spec} A$ be a derived affine scheme over $k$ and $\nu : \mathbb{L}_{X/k} \to A$ be a morphism of $A$-dg-module. Such a morphism will be called a vector field on $X$ (relative to $k$). Note that this is the same thing as a $k$-linear derivation $A \to A$. We define a graded mixed cdga $\mathbf{DR}_{\nu}$ by considering $\mathrm{Sym}_A(A[1]) = A \oplus A[1]$ as a graded cdga, and define the mixed structure by the derivation $\nu : A \to A$. The weight zero part is equal here to $A$, and thus we have a natural induced morphism of graded mixed cdga's $\mathbf{DR}(A/k) \to \mathbf{DR}_{\nu}$. By construction $\mathbf{DR}_{\nu}$ defines a derived foliation on $X$ relative to $k$ whose cotangent complex is $\mathcal{O}_X$. By étale gluing this construction can be extended to any derived Deligne-Mumford stack $X$ and $\nu \in H^0(X, \mathbb{T}_{X/k})$. Extension to derived Artin stacks is trickier, can be done using formal localization techniques from [CPT$^{+}$17], but will not be discussed in this book. We can moreover extend this example by starting with a derivation $\nu : \mathbb{L}_{X/k} \to \mathcal{L}$, with $\mathcal{L}$ an arbitrary line bundle. The previous construction provides a derived foliation $F_{\nu}$ whose cotangent complex is $\mathcal{L}$ and whose sheaf of graded mixed cdga is $\mathrm{Sym}_{\mathcal{O}_X}(\mathcal{L}[1]) \simeq \mathcal{O}_X \oplus \mathcal{L}[1]$ endowed with the mixed structure defined by $\nu$.

**EXAMPLE 2.1.5.6. Group actions.** Let $X$ be a derived Artin stack and $G$ a smooth group scheme acting on $X$. We can form the quotient map $p : X \to Y = [X/G]$, which is a morphism of derived Artin stacks. The foliation $p^*(0_Y) \in \mathcal{F}ol(X/Y)$ or rather its image in $\mathcal{F}ol(X/k)$, is called the *foliation induced from the G-action*. Its big cotangent complex has values on $u : \mathbf{Spec} A \to X$ the $A$-dg-module $\mathbb{L}_{A/Y}$, and thus sits in an exact triangle

$$ \mathbb{L}_{X/Y} \to \mathbb{L}_{p^*(0_Y)}^{\text{big}} \to \mathbb{L}_{0_X}^{\text{big}}. $$

Applying the quasi-coherator we find that the natural morphism $\mathbb{L}_{X/Y} \to \mathbb{L}_{p^*(0_Y)}$ is an equivalence. Moreover, the cartesian square of derived stacks

<figure><img src="image_16.png" /></figure>

implies that $\mathbb{L}_{X/Y}$ is of the form $g^* \otimes_k \mathcal{O}_X$, where $g^* = \mathbb{L}_{*/BG}$ is the dual Lie algebra of $G$.

## 2.2. Comparison with derived Lie algebroids

For a commutative $k$-dg algebra $A$, $T_A := \mathrm{Der}_k(A, A)$ is the tangent $A$-dg module consisting of $k$-derivations $\partial : A \to A$. Note that the graded commutator $[\partial, \partial'] := \partial \circ \partial' + (-1)^{|\partial||\partial'|+1} \partial' \circ \partial$ endows $T_A$ with the structure of a dg-Lie algebra over $k$.

**DEFINITION 2.2.0.1.** Let $A$ be a commutative $k$-dg algebra. We denote by $\mathrm{dgLieAlgds}_A$ the category of dg-Lie algebroids over $A$, whose objects are pairs $(\mathfrak{g}, \rho)$ where $\mathfrak{g}$ is an (a priori, unbounded) $A$-dg module, endowed with a further structure $[\cdot, \cdot]_{\mathfrak{g}} : \mathfrak{g} \otimes_k \mathfrak{g} \to \mathfrak{g}$ of $k$-dg Lie algebras, and $\rho : \mathfrak{g} \to T_A$ is both a morphism of $A$-dg modules and of $k$-dg-Lie algebras, called
the anchor map, such that

$$ (8) \qquad [X, aY]_g = (-1)^{|X||a|} a[X, Y]_g + \rho(X)(a)Y. $$

Morphisms $(g, \rho) \to (g', \rho')$ in $dgLieAlgds_A$ are morphisms $g \to g'$ of $A$-dg modules and $k$-dg-Lie algebras commuting with the anchor maps.

When no confusion is possible, we will simply write $[ , ]$ instead of $[ , ]_g$ for the Lie bracket of a dg-Lie algebroid.

**REMARK 2.2.0.2.** Condition (8) can be viewed (and remembered) as a sort of a Leibniz rule, relating the anchor map to the failure of the bracket $[ , ]_g$ to be an $A$-linear dg-Lie bracket on $g$: the anchor map gives the correction to $A$-linearity of $[ , ]_g$ in the Leibniz rule (8).

**THEOREM 2.2.0.3.** [Nui19, Theorem 3.1] Let $A$ be a commutative $k$-dg algebra. The category $dgLieAlgds_A$ of dg-Lie algebroids over $A$ admits a right proper, tractable, semi-model structure whose fibrations are degreewise surjections, and whose weak equivalences are quasi-isomorphisms.

For the necessary background on semi-model structures, the reader is referred to [Nui19].

**DEFINITION 2.2.0.4.** Let $A$ be a commutative $k$-dg algebra, and $QA$ its cofibrant replacement. We denote by

* $dgLieAlgds_A$ the $\infty$-category obtained by Dwyer-Kan localization of $dgLieAlgds_{QA}$ along its weak-equivalences of Theorem 2.2.0.3.

* $dgLieAlgds_A^{\text{perf},\ge 0}$ the full $\infty$-subcategory of $dgLieAlgds_A$ consisting of dg-Lie algebroids $(g, \rho)$ where $g$ is a perfect $A$-dg-module cohomologically concentrated in degrees $[0, +\infty)$.

For a dg-Lie algebroid $(g, \rho)$ over $A$, we will consider its Chevalley-Eilenberg complex $\text{CE}(g, \rho)$, which we now define. Let $\text{Sym}_A(g[-1])$ be the symmetric algebra over $A$ of $g[-1]$ viewed as an $A$-dg-module.

**DEFINITION 2.2.0.5.** For a dg-Lie algebroid $(g, \rho)$ over $A$, we will denote by $\text{CE}^*(g, \rho)$ the $A$-linear dual of the $A$-dg-module $\text{Sym}_A(g[-1])$, endowed with the following structures:

* the grading $\text{CE}^*(g, \rho)^{(n)} := \text{Hom}_A(\text{Sym}_A^n(g[-1]), A)$, $n \ge 0$.

* the graded commutative $A$-dg-algebra structure induced by being the $A$-dual of the graded cocommutative coalgebra $\text{Sym}_A(g[-1])$.

* the internal differential

$$ d : (\text{CE}^*(g, \rho)^{(n)})^m \to (\text{CE}^*(g, \rho)^{(n)})^{m+1}, \omega \mapsto d_A \circ \omega - (-1)^m \omega \circ d_{\text{Sym}_A^n(g[-1])} $$

i.e. the differential of the $A$-dg module $\text{Hom}_A(\text{Sym}_A^n(g[-1]), A)$.

* the mixed structure

$$ \epsilon_{\text{CE}}^{(n)} : \text{CE}^*(g, \rho)^{(n)} \to (\text{CE}^*(g, \rho)^{(n+1})[-1] $$
is given by (omitting Koszul signs)

$$ \epsilon_{\mathrm{CE}}^{(n)}(\omega)(\xi_1, \dots, \xi_{n+1}) = \sum_i \rho(\xi_i)(\omega(\xi_1, \dots, \not\xi_i, \dots, \xi_{n+1})) - \sum_{i<j} \omega([\xi_i, \xi_j], \xi_1, \dots, \not\xi_i, \dots, \not\xi_j, \dots, \xi_{n+1}). $$

Endowed with the above structures, $\mathrm{CE}^*(\mathfrak{g}, \rho)$ becomes a graded mixed cdga over $k$.

**REMARK 2.2.0.6.** Usually, $\mathrm{CE}^*(\mathfrak{g}, \rho)$ is regarded as a cdga by taking the total differential, given by the sum of the internal differential and the mixed one. However, as we stressed out in several other points, it is quite important for our purposes to keep the two differentials separated and view it as a graded mixed cdga.

Since $\mathrm{DR}(-/k)$ is left adjoint to the weight 0 part functor (see Section 1.4), and

$$ \mathrm{CE}^*(\mathfrak{g}, \rho)^{(0)} = \mathrm{Hom}_A(A, A), $$

the identity map $A \to A$ induces a morphism of graded mixed cdga's $\mathrm{DR}(A/k) \to \mathrm{CE}^*(\mathfrak{g}, \rho)$. Therefore, if $\mathfrak{g}$ is a perfect $A$-dg-module, this yields the structure of a derived foliation on $\mathrm{CE}^*(\mathfrak{g}, \rho)$ (Definition 2.1.1.1).

The following theorem expresses the precise comparison between perfect foliations and perfect dg-Lie algebroids.

**THEOREM 2.2.0.7.** Let $A$ be a connective $k$-linear cdga of finite presentation over $k$. The $\infty$-functor sending a dg-Lie algebroid $(\mathfrak{g}, \rho)$ over $A$ to its Chevalley complex $\mathrm{CE}^*(\mathfrak{g}, \rho)$ induces a equivalence

$$ \mathrm{CE}^* : (\mathrm{dgLieAlgds}_A^{\mathrm{perf}})^{\mathrm{op}} \simeq \mathrm{Fol}^{\mathrm{perf}}(A/k), $$

where $\mathrm{dgLieAlgds}_A^{\mathrm{perf}}$ is the full sub-$\infty$-category of dg-Lie algebroids which are perfect as an $A$-dg-module, and $\mathrm{Fol}^{\mathrm{perf}}(A/k)$ is the $\infty$-category of perfect derived foliations over $A$.

**PROOF.** This is a consequence of [Fu12, Theorem 5.15] □

As an immediate consequence, we get the following corollary which relates perfect derived foliations over the point $\mathrm{Spec} k$ and dg-Lie algebras over $k$ which are perfect as $k$-dg-modules. We note also that the above theorem extends to almost perfect derived foliations under the condition that $A$ is furthermore cohomologically bounded. We can thus extract the most important case of application for us, which is the following corollary.

**COROLLARY 2.2.0.8.** The Chevalley complex construction induces an equivalence of $\infty$-categories

$$ \mathrm{dgLie}_k^{\mathrm{dap}} \simeq \mathrm{Fol}^{\mathrm{ap}}(k/k) $$

between dual almost perfect dg-Lie algebras over $k$ and almost perfect derived foliations over $k$.

In the previous corollary *dual almost perfect* means that the $k$-linear dual is almost perfect. As $k$ is a field this condition is equivalent to the condition of being cohomologically bounded on the left and with finite dimensional individual cohomology spaces.
## 2.3. Formal structures

All along this section, we will assume that our base ring $k$ is a field (of characteristic zero). We will be interested in derived foliations restricted to formal derived stacks, and their interpretations in terms of dg-Lie algebras over $k$. These results are not optimal and replacing $k$ be a more general base is also possible. However, the general notion of formal derived stacks over general bases is out of the scope of this book, and we refer the interested reader to [BMN25] for recent and pretty optimal results in that direction.

**2.3.1. Derived stacks and formal moduli problems.** In this section we recall the fundamental results of [Lur10, Lur11] (see also [Lur22b, §IV] and [Toë17]), relating pointed derived formal stacks and dg-lie algebras. For this, we first introduce $\mathrm{dgart}_k$, the $\infty$-category of commutative, connective, Artinian local dg-algebras over $k$. By definition, this is the full sub-$\infty$-category of $\mathrm{cdga}_k$ consisting of objects $A$ satisfying the following conditions:

* $H^i(A) = 0$ for all $i > 0$

* $H^0(A)$ is a local Artinian $k$-algebra

* $H^*(A)$ is finite dimensional over $k$.

Note that the third condition implies in particular that $H^i(A)$ is finite dimensional for all $i$, but also that $H^i(A) = 0$ when $i \ll 0$. In a similar manner, we denote by $\mathrm{dgart}_k^*$ the $\infty$-category of pointed objects in $\mathrm{dgart}_k$, or equivalently the $\infty$-category of augmented connective Artinian dg-algebras. Note that in particular, if $A \in \mathrm{dgart}_k^*$ the residue field of $H^0(A)$ is then automatically $k$ itself.

**DEFINITION 2.3.1.1.** A formal moduli problem is an $\infty$-functor

$$F : \mathrm{dgart}_k^* \to \mathrm{Top}$$

satisfying the following conditions.

(1) The space $F(k)$ is contractible.

(2) For any cartesian square in $\mathrm{dgart}_k^*$

$$\begin{tikzcd}
    B' \arrow[r] \arrow[d] & A' \\
    B \arrow[r] \arrow[d] & A
\end{tikzcd}$$

such that $H^0(A') \to H^0(A)$ is surjective, the square

$$\begin{tikzcd}
    F(B') \arrow[r] \arrow[d] & F(A') \\
    F(B) \arrow[r] \arrow[d] & F(A)
\end{tikzcd}$$

is cartesian in Top.
We thus have constructed an adjunction<fnref n="1" />

$$ \mathcal{M}\mathcal{C} : \mathrm{dgLie}_k \stackrel{\sim}{\to} \mathrm{dSt}_k^* : \ell_- . $$

The $\infty$-functor $\ell_-$ sends a pointed derived stack $(X, x)$ to the dg-lie algebra $\ell_{(X,x)}$ controlling the formal moduli problem $(X, x)^\wedge$, and $\mathcal{M}\mathcal{C}$ is its left adjoint. We have then the following important observation.

**PROPOSITON 2.3.1.2.** *The $\infty$-functor*

$$ \mathcal{M}\mathcal{C} : \mathrm{dgLie}_k \to \mathrm{dSt}_k^* $$

*is fully faithful.*

*Proof.* As a first reduction, we can start by noticing that $k \mapsto \mathrm{dgLie}_k$ and $k \mapsto \mathrm{dSt}_k^*$ are both stacks for the fpqc topology on $k$. Therefore, the $\infty$-functor $\mathcal{M}\mathcal{C}$ being compatible with the change of base field $k$, we can assume that $k$ is algebraically closed.

By construction, the $\infty$-functor $\mathcal{M}\mathcal{C}$ is equivalent to the composition

$$ j : \mathrm{FMP}_k \to \mathrm{Fun}^\infty(\mathrm{dgart}_k^*, \mathrm{Top}) \xrightarrow{\phi} \mathrm{dSt}_k^* $$

Here, the first $\infty$-functor is the canonical embedding and the second $\infty$-functor

$$ \phi : \mathrm{Fun}^\infty(\mathrm{dgart}_k^*, \mathrm{Top}) \to \mathrm{dSt}_k^* $$

is the composition of the left Kan extension

$$ i_! : \mathrm{Fun}^\infty(\mathrm{dgart}_k^*, \mathrm{Top}) \hookrightarrow \mathrm{Fun}^\infty(\mathrm{cdga}_k^{\le 0}, \mathrm{Top}) $$

along the inclusion $i : \mathrm{dgart}_k \hookrightarrow \mathrm{cdga}_k^{\le 0}$, with the associated stack $\infty$-functor

$$ \mathrm{Fun}^\infty(\mathrm{cdga}_k^{\le 0}, \mathrm{Top}) \to \mathrm{dSt}_k^* $$

It is therefore enough to show that $\phi$ is fully faithful. The right adjoint to $\phi$ is the restriction $\psi$ sending a pointed stack $F : \mathrm{cdga}_k^{\le 0} \to \mathrm{Top}$ to its restriction to Artinian dg-algebras. However, as $k$ is algebraically closed, any Artinian local dg-algebra is trivial for the étale topology, and thus for any prestack $F$, with associated stack $F'$, the natural map $F \to F'$ induces equivalences $F(A) \simeq F'(A)$ for any Artinian local $k$-algebra $A$. As a consequence, for any $F \in \mathrm{Fun}^\infty(\mathrm{dgart}_k^*, \mathrm{Top})$, the adjunction morphism $F \to \psi\phi(F)$, is simply equivalent to the adjunction morphims $F \to i^*i_!(F)$, which is an equivalence as the left Kan extension

$$ i_! : \mathrm{Fun}^\infty(\mathrm{dgart}_k^*, \mathrm{Top}) \hookrightarrow \mathrm{Fun}^\infty(\mathrm{cdga}_k^{\le 0}, \mathrm{Top}) $$

is fully faithful.

□

We will mainly use the previous adjunction when restricted to derived Artin stacks. We note that for such a pointed derived Artin stack $(X, x)$, the underlying complex of the dg-lie

<footnote n="1">¹The functor $\mathcal{M}\mathcal{C}$ is so named after L. Maurer and E. Cartan.</footnote>
algebra $\ell_{(X,x)}$ is functorialy quasi-isomorphic to $T_{X,x}[-1]$, the shifted tangent complex of $X$ at $x$. This follows directly from a simple unfolding of the various definitions.

Finally, because of Proposition 2.3.1.2 we will often consider formal moduli problems as fully embedded in pointed derived stacks. In particular, we will often use the notation $(X, x)^\wedge$ to denote also the pointed derived stack $\mathcal{MC}((X, x)^\wedge)$ and call it the *formal completion of X* at $x$. This formal completion $(X, x)^\wedge$ can be in fact described in the usual manner on smooth atlases of $X$. Assuming first that $X = \text{Spec } A$ is a pointed affine derived scheme, the point provides an augmentation $e: A \to k$, and we can define a pro-object in $\text{dgart}_k^*$, called the formal completion of $A$ along $e$, by

$$ \hat{A} := \lim_{A \to A'} A', $$

where $A'$ runs in the $\infty$-category of morphism of augmented cdga morphism $A \to A'$, where $A'$ is a local Artinian cdga. When $A$ is an underived $k$-algebra of finite type, then it can be shown that this pro-object is equivalent to the usual formal completion $\lim_n A/m^n$ where $m \subset A$ is the maximal ideal corresponding to the augmentation $e$ (see Proposition A.3.0.2). The pro-object $\hat{A}$ pro-represents $(X, x)^\wedge$ is the sense that one has

$$ (X, x)^\wedge \simeq \text{Spf}(\hat{A}) = \text{colim} \, \text{Spec } A'. $$

In general, up to base changing to a finite extension $k'/k$, we can find a smooth morphism $p: U \to X$ of pointed derived stacks for which $U$ is affine and such that $x$ lies in the image of $p$. We can form the nerve $U_* \to X$ of $p$. We then have

$$ (X, x)^\wedge \simeq \text{colim}_n (U_n, x)^\wedge. $$

We can this way compute the formal completion of $(X, x)^\wedge$ inductively on the geometricity of $X$. We could also replace the nerve of $p$ here by any pointed smooth hypercovering (locally around $x$) $U_* \to X$ where all $U_i$ are affine, and get this way a formula for $(X, x)^\wedge$ as the colimit of formal completions $(U_n, x)^\wedge$ which can individually be described as the formal spectrum of a pro-Artinian cdga.

**2.3.2. Derived foliations on formal moduli problems.** In this part we show how derived foliations can be defined over formal moduli problems. For this we simply use the fully faithful $\infty$-functor defined in the previous section $j: \text{FMP}_k \to \text{dSt}_k^*$.

**DEFINITION 2.3.2.1.** For a formal moduli problem $X \in \text{FMP}_k$ we set

$$ \mathcal{Fol}(X/k) := \mathcal{Fol}(j(X)/k). $$

When the base field is clear we will simply use the notation $\mathcal{Fol}(X)$.

Note that $X \mapsto \mathcal{Fol}(X)$ naturally defines an $\infty$-functor $\mathcal{Fol}^{op}: \text{FMP}_k \to \text{Cat}_\infty$. Unfolding definitions and constructions we can give the following explicit formula

$$ \mathcal{Fol}(X) \simeq \lim_{\text{Spec } A \to X} \mathcal{Fol}(A), $$
where **Spec** A runs over the $\infty$-category of $A \in \mathbf{dgart}_k^*$ endowed with a morphism **Spec** A $\to$ X
(concretely the comma $\infty$-category **Spec**/X where **Spec** : $\mathbf{dgart}_k^{op} \to \mathbf{FMP}_k$ is the Yoneda
embedding).

Note that, for any pointed derived stack $(X, x) \in \mathbf{dSt}_k^*$, there exists a canonical $\infty$-functor

$$
\mathcal{F}\mathit{ol}(X) \to \mathcal{F}\mathit{ol}((X, x)^{\wedge}).
$$

Indeed, by construction of the adjunction between $j$ and $(-)^{\wedge}$, we have a natural adjunction morphism of derived stack $i_x : (X, x)^{\wedge} \to X$, and thus an induced pull-back $\infty$-functor $i_x^* : \mathcal{F}\mathit{ol}(X) \to \mathcal{F}\mathit{ol}((X, x)^{\wedge})$.

**DEFINITION 2.3.2.2.** For a pointed derived stack $(X, x)$ the $\infty$-functor $i_x^* : \mathcal{F}\mathit{ol}(X) \to \mathcal{F}\mathit{ol}((X, x)^{\wedge})$ above is the formalization at $x$, or formal completion at $x$. It is denoted by $\mathcal{F} \mapsto \hat{\mathcal{F}}_x$.

Proposition 2.3.2.3 below shows that derived foliations which are almost perfect, can be
described explicitly on the formal completion of an affine derived scheme. Before explaining
this, let A be a connective cdga with an augmentation A $\to$ k, and assume that A is almost
of finite presentation (see appendix §A.2). Wet set X = **Spec** A endowed with its natural
point x $\in$ X(k) associated to the augmentation. We denote by $\hat{A}_x$ (or simply by $\hat{A}$) the formal
completion of A at the augmentation x : A $\to$ k. Recall (see appendix §A.3) that this is
the pro-object in **dgart**_k* given by " lim$_{A\to A'}$ A', where A' runs in the $\infty$-category of all local
Artinian cdga with an augmentation preserving morphism A $\to$ A'. We let X = **Spec** A and
(X, x)^$\wedge$ := **Spf**($\hat{A}$) be the corresponding derived affine schemes and formal completion.

We define the graded mixed cdga $\mathbf{D}\mathbf{R}(\hat{\mathcal{A}}/k)$, called the *formal de Rham algebra of $\hat{\mathcal{A}}$*, by
the following limit

$$
\mathbf{D}\mathbf{R}(\hat{\mathcal{A}}/k) := \lim_{A \to A'} \mathbf{D}\mathbf{R}(A'/k).
$$

By Corollary A.3.0.5 we know that, as graded cdga's we have

$$
\mathbf{D}\mathbf{R}(\hat{\mathcal{A}}/k) \simeq \mathrm{Sym}_{\hat{\mathcal{A}}}(\hat{\mathbb{L}}_{A/k}[1]),
$$

where $\hat{A} = \lim_{A \to A'} A'$ is the realization of the pro-object $\hat{A}$, and $\hat{\mathbb{L}}_{A/k} = \lim_{A \to A'} \mathbb{L}_{A'/k}$ is the formal cotangent complex of $\hat{A}$. Now, the limit $\infty$-functor provides a natural $\infty$-functor

$$
\psi : \mathcal{F}\mathit{ol}((X, x)^{\wedge}) = \lim_{A \to A'} \mathcal{F}\mathit{ol}(\mathit{Spec} A') \to (\mathbf{D}\mathbf{R}(\hat{A}/k) - \epsilon - \mathbf{cdga}_k^{gr})^{\mathrm{op}},
$$

from derived foliations on the formal completion of $X$ at $x$ to graded mixed commutative $\mathbf{D}\mathbf{R}(\hat{A}/k)$-algebras. Explicitly, the $\infty$-functor $\psi$ sends an object $\{\mathcal{F}'_A\} \in \lim_{A \to A'} \mathcal{F}\mathit{ol}(\mathit{Spec} A')$ to $\lim_{A \to A'} \mathbf{DR}(\mathcal{F}'_A)$, as an algebra over $\lim_{A \to A'} \mathbf{DR}(A'/k)$.

**PROPOSITON 2.3.2.3.** Let $(X, x) = \mathbf{Spec} A$ be a pointed derived affine scheme almost of finite presentation over $k$. The $\infty$-functor

$$
\psi : \mathcal{F}\mathit{ol}((X, x)^{\wedge}) = \lim_{A \to A'} \mathcal{F}\mathit{ol}(\mathit{Spec} A') \to (\hat{\mathbf{D}}\mathbf{R}(\hat{A}/k) - \epsilon - \mathbf{cdga}_k^{gr})^{\mathrm{op}}
$$
constructed above, is fully faithful when restricted to almost perfect derived foliations. Its essential image consists of all graded mixed commutative $\mathbf{DR}(\hat{A}/k)$-algebras whose underlying graded cdga's are of the form $\mathrm{Sym}_{\hat{A}}(E)$, with $E$ an almost perfect $\hat{A}$-dg-module.

*Proof.* The limit $\infty$-functor

$$ \lim_{A \to A'} (\mathbf{DR}(A'/k) - \epsilon - \mathbf{cdga}_k^{gr}) \to \mathbf{DR}(\hat{A}/k) - \epsilon - \mathbf{cdga}_k^{gr} $$

admits a left adjoint $\phi$, which sends a graded mixed commutative $\mathbf{DR}(\hat{A}/k)$-algbera $B$ to the family of objects $\mathbf{DR}(A'/k) \otimes_{\mathbf{DR}(\hat{A}/k)} B$, together with their natural transition morphisms.

We denote by $\mathcal{C} \subset \mathbf{DR}(\hat{A}/k) - \epsilon - \mathbf{cdga}^{gr}$ be the full sub-$\infty$-category consisting of $B$ which, as graded $\mathbf{DR}(\hat{A}/k)$-cdga's, are of the form $\mathrm{Sym}_{\hat{A}}(E)$ for some $E \in \mathbf{APerf}(\hat{A})$. We first observe that the essential image of $\phi$ is contained in $\mathcal{C}$. Let $\mathcal{F} \in \lim_{A \to A'} \mathcal{F}ol^{ap}Spec A'$) be a derived foliation on $(X, x)^\wedge$ which is assumed to be almost perfect It consists of a family of $\mathbf{DR}(\mathcal{F}_{A'})$ of graded mixed $\mathbf{DR}(A'/k)$-cdga, which, as graded $\mathbf{DR}(A'/k)$-algebras, are of the form $\mathrm{Sym}_{A'}(\mathbb{L}_{\mathcal{F}_{A'}}[1])$ for an almost perfect $A'$-dg-module $\mathbb{L}_{\mathcal{F}_{A'}} \in \mathbf{APerf}(A')$. We set $\mathbb{L}_{\mathcal{F}} := \lim_{A \to A'} \mathbb{L}_{\mathcal{F}_{A'}}$, which is an $\hat{A}$-module, and let $\mathcal{N}_{\mathcal{F}}^*$ be the fiber of the natural morphism $\hat{\mathbb{L}}_A \otimes_{\hat{A}} A' \to \mathbb{L}_{\mathcal{F}}$.

We consider the natural commutative diagram of exact triangles of pro-$\hat{A}$-dg-modules

$$ \begin{tikzcd} \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node (A) at (0,0) {$\mathbf{DR}(\hat{A}/k) - \epsilon - \mathbf{cdga}^{gr}$}; \node (B) at (1,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (C) at (2,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (D) at (3,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (E) at (4,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (F) at (5,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (G) at (6,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (H) at (7,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (I) at (8,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (J) at (9,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (K) at (10,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (L) at (11,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (M) at (12,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (N) at (13,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (O) at (14,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (P) at (15,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Q) at (16,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (R) at (17,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (S) at (18,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (T) at (19,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (U) at (20,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (V) at (21,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (W) at (22,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (X) at (23,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Y) at (24,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Z) at (25,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (A1) at (0,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (B1) at (1,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (C1) at (2,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (D1) at (3,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (E1) at (4,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (F1) at (5,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (G1) at (6,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (H1) at (7,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (I1) at (8,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (J1) at (9,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (K1) at (10,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (L1) at (11,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (M1) at (12,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (N1) at (13,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (O1) at (14,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (P1) at (15,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Q1) at (16,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (R1) at (17,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (S1) at (18,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (T1) at (19,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (U1) at (20,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (V1) at (21,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (W1) at (22,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (X1) at (23,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Y1) at (24,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Z1) at (25,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (A2) at (0,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (B2) at (1,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (C2) at (2,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (D2) at (3,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (E2) at (4,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (F2) at (5,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (G2) at (6,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (H2) at (7,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (I2) at (8,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (J2) at (9,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (K2) at (10,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (L2) at (11,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (M2) at (12,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (N2) at (13,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (O2) at (14,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (P2) at (15,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Q2) at (16,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (R2) at (17,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (S2) at (18,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (T2) at (19,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (U2) at (20,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (V2) at (21,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (W2) at (22,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (X2) at (23,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Y2) at (24,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Z2) at (25,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (A3) at (0,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (B3) at (1,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (C3) at (2,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (D3) at (3,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (E3) at (4,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (F3) at (5,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (G3) at (6,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (H3) at (7,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (I3) at (8,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (J3) at (9,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (K3) at (10,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (L3) at (11,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (M3) at (12,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (N3) at (13,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (O3) at (14,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (P3) at (15,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Q3) at (16,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (R3) at (17,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (S3) at (18,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (T3) at (19,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (U3) at (20,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (V3) at (21,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (W3) at (22,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (X3) at (23,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Y3) at (24,0) {$\mathbf{DR}(\hat{A}/k)$}; \node (Z3) at (25,0) {$\mathbf{DR}(\hat{A}/k)$}; \end{tikzcd} $$

The vertical left arrow is an equivalence of pro-objects thanks to theorem A.3.0.5. The right vertical arrow is also an equivalence (and even a levelwise equivalence) thanks to the fact that the natural $\infty$-functor

$$ \lim_{A \to A'} \mathbf{APerf}(A') \to \mathbf{APerf}(\hat{A}) $$

is an equivalence of $\infty$-categories (see appendix A.4.0.1) and its inverse is given by $E \mapsto \lim_{A \to A'} E \otimes_{\hat{A}} A'$. Therefore, the natural morphism $\lim_{A \to A'} (\mathbb{L}_{\mathcal{F}} \otimes_{\hat{A}} A') \to \lim_{A \to A'} \mathbb{L}_{\mathcal{F}_{A'}}$ is again an equivalence of pro-$\hat{A}$-dg-modules. As a consequence, we see that for all $p$,

$$ \lim_{A \to A'} (\wedge_{\hat{A}}^{p}(\mathbb{L}_{\mathcal{F}}) \otimes_{\hat{A}} A') \to \lim_{A \to A'} \wedge_{A'}^{p} \mathbb{L}_{\mathcal{F}_{A'}} $$

is an equivalence of pro-objects. This implies in particular that $\psi(\mathcal{F})$ lies indeed in the sub-$\infty$-category $\mathcal{C}$.

We thus have seen that the adjunction

$$ \phi : \mathbf{DR}(\hat{A}/k) - \epsilon - \mathbf{cdga}_k^{gr} \rightleftharpoons \lim_{A \to A'} \mathbf{DR}(A'/k) - \epsilon - \mathbf{cdga}_k^{gr} : \psi $$
restricts to an adjunction

$$ \phi : \mathcal{C} \underset{\sim}{\longrightarrow} \lim_{A \to A'} (\mathcal{F}ol^{ap}(\mathbf{Spec} A'/k)) : \psi. $$

To finish the proof of the proposition we need to check that the unit and counit of this adjunction are equivalence. Let $B \in \mathcal{C}$, and $\mathbb{L} := B^{(1)}[1]$ be its weight one part. It is an object in $\mathbf{APerf}(\hat{A})$ by assumption, and thus if we denote by $\mathcal{N}_{\mathbb{L}}^*$ the fiber of $\mathbb{L}_A \to \mathbb{L}$, we have that $\mathcal{N}_{\mathbb{L}}^*$ is an almost perfect $\hat{A}$-dg-module. The adjunction morphism $B\psi(\phi(B))$, on the weight one part, is equivalent to the canonical morphism $\mathbb{L} \to \lim_{A \to A'} \mathbb{L} \otimes_{\hat{A}} A'$. This is an equivalence thanks to proposition A.4.0.1. This shows that $\phi$ is fully faithful. But $\psi$ is also conservative. Indeed, if a morphism $f: \mathcal{F} \to \mathcal{F}'$ in $\lim_{A \to A'} \mathcal{F}ol^{ap}(\mathbf{Spec} A')$ induces an equivalence after applying $\phi$, then in particular the induced morphism on conormal complexes $\lim_{A \to A'} \mathcal{N}_{\mathbb{L}_{A'}}^* \to \lim_{A \to A'} \mathcal{N}_{\mathbb{L}_{A'}}^*$ is an equivalence. The limit $\infty$-functor $\lim_{A \to A'} \mathbf{APerf}(A') \to \mathbf{APerf}(\hat{A})$ is an equivalence, and is thus conservative, we have therefore that for each $A'$ the induced morphism $\mathcal{N}_{\mathbb{L}_{A'}}^* \to \mathcal{N}_{\mathbb{L}_{A'}}^*$ is an equivalence, and thus that $\mathcal{F}_{A'} \to \mathcal{F}_{A'}$ is an equivalence of derived foliations on $\mathbf{Spec} A'$. This shows that $\psi$ is conservative and thus the proof of the Proposition. $\square$

Finally, we have the formal version of Theorem 2.1.3.8. For this, recall that $X^{art}$ is the $\infty$-site of pointed morphisms $\mathbf{Spec} A \to X$ with $A$ a local Artinian connective cdga (see §A.6). It comes equipped with the canonical stack of cdga's $\mathcal{O}_{X^{art}}$, as well as its $\infty$-category of $\mathcal{O}_{X^{art}}$-dg-modules. We have the full sub-$\infty$-category $\mathbf{QCoh}(X^{art}) \subset \mathcal{O}_{X^{art}} - \mathbf{dg}$ of quasi-coherent complexes, which can canonically be identified with $\mathbf{QCoh}((X, x)^\wedge)$. Any derived foliation $\mathcal{F} \in \mathcal{F}ol((X, x)^\wedge)$ possesses a big cotangent complex $\mathbb{L}_{\mathbb{F}/k}^{\text{big}} \in \mathcal{O}_{X^{art}} - \mathbf{dg}$, and a cotangent complex $\mathbb{L}_{\mathbb{F}/k} \in \mathbf{QCoh}(X^{art})$. By definition, they are related by the formula $\mathbb{L}_{\mathbb{F}/k} = (\mathbb{L}_{\mathbb{F}/k}^{\text{big}})^{qcoh}$, where $(-)^{qcoh}: \mathcal{O}_{X^{art}} - \mathbf{dg} \to \mathbf{QCoh}(X^{art})$ is the right adjoint to the natural inclusion $\infty$-functor.

**PROPOSITON 2.3.2.4.** Let $(X, x)$ be a pointed derived stack locally almost of finite presentation and $(X, x)^\wedge$ its formal completion. Then, for any almost perfect derived foliation $\mathcal{F} \in \mathcal{F}ol^{ap}((X, x)^\wedge)$ such that $\mathcal{N}_{\mathbb{L}}^*$ is perfect, and all $i \ge 0$ the canonical morphism

$$ \wedge_{\mathcal{O}_{X^{art}}}^{i} \mathbb{L}_{\mathbb{F}} \longrightarrow (\wedge_{\mathcal{O}_{X^{art}}}^{i} \mathbb{L}_{\mathbb{F}}^{\text{big}})^{qcoh} $$

is a quasi-isomorphism.

*Proof.* This is exactly the same proof as that of Theorem 2.1.3.8, using Theorem A.6.0.1 instead of Theorem A.5.0.1. We leave the details to the reader. $\square$

**2.3.3. Formal integrability.** We will now show that almost perfect derived foliations on the formal completion of a derived Artin stack locally almost of finite presentation can be described purely in algebraic terms, using dg-Lie algebras. This classification result also implies that these derived foliations are always formally integrable at each point.
Recall that the conormal complex of a derived foliation $\mathcal{F}$ is defined by the exact triangle

$$ \mathcal{N}_{\mathcal{F}}^{*} \longrightarrow \mathbb{L}_{X} \longrightarrow \mathbb{L}_{\mathcal{F}}. $$

An important remark is that $\mathcal{N}_{\mathcal{F}}^{*}$ can be defined using either the cotangent quasi-coherent complexes or the big cotangent complexes. Indeed, the commutative square

$$ \begin{tikzcd} \mathbb{L}_{X} \arrow[r] \arrow[d] & \mathbb{L}_{\mathcal{F}} \arrow[d] \\ \mathbb{L}_{X}^{\text{big}} \arrow[r] & \mathbb{L}_{\mathcal{F}}^{\text{big}} \end{tikzcd} $$

is always cartesian (see Lemma 2.1.3.9) so that the fibers of the horizontal arrows agree. As a result, $\mathcal{N}_{\mathcal{F}}^{*}$ always makes sense as an object in $\mathcal{QCoh}(X)$, even if $X$ is merely a derived stack (i.e. not necessarily Artin). Another important fact is that conormal complexes of derived foliations are stable by pull-backs (as opposed to cotangent complexes themselves), and therefore almost perfect derived foliations form a full substack $\mathcal{Fol}^{ap} \subset \mathcal{Fol}$ of the stack of derived foliations. In particular, for any formal moduli problem $X \in \mathbf{FMP}$, we get a full sub-$\infty$-category $\mathcal{Fol}^{ap}(X) \subset \mathcal{Fol}(X)$, simply defined by $\mathcal{Fol}^{ap}(j(X))$ (Definition 2.3.2.1).

Let now $(X, x)$ be a pointed derived Artin stack locally almost of finite presentation and $\mathcal{F} \in \mathcal{Fol}((X, x)^{\wedge})$ a derived foliation on the formal completion of $X$ at $x$. The natural base point $x : \mathbf{Spec} k \to (X, x)^{\wedge}$ provides $x^{*}(\mathcal{F}) \in \mathcal{Fol}(\ast)$. Moreover, the tautological derived foliation $\mathfrak{O}_{X}$ can be formally completed to $\mathfrak{O}_{X_x} \in \mathcal{Fol}((X, x)^{\wedge})$.

**LEMMA 2.3.3.1.** *The object $\mathfrak{O}_{X_x}$ is the initial object in $\mathcal{Fol}((X, x)^{\wedge})$.*

*Proof.* The formal completion construction $\mathcal{F} \mapsto \hat{\mathcal{F}}_x$ (Definition 2.3.2.2) being obtained by restriction, the conormal complex $\mathcal{N}_{\hat{\mathcal{F}}_x}$ is simply the restriction of $\mathcal{N}_{\mathcal{F}}$. Applied to the case of $\mathcal{F} = \mathfrak{O}_X$, we see that the conormal complex of $\mathfrak{O}_{X_x}$ is identified with $\mathbb{L}_{(X,x)^{\wedge}}$, via the canonical morphism. Equivalently, $\mathbb{L}_{\mathfrak{O}_{X_x}} \simeq 0$. We then finish the proof using the exact same argument as in the non-formal case (see example 2.1.5.1, by replacing the big site $\mathbf{dAff}/X$ with $\mathbf{dgart}_k^*/X$, and using Proposition 2.1.4.6 (and Proposition 2.3.2.4 instead of Theorem 2.1.3.8). $\square$

Lemma 2.3.3.1 provides a canonical morphism $x^{*}(\mathfrak{O}_{X}) \to x^{*}(\mathcal{F})$, of derived foliations over $\mathbf{Spec} k$. This defines an $\infty$-functor $\psi : \mathcal{Fol}((X, x)^{\wedge}) \to x^{*}(\mathfrak{O}_{X})/\mathcal{Fol}(\ast)$, which can be restricted to perfect derived foliations

$$ \psi : \mathcal{Fol}^{ap}((X, x)^{\wedge}) \to x^{*}(\mathfrak{O}_{X})/\mathcal{Fol}^{ap}(\ast). $$

**THEOREM 2.3.3.2.** *With the above notations and assumptions, the $\infty$-functor*

$$ \psi : \mathcal{Fol}^{ap}((X, x)^{\wedge}) \to x^{*}(\mathfrak{O}_{X})/\mathcal{Fol}^{ap}(\ast) $$

*is an equivalence of $\infty$-categories.*
Before proving this theorem, we state its most important corollary. The target $\infty$-category of the equivalence of Theorem 2.3.3.2 can thus be identified, according to corollary 2.2.0.8, with the comma $\infty$-category of dual almost perfect dg-Lie algebras $t_X/\mathrm{dgLie}_k^{dap}$. Here $t_X$ is the dg-Lie algebra corresponding to $x^*(0_X)$ according to the equivalence of 2.2.0.8. As this equivalence is functorial, it is easy to check that $x^*(0_X)$ is the dg-Lie algebra obtained by pulling back the dg-Lie algebroid $\mathbb{T}_X$ to the point $x$, which is nothing else that $\ell_{(X,x)}$, corresponding to the formal moduli problem $(X, x)^\wedge$. The main corollary of Theorem 2.3.3.2 thus reads as follows.

**COROLLARY 2.3.3.3.** Let $(X, x)$ be a pointed derived Artin stack locally almost of finite presentation. Then, the $\infty$-functor $\psi$ combined with the equivalence of Corollary 2.2.0.8 induces an equivalence of $\infty$-categories

$$ \mathcal{F}\mathrm{ol}^{ap}((X, x)^\wedge) \simeq \ell_{(X,x)}/\mathrm{dgLie}_k^{dap}, $$

between almost perfect derived foliations on the formal completion $(X, x)^\wedge$ and the comma $\infty$-category of dual almost perfect dg-Lie algebras under $\ell_{(X,x)}$. This equivalence will be denoted by

$$ \mathcal{F} \mapsto \ell_{\mathcal{F}}. $$

The above corollary states formal integrability of derived foliations. Indeed, for any $\mathcal{F} \in \mathcal{F}\mathrm{ol}^{ap}(X)$, its formal completion at a $k$-point $x \in X(k)$ provides by the equivalence of the corollary, a morphism of dg-Lie algebras $\ell_{(X,x)} \to \ell_{\hat{\mathcal{F}}_x}$. By applying the $\infty$-functor $\mathcal{MC}: \mathrm{dgLie}_k \to \mathrm{dSt}_k^*$ we get a morphism

$$ u: \mathcal{MC}(\ell_{(X,x)}) \simeq (X, x)^\wedge \to \mathcal{MC}(\ell_{\hat{\mathcal{F}}_x}). $$

The pointed derived stack $\mathcal{MC}(\ell_{\hat{\mathcal{F}}_x})$ is, by definition, the formal leaf space of $\mathcal{F}$ at $x$. It is possible to show that the morphism $u$ integrates $\hat{\mathcal{F}}_x$ in the sense that $\hat{\mathcal{F}}_x \simeq u^*(0)$, but we will not do this here. We will however prove this in the next section, in the specific case where $X$ is a smooth variety and $\mathcal{F}$ is rigid and quasi-smooth (see §2.3.4). For now, we gather some terminology related to Corollary 2.3.3.3.

**DEFINITION 2.3.3.4.** Let $(X, x)$ be a pointed derived Artin stack locally almost of finite presentation and $\mathcal{F} \in \mathcal{F}\mathrm{ol}^{ap}(X)$ be an almost perfect derived foliation.

(1) The formal leaf space of $\mathcal{F}$ at $x$ is the formal moduli problem corresponding to the dg-Lie algebra $\ell_{\hat{\mathcal{F}}_x}$. It is denoted by $\hat{\mathcal{M}}_x(\mathcal{F})$.

(2) The formal leaf of $\mathcal{F}$ passing through $x$ is the formal moduli problem corresponding to the dg-Lie algebra which is the fiber of the canonical morphism $\ell_{(X,x)} \to \ell_{\hat{\mathcal{F}}_x}$. It is denoted by $\hat{\mathcal{L}}_x(\mathcal{F})$.

We finish this section by proving Theorem 2.3.3.2.

*Proof of Theorem 2.3.3.2.* As a first reduction, by a standard Galois descent argument, we may assume that $k$ is algebraically closed. We then proceed by induction on the geometricity
of $X$. To start with, assume that $X$ is affine, and thus of the form $X = \mathbf{Spec} A$ for $A$ a connective cdga almost of finite presentation over $k$. We let $\hat{A}$ be the formal completion of $A$ at the point $x : \mathbf{Spec} A \to k$, and consider $\hat{\mathbf{DR}}(\hat{A}/k)$ the associated formal de Rham algebra. By Proposition 2.3.2.3 $\mathcal{Fol}^{ap}((X, x)^\wedge)$ is naturally equivalent to the (opposite) $\infty$-category of graded mixed $\hat{\mathbf{DR}}(\hat{A}/k)$-cdga's which are equivalent, as graded $\hat{\mathbf{DR}}(\hat{A}/k)$-cdga, to $\mathrm{Sym}_{\hat{A}}(E)$ for $E$ an almost perfect $\hat{A}$-dg-module. Tracing back the various equivalences of $\infty$-categories, we obtain a canonical equivalence of graded mixed cdga's

$$ \mathbf{DR}(x^*(0_x)) \simeq \hat{A} \otimes_{\mathbf{DR}(\hat{A}/k)} k, $$

and the $\infty$-functor of the theorem $\psi$ is then induced by

$$ \psi : \hat{\mathbf{DR}}(\hat{A}/k) - \epsilon - \mathbf{cdga}^{gr} \to \epsilon - \mathbf{cdga}^{gr} / (A \otimes_{\hat{\mathbf{DR}}(\hat{A}/k)} k) $$

which sends $\hat{\mathbf{DR}}(\hat{A}/k) \to B$ to $B \otimes_{\hat{\mathbf{DR}}(\hat{A}/k)} k$ together with its natural augmentation to $\hat{A} \otimes_{\hat{\mathbf{DR}}(\hat{A}/k)} k$ induced by the natural morphism $B \to \hat{A}$. The $\infty$-functor $\psi$ is thus subsumed by the following commutative diagrams of graded mixed cdga's with cocartesian squares

<figure><img src="image_17.png" /></figure>

We construct an $\infty$-functor $\phi$ in the other direction by using de Rham cohomology of the first kind explained in Section §A.8. For this, we first prove the following lemma.

**LEMMA 2.3.3.5.** *The canonical morphism $k \to \hat{\mathbf{DR}}(\hat{A}/k)$ induces an equivalence after taking realizations*

$$ k \simeq |\hat{\mathbf{DR}}(\hat{A}/k)|. $$

*Proof of the lemma.* By definition $|\hat{\mathbf{DR}}(\hat{A}/k)|$ is the derived de Rham cohomology of the formal completion of $\mathbf{Spec} A$ at $x$, and the lemma simply states that derived de Rham cohomology is trivial on formal completion. By definition $\hat{\mathbf{DR}}(\hat{A}/k) \simeq \lim_{A \to A'} \mathbf{DR}(A'/k)$. As $| - |$ is a right adjoint it commutes with limits, and thus we are reduced to show that, for any local augmented dg-Artinian algebra $A$, the natural morphism $k \to \hat{C}_{DR}(A/k)$ is a quasi-isomorphism. The lemma then follows from proposition A.8.0.3. $\square$

**LEMMA 2.3.3.6.** *There exists a canonical equivalence of graded mixed cdga's*

$$ \hat{\mathbf{DR}}(\hat{A}/k) \simeq \hat{C}_{DR}^I(\hat{A} \otimes_{\hat{\mathbf{DR}}(\hat{A}/k)} k/k). $$

*Proof of the lemma.* Let $D := \hat{\mathbf{DR}}(\hat{A}/k)$. By base change, we have a canonical equivalence of double graded mixed cdga's

$$ \mathbf{DR}^{\mathrm{int}}(\hat{A}/D) \otimes_D k \simeq \mathbf{DR}^{\mathrm{int}}(\hat{A} \otimes_D k/k). $$
We therefore have a natural morphism of double graded mixed cdga's

$$ \mathbf{DR}^{\text{int}}(\hat{A}/D) \to \mathbf{DR}^{\text{int}}(\hat{A}/D) \otimes_D k \simeq \mathbf{DR}^{\text{int}}(\hat{A} \otimes_D k/k), $$

where $D$ is endowed with the trivial mixed structure as its second graded mixed structure. By applying the first realization functor we get a morphism of graded mixed cdga's

$$ \hat{C}_{DR}^{I}(\hat{A}/D) \to \hat{C}_{DR}^{I}(\hat{A}/D) \hat{\otimes}_{|D|_1} k \simeq \hat{C}_{DR}^{I}(\hat{A} \otimes_D k/k), $$

where $\hat{\otimes}$ is the completed tensor product of complete filtered complexes (see §1.3.4). Now, Lemma 2.3.3.5 precisely states that $|D|_1 \simeq k$, and thus the above morphism turns out to be an equivalence of graded mixed cdga's

$$ \hat{C}_{DR}^{I}(\hat{A}/D) \simeq \hat{C}_{DR}^{I}(\hat{A} \otimes_D k/k). $$

To finish the proof of the lemma we apply proposition A.8.0.5 of Appendix A.8 showing the existence of a canonical equivalence of graded mixed cdga's $D \simeq \hat{C}_{DR}^{I}(\hat{A}/D)$. $\square$

Using Lemma 2.3.3.6, we can define an $\infty$-functor

$$ \phi : \epsilon - \mathbf{cdga}^{gr} / (\hat{A} \otimes_{\mathbf{DR}(\hat{A}/k)} k) \to \mathbf{DR}(\hat{A}/k) - \epsilon - \mathbf{cdga}^{gr} $$

by sending an object $B' \to (\hat{A} \otimes_{\mathbf{DR}(\hat{A}/k)} k)$ to the induced morphism in de Rham cohomology of the first kind

$$ \hat{C}_{DR}^{I}(\hat{A} \otimes_{\mathbf{DR}(\hat{A}/k)} k/k) \to \hat{C}_{DR}^{I}(\hat{A} \otimes_{\mathbf{DR}(\hat{A}/k)} k/B'). $$

We claim that $\psi$ and $\phi$ are inverse to each others when restricted to the appropriate full sub-$\infty$-categories.

We start by showing that $\phi \circ \psi$ is equivalent to the identity. Let $\mathbf{DR}(\hat{A}/k) \to B$ be an object in $\mathbf{DR}(\hat{A}/k) - \epsilon - \mathbf{cdga}^{gr}$ corresponding to an almost perfect derived foliation on $(X, x)^\wedge$. By definition, the underlying graded cdga of $B$ is of the form $\text{Sym}_{\hat{A}}(\mathbb{L}[1])$, for an almost perfect $\hat{A}$-dg-module $\mathbb{L}$. The object $\phi \circ \psi(B)$ is $\hat{C}_{DR}^{I}(\hat{A} \otimes_D k/B \otimes_D k)$, where we keep using the notation $D := \mathbf{DR}(\hat{A}/k)$. We have to produce an equivalence of graded mixed cdga's $u : B \simeq \hat{C}_{DR}^{I}(\hat{A} \otimes_D k/B \otimes_D k)$, functorial in $B$. For this, we consider the commutative diagram with push-out squares in graded mixed cdga's

$$ \begin{tikzcd} \mathbf{DR}(\hat{A}/k) \arrow[r] \arrow[d] & \mathbf{DR}^{\text{int}}(\hat{A}/B) \arrow[r] \arrow[d] & \mathbf{DR}^{\text{int}}(\hat{A} \otimes_D k/B \otimes_D k) \arrow[d] \\ \mathbf{DR}^{\text{int}}(B/k) \arrow[r] \arrow[d] & B \arrow[r] \arrow[d] & k \arrow[d] \end{tikzcd} $$

where $\mathbf{DR}^{\text{int}}(B/k)$ is by definition $\lim_{A \to A'} \mathbf{DR}^{\text{int}}(B \otimes_D \mathbf{DR}(A'/k)/k)$. By passing to the realizations along the first mixed structure, and applying proposition A.8.0.5, we obtain a natural morphism

$$ (9) \qquad B \simeq \hat{C}_{DR}^{I}(\hat{A}/B) \to \hat{C}_{DR}^{I}(\hat{A} \otimes_D k/B \otimes_D k). $$
It is then easy to check that this morphism is an equivalence, simply by considering the induced morphism on each graded pieces for the natural (complete) filtrations. Assume that $B$ is of the form $Sym_{\hat{A}}(E)$ as a graded cdga, then in weight 1, the above morphism is the natural projection

$$E \to E \otimes_{\hat{A}} |\hat{A} \otimes_D k|.$$

But, by considering the lemma 2.3.3.6 for the weight 0 pieces, we have $|\hat{A} \otimes_D k| \simeq \hat{A}$, and thus this morphism is an equivalence of complexes. The same argument shows that (9) is an equivalence also in arbitrary weights $i > 0$, being the canonical morphism

$$Sym_{\hat{A}}^i(E) \to Sym_{\hat{A}}^i(E) \otimes_{\hat{A}} |\hat{A} \otimes_D k|.$$

In the other direction, to see that $\psi \circ \phi \simeq id$, we simply observe that, for any diagram of graded mixed cdga's $k \to B' \to \hat{A} \otimes_D k$, with $B'$ of the form $Sym_k(L[1])$ as a graded cdga, the following square

$$\begin{tikzcd}
D \arrow[r] \arrow[d] & \hat{C}_{DR}^I (\hat{A} \otimes_D k/B') \\
k \arrow[u] \arrow[r] & B'
\end{tikzcd}$$

is a push-out diagram of graded cdga's: this can be easily checked by considering the various cotangent complexes.

We have therefore finished the proof of the theorem for $X$ affine, and it remains to extend the statement to the case of an arbitrary derived Artin stack $X$. For this we proceed on induction on the geometricity of $X$, as we have already seen that the theorem is true when $X$ is affine, or more generally a disjoint union of affines. We fix a smooth atlas $U \to X$, where $U$ is a disjoint union of of affine derived schemes locally of finite presentation. As $k$ is algebraically closed we can lift the point $x : \text{Spec } k \to X$ to a point $x_0 : \text{Spec } k \to U$. We consider the nerve $U_* \to X$ of $U \to X$, which comes equipped with a $k$-point $x_* : \text{Spec } k \to U_*$ compatible with the point $x$. The theorem is true for all $U_i = U \times_X \cdots \times_X U$ by our induction hypothesis. In order to deduce the result for $X$, we consider the following commutative square of $\infty$-categories

$$\begin{tikzcd}
\mathcal{F}ol^{ap}((X, x)^\wedge) \arrow[r] & \ell_{X,x}/\mathrm{dgLie}_k^{dap} \\
\lim_i \mathcal{F}ol^{ap}((U_i, x_i)^\wedge) \arrow[u] \arrow[r] & \lim_i \ell_{U_i,x_i}/\mathrm{dgLie}_k^{dad}.
\end{tikzcd}$$

The bottom horizontal $\infty$-functor is an equivalence by our induction assumption. The vertical $\infty$-functor on the left is an equivalence, because $(U_*, u_*)^\wedge \to (X, x)^\wedge$ is a hypercovering of derived stacks. Finally, the induced augmented simplicial object $\ell_{U_*, u_*} \to \ell_{X, x}$ is a smooth hypercovering of dg-Lie algebras in the sense of [Lur11, Prop. 1.5.8], and thus induces an
equivalence of dg-Lie algebras

$$ \mathrm{colim}_i \ell_{U_*,u_*} \simeq \ell_{X,x}. $$

This implies that also the vertical $\infty$-functor on the right is an equivalence of $\infty$-categories. $\square$

### 2.3.4. Transversely smooth and rigid derived foliations.
In this section we prove a stronger formal integrability statement for quasi-smooth and rigid derived foliations, which is not only valid formally at each point but globally on $X$. This result provides an important connection between derived foliations and classical notions such as Gobdillon-Vey sequences that will be explored later on.

For this section we fix $X$ a smooth DM-stack over $k$. The results of these sections can certainly be generalized to the more general setting, but we restrict to this case for simplicity.

**DEFINITION 2.3.4.1.** A derived foliation $\mathcal{F} \in \mathcal{Fol}(X)$ is transversely smooth and rigid if its normal complex $\mathcal{N}_{\mathcal{F}}$ is a vector bundle.

Note that because $X$ is itself a smooth DM-stack then $\mathbb{L}_{\mathcal{F}}$ is quasi-isomorphic to the 2-term complex of vector bundles $\mathcal{N}_{\mathcal{F}}^* \to \Omega_X^1$, and that $\mathcal{F}$ is automatically quasi-smooth in the sense of the general Definition 2.1.4.5. The term "rigid" in Definition 2.3.4.1 refers here to the fact that $\Omega_X^1 \to H^0(\mathbb{L}_{\mathcal{F}})$ must be an epimorphism of sheaves, so that $\mathcal{F}$ does not have any "infinitesimal automorphisms" or no "fixed points". As an example, the foliation induced by a group scheme $G$ acting on $X$ (see Example 2.1.5.6) is transversely smooth and rigid if and only if the stabilizers of the action are all discrete.

We start by constructing a canonical formal thickening $X \hookrightarrow \mathcal{J}_{\mathcal{F}}$ associated to a transversely smooth and rigid derived foliation $\mathcal{F}$. For this, we consider the push-out of graded mixed cdga's over $X$

<figure><img src="image_18.png" /></figure>

The graded mixed cdga $B$ is $\mathcal{O}_X$-linear via its canonical morphism $\mathcal{O}_X \to B$, and as a graded $\mathcal{O}_X$-linear cdga it is equivalent to $\mathrm{Sym}_{\mathcal{O}_X}(\mathcal{N}_{\mathcal{F}}^*[2])$. Using the equivalence between graded mixed complexes and complete filtered complexes of Corollary 1.3.4.4, we associate to $B$ a complete filtered $\mathcal{O}_X$-linear cdga $\mathcal{B}_{\mathcal{F}} := |B|^t$. The associated graded of $B$ is canonical equivalent to $\mathrm{Sym}_{\mathcal{O}_X}(\mathcal{N}_{\mathcal{F}}^*)$. Moreover, as $B$ comes equipped with an augmentation $B \to \mathcal{O}_X$, the filtered cdga $\mathcal{B}_{\mathcal{F}}$ is itself canonically augmented $\mathcal{B}_{\mathcal{F}} \to \mathcal{O}_X$.

Let us denote the filtration on $\mathcal{B}_{\mathcal{F}}$ by

$$ \dots \longrightarrow F^i \mathcal{B}_{\mathcal{F}} \longrightarrow F^{i-1} \mathcal{B}_{\mathcal{F}} \longrightarrow \dots \quad F^0 \mathcal{B}_{\mathcal{F}} = \mathcal{B}_{\mathcal{F}}, $$
and note that $F^i\mathcal{B}_F = F^{i+1}\mathcal{B}_F = \mathcal{B}_F$ for all $i \ge 0$. We thus have a pro-objects in the $\infty$-category of quasi-coherent $\mathcal{O}_X$-linear cdga's

$$ \hat{\mathcal{B}}_F = \lim_{i} \mathcal{B}_F / (F^i \mathcal{B}_F). $$

This pro-object possesses a formal spectrum

$$ \mathcal{J}_F := \mathrm{Spf}(\hat{\mathcal{B}}_F) = \mathrm{colim}_i \mathrm{Spec}(\mathcal{B}_F / (F^i \mathcal{B}_F)) \to X, $$

which is an affine formal scheme over $X$. The formal scheme $\mathcal{J}_F$ comes equipped with a canonical closed embedding $X \to \mathcal{J}_F$ induced from the augmentation $\mathcal{B}_F \to \mathcal{O}_X$, in such a way that $X$ is a retract of $\mathcal{J}_F$ because of the $\mathcal{O}_X$-linear structure on $\hat{\mathcal{B}}_F$. Note also that $\mathcal{J}_F \to X$ is locally on $X$ of the form $X \times \hat{\mathbb{A}}^d$, where $d$ is the codimension of $\mathcal{F}$ (i.e. the rank of the normal bundle $\mathcal{N}_F$). Indeed, as $\mathcal{J}_F$ is a relative formal spectrum, its construction is stable by base change, so we may assume that $X$ is affine of the form $\mathrm{Spec} A$. In this case, the filtered $A$-cdga $\mathcal{B}_F$ is such that $Gr^*(\mathcal{B}_F) \simeq \mathrm{Sym}_A(\mathcal{N}_F^*)$, where $\mathcal{N}_F^*$ is of weight 1. Since $\mathcal{N}_F^*$ is a vector bundle and $X$ is affine, there are no obstructions to lift the canonical morphism of $A$-dg-modules

$$ \mathcal{N}_F^* \to \mathcal{B}_F / F^2 \simeq A \oplus \mathcal{N}_F^* $$

to a morphism of $A$-dg-modules $\mathcal{N}_F^* \to \mathcal{B}_F$. This extends by multiplicativity to a morphism of completed $A$-algebras $S\hat{\mathrm{ym}}_A(\mathcal{N}_F^*) \to \mathcal{B}_F$ which is clearly an equivalence. This shows that $\mathcal{J}_F$ is thus equivalent, as a formal affine scheme over $X$, to $\mathcal{V}(\hat{\mathcal{N}}_F) \to X$, the formal completion at the zero section of the total space of the vector bundle $\mathcal{N}_F$ over $X$. Note that however this is not a functorial description, and therefore the two formal schemes $\mathcal{V}(\hat{\mathcal{N}}_F)$ and $\mathcal{J}_F$ are only locally (not globally) equivalent, for the étale topology on $X$, and the difference between these two objects is understood globally via the filtration on $\hat{\mathcal{B}}_F$ interpolating between them.

So we have constructed, for any transversely smooth and rigid derived foliation $\mathcal{F}$ on $X$, a formal scheme $\mathcal{J}_F \to X$ together with a section $j: X \to \mathcal{J}_F$, which is moreover obviously functorial in the pair $(X, \mathcal{F})$.

**DEFINITION 2.3.4.2.** For a transversely smooth and rigid $\mathcal{F} \in \mathrm{Fol}(X)$, the relative formal scheme $\mathcal{J}_F \to X$ constructed above is called the (transversal) jet space of $\mathcal{F}$.

The importance of the transversal jet space lies in the following integrability result.

**PROPOSITON 2.3.4.3.** Let $X$ be a smooth DM-stack, and $\mathcal{F} \in \mathrm{Fol}(X)$ transversely smooth and rigid with transverse jet space $\mathcal{J}_F \to X$. There exists a canonical smooth, transversely smooth and rigid derived foliation $\mathcal{F}' \in \mathrm{Fol}(\mathcal{J}_F)$ such that $j^*(\mathcal{F}')$ is canonically equivalent to $\mathcal{F}$ in $\mathrm{Fol}(X)$.

*Proof.* It is enough to prove the Proposition on $X = \mathrm{Spec} A$ in such a way that the construction of $\mathcal{F}'$, and the equivalence $j^*(\mathcal{F}') \simeq \mathcal{F}$, are both stable by base change along any étale morphism $Y \to X$ between smooth affine schemes. We are thus essentially reduced to the case where $X$ is smooth and affine.
Let $\mathbf{DR}(\mathcal{F}/k)$ be the graded mixed $\mathbf{DR}(A/k)$-cdga corresponding to $\mathcal{F}$. We have an exact triangle of complexes $N_{\mathcal{F}}^* \to \Omega_{A/k}^1 \to \mathbb{L}_{\mathcal{F}}$. Equivalently, $\mathbb{L}_{\mathcal{F}}$ identifies canonically with the two terms complex $N_{\mathcal{F}}^* \to \Omega_{A/k}^1$ where $\Omega_{A/k}^1$ sits in degree 0. The formal scheme $\mathcal{J}_{\mathcal{F}}$ is here the formal spectrum of the filtered $A$-algebra $\mathcal{B}_{\mathcal{F}} = |B|^t$, where $B$ is the $A$-linear graded mixed cdga defined by $B := \mathbf{DR}(\mathcal{F}/k) \otimes_{\mathbf{DR}(A/k)} A$. Note that $\mathcal{B}_{\mathcal{F}}$ is non-canonically isomorphic to $S\hat{y}m_A(\mathcal{N}_{\mathcal{F}})$.

We let $\hat{\mathbf{DR}}(\mathcal{J}_{\mathcal{F}}/k)$ be defined by

$$ \hat{\mathbf{DR}}(\mathcal{J}_{\mathcal{F}}/k) := \lim_{n} \mathbf{DR}((\mathcal{B}_{\mathcal{F}}/F^n)/k) $$

be the completed graded mixed cdga associated to the formal scheme $\mathcal{J}_{\mathcal{F}}$. We have the following statement concerning perfect derived foliations on $\mathcal{J}_{\mathcal{F}}$, whose proof is very much similar (actually simpler) than proposition 2.3.2.3, and is left to the reader as an exercise.

LEMMA 2.3.4.4. There is a natural equivalence between the following two $\infty$-categories:

* $\mathcal{F}ol^{ap}(\mathcal{J}_{\mathcal{F}})$, the $\infty$-category of perfect derived foliations on $\mathcal{J}_{\mathcal{F}}$

* the $\infty$-category of graded mixed $\hat{\mathbf{DR}}(\mathcal{J}_{\mathcal{F}}/k)$-cdga $D$, of the form $\mathrm{Sym}_{\mathcal{B}_{\mathcal{F}}}(V[1])$ as graded cdga where $V$ is an almost perfect $\mathcal{B}_{\mathcal{F}}$-dg-module.

To achieve the proof proposition 2.3.4.3, it is enough to construct a smooth derived foliation $\mathcal{F}' \in \mathcal{F}ol(\mathcal{J}_{\mathcal{F}})$ together with an equivalence $j^*(\mathcal{F}') \simeq \mathcal{F}$, which is functorial under étale base change on $X$. Indeed, as $\mathcal{F}$ is rigid, such an $\mathcal{F}'$ must automatically be rigid as well: the conormal complex $N_{\mathcal{F}'}^*$ being perfect on $\mathcal{J}_{\mathcal{F}}$ is such that $j^*(N_{\mathcal{F}'}^*) \simeq N_{\mathcal{F}}^*$ is vector bundle, and thus $N_{\mathcal{F}'}^*$ must already be a vector bundle on the formal thickening $\mathcal{J}_{\mathcal{F}}$. As a result, such a smooth $\mathcal{F}'$ with $j^*(\mathcal{F}') \simeq \mathcal{F}$ must be transversally smooth, and thus is automatically rigid because the short exact sequence of bundles $0 \to N_{\mathcal{F}'}^* \to \Omega_{\mathcal{J}_{\mathcal{F}}}^1 \to \mathbb{L}_{\mathcal{F}'} \to 0$ locally splits.

In order to construct $\mathcal{F}'$, we use our notion of de Rham cohomology of the first kind explained in §A.8. We recall the push-out square of graded mixed cdga's

<figure><img src="image_19.png" /></figure>

and we consider $\hat{C}_{DR}^I(B/\mathbf{DR}(\mathcal{F}/k))$, the de Rham cohomology of the first kind. A direct computation of cotangent complexes shows that, as a graded algebra, $\hat{C}_{DR}^I(B/\mathbf{DR}(\mathcal{F}/k))$ is of the form $\mathrm{Sym}_{\mathcal{B}_{\mathcal{F}}}(\Omega_X^1 \otimes_A \mathcal{B}_{\mathcal{F}}[1])$. Therefore, by the equivalence of $\infty$-categories describing $\mathcal{F}ol(\mathcal{J}_{\mathcal{F}})$, $\hat{C}_{DR}^I(B/\mathbf{DR}(\mathcal{F}/k))$ defines a perfect derived foliation $\mathcal{F}' \in \mathcal{F}ol(\mathcal{J}_{\mathcal{F}}/k)$. Moreover, the cotangent complex of $\mathcal{F}'$ is $p^*(\Omega_{X/k}^1)$, where $p: \mathcal{J}_{\mathcal{F}} \to X$ is the natural projection, and thus $\mathcal{F}'$ is smooth.

We consider now the canonical morphism $\hat{C}_{DR}^I(B/\mathbf{DR}(\mathcal{F}/k)) \to \hat{C}_{DR}^I(A/\mathbf{DR}(\mathcal{F}/k))$, via the augmentation $B \to A$. By Proposition A.8.0.5, we know that the right hand side is
naturally equivalent to DR($\mathcal{F}/k$), and we thus have a morphism of graded mixed cdga's
$\hat{C}_{DR}^I(B/DR(\mathcal{F}/k)) \to DR(\mathcal{F}/k)$, making the following diagram commutative

<figure><img src="image_20.png" /></figure>

This produces a morphism

$$
\hat{C}_{DR}^{I}(B/\mathbf{DR}(\mathcal{F}/k)) \otimes_{\hat{\mathbf{DR}}(\mathcal{J}_{\mathcal{F}}/k)} \mathbf{DR}(A/k) \to \mathbf{DR}(\mathcal{F}/k)
$$

which is easily seen to be an equivalence by computing the cotangent complexes. By construc-
tion, this is the required equivalence $j^*(\mathcal{F}') \simeq \mathcal{F}$. $\square$

REMARK 2.3.4.5. (1) An important point of proposition 2.3.4.3 is that the association $\mathcal{F} \mapsto (\mathcal{J}_{\mathcal{F}}, \mathcal{F}')$ is canonical. The foliation $\mathcal{F}'$ will be called the *canonical spread* of the derived foliation $\mathcal{F}$. We will see below that the pair $(\mathcal{J}_{\mathcal{F}}, \mathcal{F}')$ is closely related to the classical notion of Godbillon-Vey sequences (see for instance [Cas02] and [Mal77, (3.4)] for classical references).

(2) In the special case where, in the statement of the proposition 2.3.4.3 $\mathcal{F}$ is integrable, the objects $\mathcal{J}_{\mathcal{F}}$ and $\mathcal{F}'$ can be described explicitly as follows. Let $f: X \to Y$ be a morphism of smooth separated schemes such that $f^*(0_Y) \simeq \mathcal{F}$. Then $\mathcal{J}_{\mathcal{F}}$ is identified with the formal completion of $X \times_k Y$ along the graph of $f$, and $\mathcal{F}'$ is $q^*(0_Y)$ where $q: X \times_k Y \to Y$ is the second projection.

(3) Being at the same time smooth and transversally smooth, the derived foliation $\mathcal{F}'$ is a genuine foliation on $\mathcal{J}_{\mathcal{F}}$, given by an integrable sub-bundle $\mathbb{T}_{\mathcal{F}}' \subset \mathbb{T}_{\mathcal{J}_{\mathcal{F}}}$ of the tangent sheaf.

One important corollary of proposition 2.3.4.3 is the following formal integrability result, that could also be obtained from the more general theorems 2.3.3.2 and 2.3.3.3.

COROLLARY 2.3.4.6. Let $X$ be a smooth DM-stack and $x : \text{Spec } k \to X$ a global point. Let $\mathcal{F} \in \text{Fol}(X/k)$ be a transversely smooth and rigid derived foliation on $X$. There exists a morphism of formal schemes $f : (X, x)^\wedge \to \hat{\mathbb{A}}^d$, such that $\hat{F}_x \simeq f^*(0)$ in $\text{Fol}((X, x)^\wedge)$. In other words, $\mathcal{F}$ is formally integrable at each point on $X$.

Proof of the Corollary. The statement is local at $x$, so we can assume that $X$ is affine, and moreover that the normal bundle $\mathcal{N}_{\mathcal{F}} \simeq \mathcal{O}_X^d$ is trivial, as well as $\Omega_{X/k}^1$. We use the canonical spread $j : X \hookrightarrow X \times_{\hat{\mathbb{A}}^d} \hat{\mathbb{A}}^d$, with $\mathcal{F}' \in \text{Fol}(X \times_{\hat{\mathbb{A}}^d} \hat{\mathbb{A}}^d)$, $j^*(\mathcal{F}') \simeq \mathcal{F}$. We formally complete at $x$ to get

$$
\hat{j} : (X, x)^\wedge \to (X \times \mathbb{A}^d, (x, 0))^\wedge.
$$

The formal completion at $(x,0)$ defines a foliation $\hat{\mathcal{F}}' \in \mathcal{F}ol((X \times \mathbb{A}^d, (x,0))^\wedge)$, which is here given by a family of formal vector fields $\nu_1, \dots, \nu_n$ on the formal scheme $(X \times \mathbb{A}^d, (x,0))^\wedge \simeq \hat{\mathbb{A}}^{n+d}$,
where $n = Dim_k X$, satisfying the classical Frobenius condition: each $[\nu_i, \nu_j]$ can be written as a linear combination of $\nu_k$'s. The formal Frobenius lemma (see e.g. [BZ09]) then tells us that there is a morphism of formal schemes $p : (X \times A^d, (x, 0))^\wedge \to \hat{A}^d$, such that $\hat{\mathcal{F}}' \simeq p^*(0)$. We can thus take $f = p \circ \hat{j}$. $\square$

## 2.4. Existence of derived enhancements

In this section we study the relations between the notion of derived foliation on a derived scheme $X$, and the classical notion of differential ideal on the truncation $\tau_0(X)$. We show that any derived foliation underlies a classical differential ideal. However, not every differential ideal arises this way, and in the case where $X$ is a smooth variety, we provide a criterion for the existence of a derived foliation enhancing a given classical differential ideal, based on the classical notion of Gobdillon-Vey sequences (see for instance [Cas02, Mal77]).

**2.4.1. Derived foliations and differential ideals.** Let $X$ be a derived scheme locally almost of finite presentation over $k$, and $\mathcal{F} \in \mathcal{F}ol(X/k)$ be a derived foliation on $X$. We assume that $\mathcal{F}$ is connective in the sense of Definition 2.1.1.1, i.e. $H^i(\mathbb{L}_{\mathcal{F}/k}) \simeq 0$ for $i > 0$. We consider the corresponding sheaf of graded mixed $\mathbf{DR}_{X/k}$-cdga $\mathbf{DR}_{\mathcal{F}}$. We have the canonical morphism of quasi-coherent complexes

$$a : \mathbb{L}_{X/k} \to \mathbb{L}_{\mathcal{F}/k}$$

from which we form the kernel of the induced morphism on $H^0$

$$K_{\mathcal{F}} := \mathrm{Ker}(H^0(\mathbb{L}_{X/k}) \to H^0(\mathbb{L}_{\mathcal{F}/k})).$$

The sheaf $K_{\mathcal{F}}$ is a coherent subsheaf of $H^0(\mathbb{L}_{X/k}) \simeq \Omega_{X_0/k}^1$, where $X_0 = \tau_0(X)$ is the underived scheme obtained from $X$ by truncation (see Appendix A.2).

**LEMMA 2.4.1.1.** *The subsheaf $K_{\mathcal{F}} \subset \Omega_{X_0/k}^1$ is a coherent differential ideal:*

$$dR(D_{\mathcal{F}}) \subset \mathrm{Im}(D_{\mathcal{F}} \otimes \Omega_{X_0/k}^1 \to \Omega_{X_0/k}^2).$$

*Proof.* Since the graded mixed structure on $\mathbf{DR}_{\mathcal{F}} \simeq \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\mathcal{F}/k}[1])$ is compatible with the de Rham differential on $\mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{X/k})$, we have a commutative diagram of sheaves of $k$-modules on $X$

$$ \begin{tikzcd} H^0(\mathbb{L}_{X_0}/k) \arrow[r, "dR"] \arrow[d, "a"] & H^0(\wedge_{\mathcal{O}_X}^2 \mathbb{L}_{X_0}/k) \\ H^0(\mathbb{L}_{\mathcal{F}/k}) \arrow[r, "\epsilon"] \arrow[d, "\wedge^2 a"] & H^0(\wedge_{\mathcal{O}_X}^2 \mathbb{L}_{\mathcal{F}/k}). \end{tikzcd} $$
Because $\mathbb{L}_{X_0/k}$ and $\mathbb{L}_{\mathcal{F}/k}$ are both connective, the above diagram is isomorphic to

$$ \begin{tikzcd} \Omega_{X_0/k}^1 \arrow[r, "dR"] \arrow[d, "a"] & \Omega_{X_0/k}^2 \arrow[d, "\wedge^2 a"'] \\ H^0(\mathbb{L}_{\mathcal{F}/k}) \arrow[r, "\epsilon", "\wedge \Omega_X^2"] & H^0(\mathbb{L}_{\mathcal{F}/k}). \end{tikzcd} $$

The lemma thus follows by considering the induced morphism on the kernels of the vertical morphisms. $\square$

**DEFINITION 2.4.1.2.** (1) With the notations and conditions above, the differential ideal $K_{\mathcal{F}} \subset \Omega_{X_0/k}^1$ is called the classical truncation of the derived foliation $\mathcal{F} \in \mathcal{F}ol(X/k)$.

(2) For a coherent differential ideal $K \subset \Omega_{X_0/k}^1$, a derived enhancement of $D$ is a connective derived foliations $\mathcal{F} \in \mathcal{F}ol(X/k)$ such that $K = K_{\mathcal{F}}$.

Note that, if $\mathcal{F} \to \mathcal{F}'$ is a morphism of connective derived foliations, then we have a natural inclusion of differential ideals $K'_{\mathcal{F}} \subset K_{\mathcal{F}} \subset \Omega_{X_0/k}^1$. Therefore, if we denote by $IDiff(X_0)$ the category of coherent differential ideals on $X_0$ (with morphisms given by inclusions), the classical truncation is an $\infty$-functor

$$ K_- : \mathcal{F}ol^c(X/k)^{op} \to IDiff(X_0), $$

from connective derived foliations on $X$ to differential ideals on $X_0$. As we will now explain, there are no reasons to expect that the above $\infty$-functor is essentially surjective, even in the case where $X$ is a nice smooth scheme over $k$. Indeed, if $K \subset \Omega_{X/k}^1$ is a coherent differential ideal, we can certainly form a graded mixed cdga by considering $Sym_{\mathcal{O}_X}^{st}(E[1])$, where $E$ is the coherent sheaf $\Omega_{X/k}^1/K$, and where the mixed structure is defined by the unique possible extension of the de Rham differential along the quotient map $\mathbf{DR}(X/k) \to Sym_{\mathcal{O}_X}^{st}(E[1])$.

It is important to note that here $Sym^{st}$, as opposed to notation $Sym$, denotes the strict or naive symmetric algebra, defined in the tensor abelian category of coherent sheaves on $X$. This makes a difference as $E$ is in general not a flat $\mathcal{O}_X$-module, and thus the canonical morphism $Sym_{\mathcal{O}_X}(E[1]) \to Sym_{\mathcal{O}_X}^{st}(E[1])$, from the derived $Sym$-algebra to the naive $Sym$-algebra is not a quasi-isomorphism. In particular, the graded mixed cdga $Sym_{\mathcal{O}_X}^{st}(E[1])$ does not satisfy the condition appearing in Definition 2.1.1.1, and thus does not define a derived foliation on $X$. In order to promote $Sym_{\mathcal{O}_X}^{st}(E[1])$ into a derived foliation, we would have to lift the de Rham differential along the natural morphism $Sym_{\mathcal{O}_X}(E[1]) \to Sym_{\mathcal{O}_X}^{st}(E[1])$. We will see in the next section that there are obstructions to the existence of such a lift in general, and that these obstructions can be expressed explicitly in simple terms.

The main reason for expecting that the classical truncation $K_-$ is not essentially surjective comes from integrability results for derived foliations, such as Theorem 2.3.3.3 or proposition 2.3.4.3, which indicate that derived foliations are always formally integrable at each point.
There are however example of differential ideals, or even analytic germs of differential ideals, which are not formally integrable (see remark 2.4.1.3 (1) below). Of course, the notion of derived enhancement of Definition 2.4.1.2 is extremely general as we do not restrict the type of derived foliation $\mathcal{F}$, so theorem 2.3.3.3 or proposition 2.3.4.3 are not strictly speaking counterexamples to the existence of derived enhancements, and possibly the questions of their existence is complicated and not well posed. However, when restricting the type of derived foliations allowed, the question of existence of derived enhancements can be treated in great detail. This is what we will do in the next section for *transversely smooth and rigid* derived foliations.

**REMARK 2.4.1.3.**
* Examples of not formally integrable differential ideals already exists in dimension 2. Indeed, for a germ $\omega$ of holomorphic 1-differential with an isolated zero at $0 \in \mathbb{C}^2$ and satisfying $d\omega \wedge \omega = 0$, formal integrability and holomorphic integrability are equivalent conditions by the main result of [Mal76]. Moreover, it is well known that holomorphic integrability is restricted by topological obstructions, see for instance [MM80, Theorem B]. This provides examples of (analytic germs of) differential ideal which do not admit derived enhancements.

* It is worth mentioning that the truncation functor $K_-$ of Definition 2.4.1.2 can also be understood in terms of a natural $t$-structure on graded mixed complexes. Indeed, we can introduce a $t$-structure on graded mixed complexes by declaring that $E \in \epsilon - \mathrm{dg}_k^{gr}$ is *connective* if for each $n \in \mathbb{Z}$, the part $E^{(n)}$ of weight $n$ is $n$-connective: $H^i(E^{(n)}) = 0$ for $i < -n$. This $t$-structure is compatible with the symmetric monoidal structure, and thus $t$-truncations functors extends to commutative algebra objects. A connective derived foliation $\mathcal{F} \in \mathcal{F}ol^c(X/k)$ is such that $\mathrm{DR}(\mathcal{F})$ is a connective graded mixed cdga, and its 0-truncation for the above mentioned $t$-structure is precisely $\mathrm{Sym}^{st}_{\mathcal{O}_X}(E[1])$ where $E = \Omega_{X_0/k}^1/K_{\mathcal{F}}$.

**2.4.2. Derived enhancements and Godbillon-Vey sequences.** Let $X = \mathrm{Spec} A$ be a smooth affine scheme over $k$, and $K \subset \Omega_{X/k}^1$ be a differential ideal and we assume that $K$ is a free an $A$-module (but we do not assume that it is a sub-bundled eof $\Omega_{X/k}^1$). We chose a set of generators $(\omega_1, \dots, \omega_d)$ of $K$ as an $A$-module, which is a basis. We recall from [Mal77, (3.4)] that a *Godbillon-Vey* (or simply *GV*) sequence of forms adapted to $(\omega_1, \dots, \omega_d)$ consists of a sequence of elements $\omega_{i,\alpha} \in \Omega_{X/k}^1$, where $\alpha \in \mathbb{N}^p$, satisfying the equations

$$ (10) \qquad dR(\omega_{i,\alpha}) = \sum_{j,\beta} \binom{\alpha}{\beta} \omega_{j,\beta} \wedge \omega_{i,\alpha-\beta+e_j} \qquad \omega_{i,0} = \omega_i $$

where $e_j = (0, \dots, 0, 1, 0, \dots, 0) \in \mathbb{N}^p$ is the $j$-th element of the standard basis. The GV sequences can also be interpreted as follows.

**LEMMA 2.4.2.1.** With the notations above, there is a bijection between the following sets.

* The set of GV sequences adapted to $(\omega_1, \dots, \omega_d)$.
• Families of differential forms $(\tilde{\omega}_1, \dots, \tilde{\omega}_d)$ on the formal scheme $X \times \hat{\mathbb{A}}^d$ with the following conditions satisfied.

(1) $j^*(\tilde{\omega}_i) = \omega_i$, where $j: X \hookrightarrow X \times \hat{\mathbb{A}}^d$ is the embedding at 0.

(2) For each $i$, the form $\tilde{\omega}_i$ can be written as

$$ \tilde{\omega}_i = dR(t_i) + \sum_k a_k.p^*(\lambda_k) $$

with $\lambda_k \in \Omega_{X/k}^1$, $p: X \times \hat{\mathbb{A}}^d \to X$ being the projection onto the first factor, $(t_1, \dots, t_p)$ the standard coordinates on $\hat{\mathbb{A}}^d$, and $a_k \in A[[t_1, \dots, t_p]]$ formal functions on $X \times \hat{\mathbb{A}}^d$.

(3) The forms $(\tilde{\omega}_1, \dots, \tilde{\omega}_d)$ define a differential ideal in $\hat{\Omega}_{X \times \hat{\mathbb{A}}^d}^1$.

*Proof.* This is almost straightforward: to a GV sequence $\omega_{i,\alpha}$ we associate the forms

$$ \tilde{\omega}_i := dR(t_i) + \sum_{i,\alpha} p^*(\omega_{i,\alpha}) \cdot \frac{t^\alpha}{\alpha!} $$

We refer to [Mal77, Prop. 3.2] for more details. □

It is important to note that because of condition (2), the forms $(\tilde{\omega}_1, \dots, \tilde{\omega}_d)$ are automatically linearly independent everywhere, and thus the differential ideal they generate is a free subbundle of rank $p$ in $\hat{\Omega}_{X \times \hat{\mathbb{A}}^d}^1$. Therefore, they define a smooth, transversely smooth and rigid foliation (i.e. a classical foliation in the usual sense) on the formal scheme $X \times \hat{\mathbb{A}}^d$ exactly as in proposition 2.3.4.3. In fact, proposition 2.3.4.3 can be used in order to prove the following result, which expresses that GV sequences and transversely smooth and rigid derived enhancements are essentially the same thing.

**PROPOSITON 2.4.2.2.** Let $X$ be a smooth scheme over $k$ and $K \subset \Omega_{X/k}^1$ be a coherent differential ideal such that $K$ is a vector bundle. The following two conditions are equivalent:

(1) Locally for the Zariski topology on $X$, there exists a Godbillon-Vey sequence (adapted to some choice of generators $(\omega_1, \dots, \omega_p)$ of $K$).

(2) Locally for the Zariski topology on $X$ there exists a transversely smooth and rigid derived foliation $\mathcal{F} \in \mathcal{Fol}(X/k)$ such that $K = K_{\mathcal{F}}$.

*Proof.* We have already seen that (1) ⇒ (2). Indeed, a GV sequence for $K \in \mathrm{IDiff}(X)$ defines a smooth, transversely smooth and rigid derived foliation $\mathcal{F}'$ on $X \times \hat{\mathbb{A}}^d$. We let $\mathcal{F} := j^*(\mathcal{F}') \in \mathcal{Fol}(X/k)$. As pull-backs of derived foliations preserve conormal complexes, we see that $\mathcal{F}$ is transversely smooth and rigid. Moreover, by construction, the differential ideal $K_{\mathcal{F}}$ is generated by $(\omega_1, \dots, \omega_d)$ and thus coincides with $K$.

Conversely, suppose that $K$ is of the form $K_{\mathcal{F}}$. We apply proposition 2.3.2.3 to $\mathcal{F}$. As the statement is local, we can even assume that the normal bundle of $\mathcal{F}$ is trivialized, and the proposition gives us $\mathcal{F}' \in \mathcal{Fol}(X \times \hat{\mathbb{A}}^d/k)$ such that $j^*(\mathcal{F}') = \mathcal{F}$. Moreover, by observing the
proof of proposition 2.3.2.3, it is easy to see that $\mathcal{F}'$ is given by a differential ideal $(\tilde{\omega}_1, \dots, \tilde{\omega}_d) \subset \hat{\Omega}_{X \times \hat{\mathbb{A}}^d}^1$, where each $\tilde{\omega}_i$ is of the form

$$ \tilde{\omega}_i = \sum_j f_{i,j} \cdot dR(t_j) + \sum_k a_k \cdot p^*(\lambda_k) $$

with $f_{i,j} \in A[[t]] = A[[t_1, \dots, t_p]]$ a formal function such that the $p \times p$ matrix $M := (f_{i,j})$ is invertible. Indeed, it is easy to see by construction that the cotangent complex of $\mathcal{F}'$ is a vector bundle of rank $\dim X$, and that the anchor map

$$ \hat{\Omega}_{X \times \hat{\mathbb{A}}^d}^1 \to \mathbb{L}_{\mathcal{F}'} $$

precomposed with $p^*(\Omega_{X/k}^1) \to \hat{\Omega}_{X \times \hat{\mathbb{A}}^d}^1$ induces an isomorphism $p^*(\Omega_{X/k}^1) \simeq \mathbb{L}_{\mathcal{F}'}$. As

$$ \hat{\Omega}_{X \times \hat{\mathbb{A}}^d}^1 \simeq p^*(\Omega_{X/k}^1) \oplus \bigoplus_i A[[t]].dR(t_i), $$

we see that the kernel of the anchor map, which is the subsheaf generated by the $\tilde{\omega}_i$'s, becomes isomorphic to $\bigoplus_i A[[t]].dR(t_i)$ via the projection $\hat{\Omega}_{X \times \hat{\mathbb{A}}^d}^1 \to \bigoplus_i A[[t]].dR(t_i)$. We can thus change the basis $(\tilde{\omega}_1, \dots, \tilde{\omega}_d)$ by $M^{-1}(\tilde{\omega}_1, \dots, \tilde{\omega}_d)$, which defines the same foliation $\mathcal{F}'$, and thus simply assume that $f_{i,j} = \delta_{i,j}$. By Lemma 2.4.2.1, the family $(\tilde{\omega}_1, \dots, \tilde{\omega}_d)$ defines a GV sequence for the differential ideal $K$. $\square$

The proposition 2.4.2.2 has an important consequence concerning existence of derived enhancement under appropriate codimension conditions. The question of uniqueness of a derived enhancement is more complicated and will be studied later using analytic methods (see §5.4).

**COROLLARY 2.4.2.3.** Let $X = \text{Spec } A$ be a smooth affine scheme over $k$ and $K \subset \Omega_{X/k}^1$ be a differential ideal. We assume that the coherent sheaf $\Omega_{X/k}^1/K$ is a vector bundle on a dense open $U \subset X$, whose closed complement $Z$ is of codimension at least 3 in $X$. Then, locally for the Zariski topology on $X$, $K$ is of the form $K = K_{\mathcal{F}}$ for a transversely smooth and rigid derived foliation $\mathcal{F} \in \mathcal{Fol}(X/k)$.

*Proof of the Corollary.* We first notice that $K$ must be a vector bundle on $U$, and thus on $X$ by Hartogs. Since the statement is local for the Zariski topology on $X$, we can assume the existence of an isomorphism of coherent sheaves $\mathcal{O}_X^d \simeq K$. We consider the induced injective morphism $\mathcal{O}_X^d \hookrightarrow \Omega_{X/k}^1$. This defines a complex of vector bundles concentrated in cohomological degrees $[-1, 0]$ that we denote by $\mathbb{L}$. We then use the following well-known vanishing lemma.

**LEMMA 2.4.2.4.** Under the conditions above, for all $n \ge 0$ we have $H^i(\wedge_{\mathcal{O}_X}^{n+2}(\mathbb{L})) \simeq 0$ for all $i \le -n$.

*Proof of the lemma.* The complex $\wedge_{\mathcal{O}_X}^{n+2}(\mathbb{L})$ is a perfect complex which is a direct factor of $\mathbb{L}^{\otimes n+2}$, which is itself perfect and represented by the genuine complex of vector bundles $(\mathcal{O}_X^d \to \Omega_{X/k}^1)^{\otimes n+2}$. This is a complex of vector bundles sitting in cohomological degrees
$[-n - 2, 0]$. We form the dual perfect complex $\mathbb{L}^{\vee}$ which is a bounded complex of vector bundles concentrated in degrees $[0, n + 2]$. Moreover, on the open $U$ the complex $\mathbb{L}$ is equivalent to a vector bundle, and this all the cohomology sheaves $H_i := \mathcal{H}^i(\mathbb{L}^{\vee})$ vanishes on $U$ as soon as $i > 0$. In particular, by Grothendieck duality the ext-sheaves $\mathcal{E}xt_{\mathcal{O}_X}^i(H_i, \mathcal{O}_X)$ are zero for $i = 0, i = 1$ and $i = 2$. This clearly implies, by postnikov induction on $\mathbb{L}^{\vee}$, that $\mathbb{L} \simeq \mathbb{R}\operatorname{Hom}_{\mathcal{O}_X}(\mathbb{L}^{\vee}, \mathcal{O}_X)$ has cohomology sheaves concentrated in cohomological degrees $[-n, 0]$ as wanted. $\square$

Coming back to the proof of the Corollary, by proposition 2.4.2.2 it is enough to show the existence of a GV sequence $\omega_{i,\alpha}$ adapted to the basis $(\omega_1, \dots, \omega_d)$ of $K$ coming from our choice of isomorphism $\mathcal{O}_X^d \simeq K$. We construct the $\omega_{i,\alpha}$ by induction on the multi-index $\alpha$. Suppose that we have already constructed $\omega_{i,\alpha}$ for all $|\alpha| \le n$, such that equations (10) are satisfied for all $|\alpha| < n$. For all $i$, and a multi-index $\alpha$ with $|\alpha| = n$, consider the equation (10) in the following form

$$ (11) \qquad \sum_j \omega_{i,0} \wedge \omega_{i,\alpha+e_j} = dR(\omega_{i,\alpha}) - \sum_{j,\beta \ge 1} \binom{\alpha}{\beta} \omega_{j,\beta} \wedge \omega_{i,\alpha-\beta+e_j}, $$

where $\omega_{i,\alpha+e_j}$ are to be determined. The lemma 2.4.2.4 tells us that the complex $\wedge_{\mathcal{O}_X}^{n+2}(\mathbb{L})$ is exact in degree $[n - 2, n]$, and thus, in particular, that the sequence

$$ \mathrm{Sym}^{n+1}(\mathcal{O}_X^d) \otimes \Omega_{X/k}^1 \to \mathrm{Sym}^n(\mathcal{O}_X^d) \otimes \Omega_{X/k}^2 \to \mathrm{Sym}^{n-1}(\mathcal{O}_X^d) \otimes \Omega_{X/k}^3 $$

is exact. The right hand side of (11) can be considered as cocycle in $\mathrm{Sym}^n(\mathcal{O}_X^d) \otimes \Omega_{X/k}^2$, because equations (10) are satisfied for $|\alpha| < n$. The condition that this cocycle is in fact a coboundary precisely says that there exists forms $\omega_{i,\alpha+e_j}$ such that the above equation (11) is satisfied. This finishes the proof of the Corollary. $\square$

We finish by emphasising an important consequence of lemma 2.4.2.4 that will be used later on. Let $\mathcal{F}$ be a derived foliation which is rigid and quasi-smooth, and $K_{\mathcal{F}}$ the corresponding ideal. We can compare the derived de Rham cohomology of $\mathcal{F}$ with the naive de Rham cohomology of $K_{\mathcal{F}}$ by means of the natural projection

$$ p : \mathbf{DR}_{\mathcal{F}} \to \mathbf{DR}_{\mathcal{F}}^{st} = \mathrm{Sym}_{\mathcal{O}_X}^{st}(\Omega_{\mathcal{F}}^1[1]). $$

Here, $\Omega_{\mathcal{F}}^1 = H^0(\mathbb{L}_{\mathcal{F}})$ is the coherent sheaf of 1-forms along $\mathcal{F}$, which is quasi-isomorphic to $\mathbb{L}_{\mathcal{F}}$ via the natural truncation map $\mathbb{L}_{\mathcal{F}} \to H^0(\mathbb{L}_{\mathcal{F}}) \simeq \Omega_{\mathcal{X}}^1/\mathcal{N}_{\mathcal{F}}^*$. The graded mixed cdga $\mathrm{Sym}_{\mathcal{O}_X}^{st}(\Omega_{\mathcal{F}}^1[1])$ is the naive or strict symmetric product algebra over the coherent sheaf $\Omega_{\mathcal{F}}[1]$, where the mixed structure is induced by the de Rham differential $\mathcal{O}_X \to \Omega_{\mathcal{F}}^1$. The morphism $p$ is also the natural projection of the connective object $\mathbf{DR}_{\mathcal{F}}$ to its 0-th truncation for the Beilinson's t-structure on graded mixed objects (see 1.2.3.4). By passing to the non-Tate realization we have an induced morphism of the corresponding sheaves of de Rham cohomologies

$$ |\mathbf{DR}_{\mathcal{F}}| \simeq \hat{C}_{DR}^*(\mathcal{F})^u \to |\mathbf{DR}_{\mathcal{F}}^{st}| := C_{DR,naive}^*(\mathcal{F}). $$
Passing on hyper cohomology groups over $X$ we get an induced morphism

$$ \hat{H}_{DR}^{i}(\mathcal{F}) \to H_{DR,naive}^{i}(\mathcal{F}) $$

relating the Hodge completed derived de Rham cohomology with naive de Rham cohomology of $\mathcal{F}$.

**COROLLARY 2.4.2.5.** With the notations above, assume that $\mathbb{L}_{\mathcal{F}}$ is a vector bundle outside of a closed subset $Z \in X$ of codimension $d$. Then, the induced morphism

$$ \hat{H}_{DR}^{i}(\mathcal{F}) \to H_{DR,naive}^{i}(\mathcal{F}) $$

is an isomorphism for all $i < d - 1$.

*Proof of the corollary.* This is a direct application of the lemma 2.4.2.4 and the compatibility of the morphism $|\mathbf{DR}_{\mathcal{F}}| \simeq \hat{C}_{DR}^{*}(\mathcal{F})^{u} \to |\mathbf{DR}_{\mathcal{F}}^{st}| =: C_{DR,naive}^{*}(\mathcal{F})$ with the Hodge filtrations on both sides. $\square$

## 2.5. Direct images

Let $f: X \to Y$ be a morphism of derived stacks. The pull-back $f^{*}: \mathbf{dSt}/Y \to \mathbf{dSt}/X$, sending $(Z \to Y)$ to $(X \times_{Y} Z \to X)$, commutes with arbitrary colimits. In particular, it admits a right adjoint

$$ f_{*}: \mathbf{dSt}/X \to \mathbf{dSt}/Y, $$

called the *direct image*, or *Weil restriction* along $f$.

For any $Z \to X$, the adjunction morphism is a commutative diagram of derived stacks

<figure><img src="image_21.png" /></figure>

The adjunction counit $X \times_{Y} f_{*}(Z) \to Z$ will be denoted by $ev$ and will be called the *evaluation morphism*. The following theorem shows that, under certain properness conditions, derived foliations can be push-forward. This provides a very powerful tool to construct new derived foliations from existing ones.

**THEOREM 2.5.0.1.** Assume that $f: X \to Y$ is a morphism of derived Artin stacks (locally of finite presentation) which is quasi-smooth and representable by a proper derived algebraic space. Let $Z \to X$ be a derived Artin stack locally of finite presentation over $X$. Then, there exists an $\infty$-functor

$$ f_{*}: \mathcal{Fol}^{p}(Z/X) \to \mathcal{Fol}^{p}(f_{*}(Z)/Y) $$

from perfect derived foliations on $Z$ relative to $X$ to perfect derived foliations on $f_{*}(Z)$ relative to $Y$ (see Example 2.1.5.3). Moreover, the cotangent complex of $f_{*}(\mathcal{F})$ is given by

$$ \mathbb{L}_{f_{*}(\mathcal{F})} \simeq q_{+}(ev^{*}(\mathbb{L}_{\mathcal{F}})) $$
where $q: X \times_Y f_*(Z) \to f_*(Z)$ is the second projection and $q_+ : \mathrm{Perf}(X \times_Y f_*(Z)) \to \mathrm{Perf}(f_*(Z))$ is defined on perfect complexes by $q_+(E) := (q_*(E^\vee))^\vee$.

The above theorem is proven in [Alf25] and its proof will not be covered in this book. In the next two subsections we will simply explain how the $\infty$-functor $f_*$ is constructed, the reader will find details in [Alf25].

**2.5.1. Derived linear stacks.** For an affine derived scheme $X = \mathrm{Spec} A$, and an $A$-dg-module $E$, we can form the linear derived stack associated to $E$, denoted by $\mathcal{V}(E)$, as follows. We consider the $\infty$-functor

$$ \mathcal{V}(E) : \mathrm{cdga}^{\le 0}_A \to \mathrm{Top} $$

sending a commutative connective $A$-linear cdga $B$ to $\mathrm{Map}_{\mathrm{dg}_A}(E, B)$, the mapping space of $A$-dg-modules from $E$ to $B$. Note that, by adjunction, $\mathrm{Map}_{\mathrm{dg}_A}(E, B) \simeq \mathrm{Map}_{\mathrm{dg}_B}(E \otimes_A^L B, B)$. The group scheme $\mathbb{G}_m$ acts on $\mathcal{V}(E)$ by acting on $B$ by its weight 1 action. More concretely, $\mathbb{G}_m(B)$ can be identified with the simplicial set of self autoequivalences of $B$ as a $B$-dg-modules. This is, by definition, the simplicial sub-monoid of $\mathrm{Map}_{\mathrm{dg}_B}(B, B)$ of endomorphisms of $B$ as a $B$-dg-modules, and thus it acts naturally on $B$ as an object in the $\infty$-category $\mathrm{dg}_B$.

The linear derived stack associated to $E$, $\mathcal{V}(E)$, will be considered as an object in $\mathrm{dSt}^{\mathbb{G}_m}_X$, the $\infty$-category of derived stacks over $X$ endowed with a $\mathbb{G}_m$-action. These local constructions are obviously compatible with base change on $X$, so they can be glued to obtain, for any derived stack $X$, an $\infty$-functor

$$ \mathcal{V} : \mathrm{QCoh}(X)^{op} \to \mathrm{dSt}^{\mathbb{G}_m}_X. $$

It is shown in [Mon21] that the $\infty$-functor $\mathcal{V}$ has a left adjoint sending an $\mathbb{G}_m$-equivariant derived stack $Y \to X$ to its sheaf of functions of weight 1. Moreover, when restricted to bounded above quasi-coherent complexes, the counit of this adjunction is an equivalence, so that the induced $\infty$-functor

$$ \mathcal{V} : \mathrm{QCoh}^{<0}(X)^{op} \to \mathrm{dSt}^{\mathbb{G}_m}_X $$

is a full embedding. An object in the image of $\mathcal{V}$ will be called a *derived linear stack over* $X$. We will mainly use this notion for the more specific case of *perfect complexes* on $X$.

On a final note, when $E \in \mathrm{Perf}(X)$ is a perfect complex on $X$, then the linear derived stack $\mathcal{V}(E) \to X$ is a relative derived Artin stack which is strongly finitely presented (see [TV07, Prop. 2.23]). Such $\mathbb{G}_m$-equivariant derived stacks will be called *perfect derived linear stacks over* $X$. They form an $\infty$-category naturally equivalent to $\mathrm{Perf}(X)^{op}$ by means of the construction $E \mapsto \mathcal{V}(E)$. Note that $\mathcal{V}(E)$ is a derived Artin stack locally of finite presentation over $k$ if $X$ is so. Moreover, the projection $\mathcal{V}(E) \to X$ is smooth if and only if $E$ is of positive Tor amplitude, and is affine if and only if $E$ is connective.

**2.5.2. Derived foliations as equivariant derived linear stacks.** The importance of derived linear stacks for us is the fact that they can be used in order to provide a geometric interpretation for derived foliations. This geometric interpretation has its own interests, and we will use it for instance in order to construct the Hodge filtration for general derived foliations.
It will also be useful for the notion of derived foliations in non-zero characteristic discussed in Chapter 7.

We recall the group stack $\mathcal{H}_0$, defined as the semi-direct product of $B\mathbb{G}_a$ and $\mathbb{G}_m$ for the $\mathbb{G}_m$-action of weight $-1$ on $\mathbb{G}_a$ (see §1.5). For a derived affine scheme $X = \text{Spec } A$, we have its graded loop space $\mathcal{L}^{gr}(X/k)$, defined as the derived mapping stack

$$ \mathcal{L}^{gr}(X/k) := \underline{\text{Map}}(B\mathbb{G}_a, X) \in \text{dSt}_k. $$

The group $\mathcal{H}_0$ acts on $B\mathbb{G}_a$, where the $\mathbb{G}_m$ component acts by weight $-1$ and $B\mathbb{G}_a$ acts by translation on itself. As a result, $\mathcal{H}_0$ acts on $\mathcal{L}^{gr}(X/k)$, and we will thus consider the graded loop stack as a $\mathcal{H}_0$-equivariant derived stack. Note also that as $X$ is affine, so is $\mathcal{L}^{gr}(X/k)$, and it is explicitly given by

$$ \mathcal{L}^{gr}(X/k) \simeq \text{Spec}(\text{Sym}_A(\mathbb{L}_{A/k}[1])) \simeq \mathbb{V}(\mathbb{L}_{A/k}[1]), $$

and as a result we have a natural equivalence of graded cdga’s

$$ \mathcal{O}(\mathcal{L}^{gr}(X/k)) \simeq \mathbf{DR}(X/k). $$

Moreover, as shown in [MRT22], the $\mathcal{H}_0$-action on $\text{Sym}_A(\mathbb{L}_{A/k}[1]) = \mathbf{DR}(X/k)$ corresponds, by means of the above identification and our corollary 1.5.0.3, to its usual graded mixed structure given by the de Rham differential.

When $X$ is a general derived stack, we define the completed graded loop stack by imposing descent

$$ \hat{\mathcal{L}}^{gr}(X/k) := \text{colim}_{\text{Spec } A \to X} \mathcal{L}^{gr}(\text{Spec } A/k) \in \text{dSt}_k^{\mathcal{H}_0}. $$

Note that $\hat{\mathcal{L}}^{gr}(X/k)$ can not always be identified with the derived mapping stack $\underline{\text{Map}}(B\mathbb{G}_a, X)$. There exists a natural descent morphism $\underline{\text{Map}}(B\mathbb{G}_a, X) \to \hat{\mathcal{L}}^{gr}(X/k)$, but it is not an equivalence in general (it is so when $X$ is a derived scheme, or more generally a derived Deligne-Mumford stack). Finally, for an arbitrary morphism of derived stacks $X \to Y$, we can define the relative graded loop stack

$$ \hat{\mathcal{L}}^{gr}(X/Y) := \hat{\mathcal{L}}^{gr}(X/k) \times_{\hat{\mathcal{L}}^{gr}(Y/k)} Y $$

which is now an $\mathcal{H}_0$-equivariant derived stack over $Y$.

We are now ready to define a notion of geometric derived foliations, based on linear stacks and graded loop stacks. This definition is shown in [Alf25] to be equivalent to Definition 2.1.3.3. For any derived stack $X$, we consider its derived graded loop stack $\hat{\mathcal{L}}^{gr}(X/k)$ endowed with its natural $\mathcal{H}_0$-action. It comes equipped with a projection $\pi : \hat{\mathcal{L}}^{gr}(X/k) \to X$ by evaluating at the natural base point of $B\mathbb{G}_a$. We note here that even though $\pi$ is not $\mathcal{H}_0$-equivariant (for the trivial action of $\mathcal{H}_0$ on $X$), it is still endowed with a canonical $\mathbb{G}_m$-equivariant structure, given by considering the induced action via the canonical section $\mathbb{G}_m \to \mathcal{H}_0$.

**DEFINITION 2.5.2.1.** With the notations above, a geometric derived foliation $\mathcal{F}$ on $X$ (relative to $k$) consists of the datum of a $\mathcal{H}_0$-equivariant derived stack $\mathbb{V}(\mathcal{F})$ over $\hat{\mathcal{L}}^{gr}(X/k)$, such
the following condition is satisfied: for any derived affine scheme $S$ and morphism $S \to X$, the $\mathbb{G}_m$-equivariant morphism

$$V(\mathcal{F}) \times_{\hat{\mathcal{L}}^{gr}(X/k)} \hat{\mathcal{L}}^{gr}(S/k) \to S$$

makes $V(\mathcal{F})$ into a perfect derived linear stack over $S$.

By definition, geometric derived foliations $\mathcal{F}$ on $X$ form a full sub-$\infty$-category of $\mathrm{dSt}^{\mathcal{H}_0}_{/\hat{\mathcal{L}}^{gr}(X/k)}$, the $\infty$-category of $\mathcal{H}_0$-equivariant derived stacks over $\hat{\mathcal{L}}^{gr}(X/k)$.

As a first comment concerning the above definition, it is important to note that $V(\mathcal{F})$ is not a derived linear stack globally over $X$, in the exact same manner as the big cotangent complex of a derived foliation is not a quasi-coherent complex (see Definition 2.1.3.4). This is expressed by the strange looking condition in Definition 2.5.2.1. When $X$ is an affine derived scheme, taking $S = X$, this condition simply states that $V(\mathcal{F})$ is a derived linear stack over $X$. The same holds, more generally, whenever $X$ is a derived DM stack, since $\mathcal{L}^{gr}(-/k)$ has étale descent. It is shown in [Alf25] that the above notion of geometric derived foliations is equivalent to our original notion of derived foliations. The equivalence between the two notions passes through the interpretation of graded mixed dg-modules as quasi-coherent complexes with $\mathcal{H}_0$-action (see §1.5). We will not go into too many details here, and we will limit ourselves to mention some basic facts. As a first comment, we can always restrict to the case where $X$ is an affine derived scheme, and then proceed by gluing for general derived Artin stacks. For a derived foliation $\mathcal{F}$, on an affine derived scheme $X$, the corresponding geometric derived foliation will be given by $V(L_{\mathcal{F}/k}[1])$, the derived linear stack associated to the shifted tangent $L_{\mathcal{F}/k}[1]$. One of the result of [Alf25] states that the graded mixed structure on $\mathrm{DR}(\mathcal{F}/k)$ induces a $\mathcal{H}_0$-action on $V(L_{\mathcal{F}/k}[1])$ which makes it into a geometric derived foliation in the sense of Definition 2.5.2.1. Conversely, if $\mathcal{F}$ is a geometric derived foliation on $X$, the corresponding derived foliation can be obtained by considering the push-forward

$$\pi_*(\mathcal{O}) \in \mathrm{QCoh}(X \times B\mathbb{G}_m),$$

of the structure sheaf along the projection $V(\mathcal{F}) \to X$. This quasi-coherent sheaf is endowed with a commutative multiplication, as well as a graded mixed structure coming from the action of $\mathcal{H}_0$ (see Corollary 1.5.0.3), and is therefore a graded mixed cdga. The projection $V(\mathcal{F}) \to \hat{\mathcal{L}}^{gr}(X/k)$ induces a structure morphism $\mathrm{DR}_{X/k} \to \pi_*(\mathcal{O})$, which makes $\pi_*(\mathcal{O})$ into a derived foliation over $X$ (relative to $k$).

Definition 2.5.2.1 will be reconsidered (with an important variation) in Chapter 7 in order to define derived foliations in non-zero characteristic.
<figure><img src="image_22.png" /></figure>
CHAPTER 3

De Rham cohomology of derived foliations

In this third chapter we study the foliated de Rham cohomology of derived foliations. For this we introduce the notion of crystals along derived foliations. Crystals must be thought as quasi-coherent complexes endowed with partial flat connections along the foliation, and are natural coefficients for foliated de Rham cohomology. In the third section of the chapter we construct the Hodge filtration associated to a derived foliation $\mathcal{F} \in \mathcal{Fol}(X/k)$, which is a filtration on the de Rham cohomology of $X$. Its associated graded are identified with the foliated de Rham cohomology with coefficients in exterior powers of the conormal complex. Finally, the conormal complex of a derived foliation possesses very rich structures, and defines a Lie algebra objects inside the $\infty$-category of crystals. This can be used to study the transversal geometry of a derived foliations, such as transversal jets of functions, but also algebraic invariants related to the classical notion of holonomy.

All along this section $k$ is a fixed $\mathbb{Q}$-algebra.

3.1. Crystals along derived foliations

Let $\mathcal{F} \in \mathcal{Fol}(X/k)$ be a derived foliation on a derived stack $X$ over $k$, $\mathbf{DR}_{\mathcal{F}}$ the corresponding sheaf of graded mixed cdga's on $X$, and

$$ \mathbf{DR}_{\mathcal{F}} - \epsilon - \mathbf{dg}_{X}^{gr} := \lim_{\mathbf{Spec} A \to X} \mathbf{DR}_{\mathcal{F}}(A) - \epsilon - \mathbf{dg}_{k}^{gr} $$

the $\infty$-category of sheaves of graded mixed $\mathbf{DR}_{\mathcal{F}}$-dg-modules on $X$ (see Definition 2.1.3.5 and corollary 2.1.3.6). By definition of derived foliations, the weight 0 part of $\mathbf{DR}_{\mathcal{F}}^{(0)}$ comes equipped with a canonical identification to $\mathcal{O}_X$. Therefore, for any sheaf $E$ of graded mixed $\mathbf{DR}_{\mathcal{F}}$-dg-modules, the weight $n$ part $E^{(n)}$ is endowed with a natural $\mathcal{O}_X$-dg-module structure.

DEFINITION 3.1.0.1. With the notations above, a quasi-coherent crystal over $\mathcal{F}$ is a graded mixed $\mathbf{DR}_{\mathcal{F}}$-dg-module $E$ satisfying the following two conditions.

* The weight 0 dg-module $E^{(0)}$, considered as a $\mathbf{DR}_{\mathcal{F}}^{(0)} = \mathcal{O}_X$-dg-module, is quasi-coherent over $X$.

* For any $n \in \mathbb{Z}$, the natural morphism

$$ E(0) \otimes_{\mathcal{O}_X} \mathbf{DR}_{\mathcal{F}}^{(n)} \to E^{(n)} $$

is a quasi-isomorphisms of $\mathcal{O}_X$-dg-modules.
The $\infty$-category of quasi-coherent crystals over $\mathcal{F}$ is the full sub-$\infty$-category $\mathbf{QCoh}(\mathcal{F})$ of $\mathbf{DR}_{\mathcal{F}} - \epsilon - \mathbf{dg}_X^{gr}$ consisting of quasi-coherent crystals.

REMARK 3.1.0.2. It should be noted that quasi-coherent crystals on $X$ are not quasi-coherent sheaves on $X$ in the sense that, except for $n = 0$, the weight $n$ pieces $E^{(n)}$ are $\mathcal{O}_X$-dg-modules which are not quasi-coherent over $\mathcal{O}_X$. However, a quasi-coherent crystal is always quasi-coherent over the sheaf of graded mixed cdga's $\mathbf{DR}_{\mathcal{F}/k}$.

Various operations on crystals will be studied in Chapter 4, but we will mention here the existence of pull-backs. The $\infty$-category $\mathbf{QCoh}(\mathcal{F})$ is contravariantly functorial in the pair $(X, \mathcal{F})$ in the following sense. Suppose that we have two pairs $(X, \mathcal{F})$ and $(Y, \mathcal{G})$ consisting of derived schemes endowed with derived foliations. A morphism $f : (X, \mathcal{F}) \to (Y, \mathcal{G})$ will be a pair $(g, u)$ consisting of

* a morphism $g : X \to Y$ of derived schemes,

* a morphism $u : \mathcal{F} \to g^*(\mathcal{G})$ of derived foliations over $X$ (i.e. a morphism $\mathbf{DR}_{g^*\mathcal{G}} := \mathbf{DR}_{\mathcal{F}}$ of graded mixed cdga's over $X$).

Associated to such a morphism $f = (g, u)$ there is a pull-back $\infty$-functor

$$f^! : \mathbf{QCoh}(\mathcal{G}) \to \mathbf{QCoh}(\mathcal{F})$$

constructed as follows. By definition of pull-backs of derived foliations, we have an equivalence of graded mixed cdga's on $X$

$$\mathbf{DR}_{g^*(\mathcal{G})} \simeq \mathbf{DR}_X \otimes_{g^{-1}(\mathbf{DR}_Y)} g^{-1}(\mathbf{DR}_{\mathcal{G}}).$$

The morphism $u$ thus corresponds to a morphism of sheaves of graded mixed cdgas over $X$ under $\mathbf{DR}_X$

$$u : \mathbf{DR}_X \otimes_{g^{-1}(\mathbf{DR}_Y)} g^{-1}(\mathbf{DR}_{\mathcal{G}}) \to \mathbf{DR}_{\mathcal{F}}$$

or equivalently, to a morphism of sheaves of graded mixed cdga's under $g^{-1}(\mathbf{DR}_Y)$

$$u : g^{-1}(\mathbf{DR}_{\mathcal{G}}) \to \mathbf{DR}_{\mathcal{F}}.$$

The pull-back $\infty$-functor $f^!$ on quasi-coherent crystals is thus simply defined by the following formula (for $E$ a graded mixed $\mathbf{DR}_Y$-module)

$$f^!(E) := g^{-1}(E) \otimes_{g^{-1}(\mathbf{DR}_Y)} \mathbf{DR}_{\mathcal{F}}.$$

Clearly, the rule $((X, \mathcal{F}) \mapsto \mathbf{QCoh}(\mathcal{F}), f \mapsto f^!)$ can be promoted to an $\infty$-functor $\mathbf{QCoh}^!$ from the $\infty$-category of pairs $(X, \mathcal{F})$ to the $\infty$-category of $\infty$-categories. Note also that $\mathbf{QCoh}(\mathcal{F})$ comes equipped with a natural symmetric monoidal structure (induced by the tensor product of graded mixed $\mathbf{DR}_{\mathcal{F}}$-modules), and that the pull-back $f^!$ has a natural symmetric monoidal structure as well.

Moreover, $f^!$ is compatible with the pull-back of quasi-coherent sheaves on derived stacks in the following sense. By definition of $\mathbf{QCoh}(\mathcal{F})$, we have a forgetful $\infty$-functor

$$\mathbf{QCoh}(\mathcal{F}) \to \mathbf{QCoh}(X)$$
which sends a graded mixed $\mathbf{DR}_F$-module $E$ to its weight zero part $E^{(0)} \in \mathbf{QCoh}(X)$. For a morphism $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ as above, the compatibility of $f^!$ with the pull-back of quasi-coherent sheaves is expressed by the commutativity of the following square

<figure><img src="image_23.png" /></figure>

as this can be easily seen using the explicit formula $f^!(E) := g^{-1}(E) \otimes_{g^{-1}(\mathbf{DR}_F)} \mathbf{DR}_F$ and the condition (2) of Definition 3.1.0.1.

The following definition fixes some terminology.

**DEFINITION 3.1.0.3.** Let $X \in \mathbf{dSt}_k$ and $\mathcal{F} \in \mathcal{Fol}(X/k)$.

(1) For a quasi-coherent crystal $E \in \mathbf{QCoh}(\mathcal{F})$, the underlying quasi-coherent complex of $E$ is $E^{(0)}$.

(2) For $E' \in \mathbf{QCoh}(X)$, a crystal structure on $E'$ along $\mathcal{F}$ consists of $E \in \mathbf{QCoh}(X)$ together with an equivalence $E^{(0)} \simeq E'$.

We warn the reader that we will often refer to objects in $\mathbf{QCoh}(\mathcal{F})$ by their underlying objects in $\mathbf{QCoh}(X)$. The typical example is the unit crystal $\mathcal{O}_X$, that is the structure sheaf endowed with its canonical crystal structure. Strictly speaking this object should be written $\mathbf{DR}_F$, as it corresponds to $\mathbf{DR}_F$ as a module over itself. However, we will keep referring to this object with the symbol $\mathcal{O}_X$, to keep in mind that crystals are somehow $\mathcal{D}$-module structures (see Chapter 4 for a precise statement).

**3.1.1. Examples.** We conclude this section by listing some basic examples of the notion quasi-coherent crystals.

**Crystals over the trivial foliation.** Assume that $X$ is a derived Artin stack, so the initial derived foliation exists (see Example 2.1.5.1). Now, for $\mathcal{F} = \mathcal{O}_X$ the initial foliation, the $\infty$-category $\mathbf{QCoh}(\mathcal{F})$ clearly is equivalent to $\mathbf{QCoh}(X)$, the $\infty$-category of quasi-coherent complexes over $X$. This equivalence is realized by sending $E \in \mathbf{QCoh}(\mathcal{F})$ to its weight 0 part $E(0) \in \mathbf{QCoh}(X)$. It can be promoted to an equivalence of symmetric monoidal $\infty$-categories.

**Crystals over the tautological foliation.** Assume that $\mathcal{F} = \mathcal{O}_X$ is the final foliation. If $X$ is a smooth variety, then there is a canonical equivalence of $\infty$-categories

$$ \mathbf{QCoh}(\mathcal{O}_X) \simeq \mathcal{D}_X - \mathrm{dg}, $$

between quasi-coherent crystals along $\mathcal{O}_X$ and quasi-coherent complexes of left $\mathcal{D}_X$-modules. This equivalence will be reviewed and generalized in the next Section 3.2. This equivalence is
again compatible with the natural symmetric monoidal structures involved. When $X$ is not smooth anymore, and for instance is a derived scheme, we believe that our notion of quasi-coherent crystals on $*_{X}$ coincides with the notion of derived $\mathcal{D}_{X}$-modules of [Ber21].

**Crystals over the Dolbeault foliation.** Let $\mathcal{F} = *_{Dol}$ be the Dolbeault foliation on a smooth variety $X$, defined by $\mathrm{DR}(\mathcal{F}) := \mathrm{Sym}_{\mathcal{O}_{X}}(\Omega_{X}^{1}[1]))$ where the graded cdga $\mathrm{Sym}_{\mathcal{O}_{X}}(\Omega_{X}^{1}[1]))$ is endowed with the zero mixed structure. Then $\mathrm{QCoh}(*_{Dol})$ is naturally equivalent to the derived $\infty$-category of complexes of quasi-coherent Higgs sheaves on $X$ in the sense of [Sim96]. Dolbeault foliations and the tautological foliations can be combined together into a *Hodge foliation* denoted by $*_{Hod}$. It is a derived foliation on $X \times \mathbb{A}^{1}$ interpolating between $*_{DR}$ (fiber at 1) and $*_{Dol}$ (fiber at 0). This can be constructed using the deformation to the normal bundle of our appendix A.7 and will be reviewed in our next section concerning the Hodge filtration (see §3.3).

**Crystals over integrable foliations.** Let $f : X \to Y$ be a morphism of derived schemes and $\mathcal{F} := f^{*}(\mathcal{O}_{Y})$ be the corresponding derived foliation (see example 2.1.5.3). Then $\mathrm{QCoh}(\mathcal{F})$ is, by definition, the $\infty$-category of relative $\mathcal{D}$-modules on $X/Y$. When $f$ is a smooth morphism between smooth varieties these relative $\mathcal{D}$-modules can be written as complexes of modules over $\mathcal{D}_{X/Y}$, the sheaf (of algebras) of relative differential operators along the fibers of $f$. When $f$ is no more supposed to be smooth, we will see that $\mathcal{D}_{X/Y}$ only exists as a sheaf of dg-algebras on $X$ (see Definition 4.1.1.2).

**Representations of Lie algebras as crystals.** Let $X = \mathrm{Spec} k$ and $\mathfrak{g}$ be a dual almost perfect dg-Lie algebra over a field $k$. By theorem 2.2.0.7, $L$ determines a derived foliation $\mathcal{F}_{\mathfrak{g}} \in \mathcal{F}ol(*)$, for which the graded mixed cdga is the Chevalley-Eilenberg complex $\mathrm{DR}(\mathcal{F}) := \mathrm{CE}^{*}(\mathfrak{g})$. There exists a natural $\infty$-functor

$$ \phi : \mathfrak{g} - \mathrm{dg}_k \to \mathrm{DR}(\mathcal{F}) - \epsilon - \mathrm{dg}^{gr}, $$

from $\mathfrak{g}$-dg-modules to the $\infty$-category of graded mixed $\mathrm{DR}(\mathcal{F})$-dg-modules, given by the Chevalley-Eilenberg complex for dg-modules. As for theorem 2.2.0.7, it can be shown that the $\infty$-functor $\phi$ is an equivalence of $\infty$-categories. This example of course can be generalized to the case of dg-Lie algebroids.

## 3.2. Foliated de Rham cohomology

The notion of quasi-coherent crystals along derived foliations can be used in order to define a general notion of *foliated de Rham cohomology with coefficients*. Let $X \in \mathrm{dSt}_k$ be any derived stack and $\mathcal{F} \in \mathcal{F}ol(X/k)$ be a derived foliation over $X$. The symmetric monoidal $\infty$-category $\mathrm{QCoh}(\mathcal{F})$ sits inside $\mathrm{DR}_{\mathcal{F}} - \epsilon - \mathrm{dg}^{gr}$. We have a global section $\infty$-functor

$$ \Gamma : \mathrm{DR}_{\mathcal{F}} - \epsilon - \mathrm{dg}^{gr} \to \epsilon - \mathrm{dg}_{k}^{gr}, $$
which is a lax monoidal $\infty$-functor to graded mixed complexes. It can be composed with our Tate realization functor of §1.3.2, in order to get a lax monoidal global section $\infty$-functor

$$ \Gamma : \mathrm{DR}_{\mathcal{F}} - \epsilon - \mathrm{dg}^{gr} \to \widehat{\mathrm{dg}}_{k}^{\mathrm{fil}} $$

to complete filtered complexes.

**DEFINITION 3.2.0.1.** The above lax monoidal $\infty$-functor $\Gamma$, when restricted to $\mathrm{QCoh}(\mathcal{F})$ is denoted by

$$ \hat{C}_{DR}(\mathcal{F}, -) : \mathrm{QCoh}(\mathcal{F}) \to \widehat{\mathrm{dg}}_{k}^{\mathrm{fil}}. $$

For any object $E \in \mathrm{QCoh}(\mathcal{F})$, the complex $\hat{C}_{DR}(\mathcal{F}, E)$ is called the foliated de Rham cohomology along $\mathcal{F}$ with coefficients in $E$.

By construction, $\hat{C}_{DR}(\mathcal{F}, E)$ is a complete filtered complex, which is functorial in $E$, but also in $\mathcal{F}$ and $X$. The lax monoidal structure provides natural morphisms of complete filtered complexes

$$ \hat{C}_{DR}(\mathcal{F}, E) \hat{\otimes} \hat{C}_{DR}(\mathcal{F}, E') \to \hat{C}_{DR}(\mathcal{F}, E \otimes E'). $$

In particular, if $E$ is any (commutative) algebra object (e.g the unit crystal $\mathcal{O}_X$), we get a complete filtered (commutative) dg-algebra of cohomology $\hat{C}_{DR}(\mathcal{F}, E)$. Typically, the de Rham cohomology of the unit crystal along $\mathcal{F}$ provides a complete filtered cdga $\hat{C}_{DR}(\mathcal{F}, \mathcal{O}_X)$, where $\mathcal{O}_X$ is the structure sheaf endowed with its canonical crystal structure along $\mathcal{F}$.

Coming back to the basic examples of crystals given in §3.1.1, we can for each of them relate the foliated de Rham cohomology with more classical notions.

**Initial and final foliations.** We have $\mathrm{QCoh}(\mathcal{O}_X) \simeq \mathrm{QCoh}(X)$ and in this case foliated de Rham cohomology simply coincides with quasi-coherent cohomologies of $X$ with coefficients in quasi-coherent complexes. For the final foliation $*_X$ on a smooth variety, $\mathrm{QCoh}(*_X)$ identifies with the $\infty$-category of quasi-coherent $\mathcal{D}$-modules, and foliated de Rham cohomology simply is the usual de Rham cohomology of $X$ relative to $\mathrm{Spec} k$.

**Dolbeault foliation.** In this case $\mathrm{QCoh}(*_\mathcal{Dol})$ is identified with quasi-coherent Higgs complexes over $X$, and foliated de Rham cohomology coincide here with the *Hodge completed derived Higgs cohomology*. This is defined the exact same manner as the Hodge complete derived de Rham cohomology.

**Lie algebras representations.** For a derived foliation $\mathcal{F}$ on $\mathrm{Spec} k$ determines by a dual almost perfect dg-Lie algebra $\mathfrak{g}$, $\mathrm{QCoh}(\mathcal{F})$ identifies with the $\infty$-category of $\mathfrak{g}$-dg-modules and foliated de Rham cohomology is equivalent to the usual Lie algebra cohomology defined via the Chevalley complex.
### 3.3. The Hodge filtration

In this section we show that a derived foliation $\mathcal{F}$ on a derived stack $X$ induces a canonical filtration on the completed de Rham complex $\hat{C}_{DR}(X/k)$, which is moreover multiplicative and functorial in $X$. We call it the *Hodge filtration associated to $\mathcal{F}$*, as the usual Hodge filtration on $\hat{C}_{DR}(X/k)$ will be recovered for the special case where $\mathcal{F} = 0_X$ is the initial derived foliation. When the derived foliation $\mathcal{F}$ is integrable by a morphism $f: X \to Y$, the associated Hodge filtration is intimately related to the Gauss-Manin connection with respect to $f$.

As usual, we start by the case of an affine derived scheme $X = \text{Spec } A$, and obtain the globalization by usual gluing. Let $\mathcal{F} \in \text{Fol}(X/k)$ be a derived foliation on $X$, which we assume to be almost perfect. We consider the corresponding $\mathcal{H}_0$-equivariant derived linear stack $p: \mathcal{V}(\mathcal{F}) \to \mathcal{L}^{gr}(X/k)$ (see §2.5.2), an we perform the deformation to the normal bundle for the morphism $p$, as recalled in our appendix A.7. This provides a morphism of $\mathcal{H}_0$-equivariant derived stacks over $A = [A^1/G_m]$

$$j: \mathcal{V}(\mathcal{F}) \times A \to Def(p).$$

Over the open point $[G_m/G_m]$, the morphism $j$ recovers the original morphism $p$, whereas over the closed point we get the zero section of the shifted relative tangent stack $T(p)[1]$, or equivalently $\mathcal{V}(\mathcal{L}_p[-1])$, the derived linear stack associated to the shifted cotangent complex of the morphism $p$. More generally, $Def(p) \to A$ is itself a relative derived linear stack, associated to $\mathcal{L}_{X/k}[1] \to \mathcal{L}_{\mathcal{F}}[1]$, considered as a one step filtered quasi-coherent complexes on $X$, and thus as a quasi-coherent complexes on $A$ (see the end of §A.7). The cotangent complex $\mathcal{L}_p$ can furthermore be described thanks to the following lemma.

**LEMMA 3.3.0.1.** Let $X = \text{Spec } A$ be a derived affine $k$-scheme, $E$ be a bounded above quasi-coherent complex on $X$ and $\pi: \mathcal{V}(E) \to X$ the induced derived linear stack. Then, we have a functorial (in $E$ and $X$) equivalence in $\text{QCoh}(\mathcal{V}(E))$

$$\mathcal{L}_{\pi} \simeq \pi^{*}(E).$$

*Proof of the lemma.* For any derived linear stack $\pi: \mathcal{V}(E) \to X$ the relative cotangent complex $\mathcal{L}_{\pi}$, by definition, satisfies the following property: for any $A \to B$ of connective cdga's, andy $E \to B$ morphism of $A$-dg-modules corresponding to $u: \text{Spec } B \to \mathcal{V}(E)$, and any connective $M \in \text{dg}_B$, we have functorial (in $B$ and $M$) equivalences

$$\text{Map}_{\text{dg}_B}(u^*(\mathcal{L}_{\pi}), M) \simeq \text{Map}_{\text{dg}_{A/B}}(E, B \oplus M) \simeq \text{Map}_{\text{dg}_A}(E, M).$$

This shows that $u^*(\mathcal{L}_{\pi}) \simeq B \otimes_A E$ functorially in $B$, and thus that $\mathcal{L}_{\pi} \simeq \pi^*(E)$. This identification is furthermore functorial in $E$ in an obvious manner. $\square$

By passing to $\mathcal{H}_0$-equivariant global functions on $Def(p)$, we get this way a cdga endowed with a complete filtration. The underlying object of the filtration is given by $\mathcal{H}_0$-equivariant functions on $\mathcal{L}^{gr}(X/k)$, i.e. by the completed de Rham complex $\hat{C}_{DR}(X/k)$. The associated
graded for this filtration is given by $\mathcal{H}_0$-equivariant functions on $\mathcal{V}(\mathbb{L}_p[-1])$, and can be written as foliated de Rham cohomology with coefficient in some crystals as follows.

The graded cdga of graded functions on $\mathcal{V}(\mathbb{L}_p[-1])$, thanks to Lemma 3.3.0.1 is given, as a graded $\mathbf{DR}(\mathcal{F})$-cdga, by

$$ \mathrm{Sym}_{\mathbf{DR}(\mathcal{F})}(\mathbb{L}_p[-1]) \simeq \mathrm{Sym}_A(\mathcal{N}_{\mathcal{F}}^*[1]) \otimes_A \mathbf{DR}(\mathcal{F}), $$

where $\mathcal{N}_{\mathcal{F}}^*$ is the conormal complex of $\mathcal{F}$ defined as usual as being the fiber of $\mathbb{L}_{X/k} \to \mathbb{L}_{\mathcal{F}/k}$. We need to be slightly more precise here. The cdga $\mathrm{Sym}_{\mathbf{DR}(\mathcal{F})}(\mathbb{L}_p[-1]) \simeq \mathrm{Sym}_A(\mathcal{N}_{\mathcal{F}}^*[1]) \otimes_A \mathbf{DR}(\mathcal{F})$ has an action of $\mathcal{H}_0$, its natural graded mixed structure, and a second action of $\mathbb{G}_m$, as $\mathcal{V}(\mathbb{L}_p[-1])$ lives over $\mathbb{B}\mathbb{G}_m$. These two actions combined in an action of $\mathbb{G}_m \times \mathcal{H}_0$, and defines a graded objects inside the $\infty$-category of graded mixed $\mathbf{DR}(\mathcal{F})$-dg-modules. As such we have a decomposition of graded mixed $\mathbf{DR}(\mathcal{F})$-dg-modules

$$ \mathrm{Sym}_{\mathbf{DR}(\mathcal{F})}(\mathbb{L}_p[-1]) \simeq \bigoplus_{i \ge 0} ((\wedge^i \mathcal{N}_{\mathcal{F}}^*) \otimes_A \mathbf{DR}(\mathcal{F})) [i], $$

but for which $(\wedge^i \mathcal{N}_{\mathcal{F}}^*)$ is of weight $i$. In other words, the graded mixed structure on each piece $(\wedge^i \mathcal{N}_{\mathcal{F}}^*) \otimes_A \mathbf{DR}(\mathcal{F})$ is not quite a crystal in the sense of Definition 3.1.0.1 because it is not of weight 0. Rather, the graded mixed $\mathbf{DR}(\mathcal{F})$-dg-modules $(\wedge^i \mathcal{N}_{\mathcal{F}}^*) \otimes_A \mathbf{DR}(\mathcal{F})$ are such that their weight shifts ("Tate twists")

$$ ((\wedge^i \mathcal{N}_{\mathcal{F}}^*) \otimes_A \mathbf{DR}(\mathcal{F}))(-i) $$

belongs to $\mathbf{QCoh}(\mathcal{F})$.

Therefore, the $\mathcal{H}_0$-action on each piece $((\wedge^i \mathcal{N}_{\mathcal{F}}^*) \otimes_A \mathbf{DR}(\mathcal{F}))(-i)$ induces a canonical crystal structure on the quasi-coherent complexes $\wedge^i \mathcal{N}_{\mathcal{F}}^*$. The associated graded of the filtration on $\hat{C}_{DR}(X/k)$ can thus be written as

$$ |\mathrm{Sym}_{\mathbf{DR}(\mathcal{F})}(\mathbb{L}_p[-1])|^t \simeq \bigoplus_{i \ge 0} |(\wedge^i \mathcal{N}_{\mathcal{F}}^*) \otimes_A \mathbf{DR}(\mathcal{F})|^t \simeq \bigoplus_{\ge 0} \hat{C}_{DR}(\mathcal{F}, \wedge^i \mathcal{N}_{\mathcal{F}}^*)[-i], $$

where the extra shifts $[-i]$ comes from the fact that $(\wedge^i \mathcal{N}_{\mathcal{F}}^*) \otimes_A \mathbf{DR}(\mathcal{F})$ is of weight $i$ rather than weight 0 and thus its Tate realization is shifted by $-2i$ according its definition (see §1.3.2). We gather the above construction into the following proposition.

**PROPOSITON 3.3.0.2.** For a perfect derived foliation $\mathcal{F}$ on a derived Artin stack $X$.

(1) The wedge powers of the cotangent complex $\wedge_{O_X} \mathcal{N}_{\mathcal{F}}^*$ admit canonical crystal structure along $\mathcal{F}$.

(2) There exists a canonical (descending) filtration $F_{\mathcal{F}}^*$ on the (complete filtered) cdga $\hat{C}_{DR}(X/k)$ of completed de Rham cohomology on $X$. It possesses moreover the following properties.

(a) The filtration $F_{\mathcal{F}}^*$ is functorial in $X$ and $\mathcal{F}$.

(b) The associated graded, as a graded cdga, is given by for $i \le 0$

$$ \mathrm{Gr}_{F_{\mathcal{F}}^*}^i(\hat{C}_{DR}(X/k)) \simeq \hat{C}_{DR}(\mathcal{F}, \wedge^{-i} \mathcal{N}_{\mathcal{F}}^*)[i]. $$
In the above proposition, we have two extreme cases: when $\mathcal{F} = 0_X$ is the initial derived foliation the filtration $F_{\mathcal{F}}^*$ is the usual Hodge filtration, and when $\mathcal{F} = *_{X}$ is the final foliation then the filtration $F_{\mathcal{F}}^*$ is the trivial filtration.

Interesting examples of the Hodge filtration already appear for derived foliations over the point $*= \text{Spec } k$ for a field $k$. By corollary 2.2.0.8, we know that an almost perfect derived foliation $\mathcal{F}$ on $\text{Spec } k$ (relative to $k$) is given by a dg-Lie algebra $\mathfrak{g}^\vee$ which is dual almost perfect over $k$. The $\infty$-category of quasi-coherent crystals on $\mathcal{F}$ is equivalent to the $\infty$-category of $\mathfrak{g}$ dg-modules, and de Rham cohomology is given by the Lie algebra cohomology. The completed de Rham complex of $X$ reduces here to $k$. The conormal complex $N_{\mathcal{F}}^*$ naturally identifies with $\mathfrak{g}[-1]$, and its crystal structure is given by the coadjoint action. The Hodge filtration is then a filtration on $k$ whose associated graded are given by $C^*(\mathfrak{g}, \text{Sym}^i\mathfrak{g}^\vee)[-2i]$, the Lie algebra cohomology with coefficients in the wedge powers of the coadjoint representation. One way to see this filtration in more geometric terms, is to consider the formal derived stack $\mathcal{M}\mathcal{C}(\mathfrak{g}) \in \text{dSt}_k^*$ associated to $\mathfrak{g}$, as explained in §2.3.1. By corollary A.8.0.3, we have $\hat{C}_{DR}(\mathcal{M}\mathcal{C}(\mathfrak{g})) \simeq k$, and the usual Hodge filtration on the completed de Rham cohomology provides the Hodge filtration for the derived foliation $\mathcal{F}$.

We finish this part by mentioning another construction of a Hodge filtration, which we will call the *longitudinal Hodge filtration* in order to distinguish it from what we have seen in proposition 3.3.0.2. For any graded mixed cdga $D$, we define a new graded mixed cdga $D_{\text{Dol}}$ as follows. The underlying graded cdga of $D_{\text{Dol}}$ will be $D \otimes_k k[T]$ where the new variable $T$ has weight 0, and the new mixed structure will be $k[T]$-linear and given by $T \cdot \epsilon$, where $\epsilon$ is the mixed structure on $D$. This new graded mixed structure lives in $\epsilon - \text{cdga}^{gr}_{k[T]}$ and is equipped with a canonical compatible $\mathbb{G}_m$-action for which $T^n \cdot D \subset D \otimes_k k[T]$ is of weight $n$. Finally, the weight 0-part of $D_{\text{Dol}}$ is $D^{(0)}[T]$.

We now apply this to a derived foliation $\mathcal{F}$ on a derived affine $k$-scheme $X = \text{Spec } A$ and its corresponding graded mixed cdga $\text{DR}(\mathcal{F}/k)$. The graded mixed cdga $\text{DR}(\mathcal{F}/k)_{\text{Dol}}$ is, as a graded cdga, equivalent to $\text{Sym}_{A[T]}((\mathbb{L}_{\mathcal{F}} \otimes_A A[T]))[1]$, and thus defines a new derived foliation on $X \times A^1$. Moreover, the $\mathbb{G}_m$-action defines a descent data as a derived foliation on $X \times A$. This new derived foliation will be denoted by $\mathcal{F}_{\text{Hod}}$.

**DEFINITION 3.3.0.3.** The Dolbeault foliation associated to $\mathcal{F}$ is $\mathcal{F}_{\text{Hod}} \in \text{Fol}(X \times A)$ defined above.

Note that the pull-back of $\mathcal{F}_{\text{Hod}}$ at $X = [X \times \mathbb{G}_m/\mathbb{G}_m] \hookrightarrow X \times A$ recovers the original $\mathcal{F}$. On the other hand, the pull-back at the origin $X \to X \times A$ is given by the graded mixed cdga $\text{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$ with the zero mixed structure, with the induced morphism $\text{DR}(X/k) \to \text{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$ being the factorization $\text{DR}(X/k) \to \mathcal{O}_X \to \text{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$. This derived foliation will be denoted by $\mathcal{F}_{\text{Dol}}$ and is called the Dolbeault foliation associated to $\mathcal{F}$. When $\mathcal{F} = *_{DR}$, the Dolbeault foliation $\mathcal{F}_{\text{Dol}}$ is naturally equivalent to $*_{\text{Dol}}$ described in our previous section.
Note that $(*_{DR})_{Hod}$, which we will denote by $*_{Hod}$, provides an interpolation between $*_{DR}$ and $*_{Dol}$.

Finally, on the level of foliated cohomology, $\hat{C}_{DR}(\mathcal{F}_{Hod}, \mathcal{O})$ now has an induced filtration, called the *longitudinal Hodge filtration*, whose associated graded in weight $i$ is $C^*(X, \wedge^i \mathbb{L}_{\mathcal{F}})[-i]$.
We gather the previous constructions in the following proposition.

**PROPOSITON 3.3.0.4.** For any derived foliation $\mathcal{F}$ on a derived stack $X$, we can associated functorially $\mathcal{F}_{Hod} \in \mathcal{F}_{ol}(X \times \mathcal{A})$. The pull-back of $\mathcal{F}_{Hod}$ on $X = [X \times \mathbb{G}_m/\mathbb{G}_m] \hookrightarrow X \times \mathcal{A}$ canonically identifies with $\mathcal{F}$, whereas the pull-back of $\mathcal{F}_{Hod}$ at the origin $X \to X \times \mathcal{A}$ is $\mathcal{F}_{Dol}$.

The foliated de Rham cohomology of $F_{Hod}$ relative to $\mathcal{A}$ defines a (descending) filtration on $\hat{C}_{DR}(\mathcal{F}, \mathcal{O})$ whose associated graded in weight $i \le 0$ is $C^*(X, \wedge^{-i}\mathbb{L}_{\mathcal{F}})[i]$.

As a final comment, the two Hodge filtrations given by propositions 3.3.0.2 and 3.3.0.4 are different by are produced using similar constructions based on the exact triangle

$$ \mathcal{N}_{\mathcal{F}}^* \to \mathbb{L}_X \to \mathbb{L}_{\mathcal{F}} $$

relating the cotangent complexes and the conormal complex. The filtration of 3.3.0.2 is closely related by the natural $\mathbb{A}^1$-homotopy $\lambda.b$, where $b: \mathbb{L}_{\mathcal{F}} \to \mathcal{N}_{\mathcal{F}}^*[1]$ is the boundary map. Therefore, at the linear level, this filtration is simply induced by the one step filtration $\mathcal{N}_{\mathcal{F}}^* \to \mathbb{L}_X$ on $\mathbb{L}_X$. On the other side the filtration 3.3.0.4 is related to the $\mathbb{A}^1$-homotopy $\lambda.a$ where now $a: \mathbb{L}_X \to \mathbb{L}_{\mathcal{F}}$ is the anchor map. Therefore at the linear level it is given by the one step filtration $\mathbb{L}_{\mathcal{F}}[-1] \to \mathcal{N}_{\mathcal{F}}^*$ on the conormal complex. These filtrations are both on some independent interest, depending if one wants to study foliated de Rham cohomology, or rather absolute de Rham cohomology. The Hodge filtration will be used later in the study of characteristic classes associated to foliations (see §7.4).

## 3.4. Transversal geometry

In the previous section we have seen that conormal complex $\mathcal{N}_{\mathcal{F}}^*$ of a perfect derived foliation $\mathcal{F}$ on a derived Artin stack $X$ carries a canonical crystal structure along $\mathcal{F}$ and thus lifts to an object in $\mathbf{QCoh}(\mathcal{F})$. This object reflects the transversal geometry of $\mathcal{F}$, as when $\mathcal{F}$ is integrable by a morphism $f: X \to Y$ the conormal complex $\mathcal{N}_{\mathcal{F}}^*$ is given by $f^*(\mathbb{L}_Y)$, the pull-back of the cotangent complex of $Y$, equipped with its natural crystal structure relative to $f$ (see 3.1). In this section we will show moreover that the object $\mathcal{N}_{\mathcal{F}}^*(-1)$ comes equipped with a natural Lie bracket, encoded in a graded mixed structure on $\mathrm{Sym}_{\mathcal{O}_X}(\mathcal{N}_{\mathcal{F}}^*[2])$. More importantly this Lie bracket is compatible with the crystal structure and makes $\mathcal{N}_{\mathcal{F}}^*(-1)$ into a Lie algebra object in $\mathbf{QCoh}(\mathcal{F})$. The Koszul dual to this Lie object will be by definition the sheaf of transversal jets, which should be understood as the sheaf of formal functions in a transversal direction along $\mathcal{F}$.

To start with let us assume as usual that $X = \mathrm{Spec} \mathcal{A}$ is a derived affine scheme of finite presentation, and that $\mathcal{F}$ is a perfect derived foliation on $X$ (relative to $k$). We consider the morphism of graded mixed cdga's $\mathrm{DR}(X) \to \mathrm{DR}(\mathcal{F})$ and its internal de Rham algebra $\mathrm{DR}^{gr}(\mathrm{DR}(\mathcal{F})/\mathrm{DR}(X))$ as introduced in our Appendix A.8. This is a double graded mixed
cdga whose underlying bi-graded cdga is given by

$$ \mathbf{DR}^{gr}(\mathbf{DR}(\mathcal{F})/\mathbf{DR}(X)) \simeq \mathrm{Sym}_{\mathbf{DR}(\mathcal{F})}(\mathbf{DR}(\mathcal{F}) \otimes_A \mathcal{N}_{\mathcal{F}}^*[2]) \simeq \mathbf{DR}(\mathcal{F}) \otimes_A \mathrm{Sym}_A(\mathcal{N}_{\mathcal{F}}^*[2]). $$

This bi-graded cdga is endowed with two different graded mixed structure. This first one is induced by the mixed structure on $\mathbf{DR}(X) \to \mathbf{DR}(F)$, whereas the second one is the de Rham differential in $\mathbf{DR}^{gr}(\mathbf{DR}(\mathcal{F})/\mathbf{DR}(X))$. When considered as a graded mixed structure for the former, $\mathbf{DR}(\mathcal{F}) \otimes_A \mathrm{Sym}_A(\mathcal{N}_{\mathcal{F}}^*[2])$ defines a commutative algebra object inside $\mathbf{QCoh}(\mathcal{F})$, the symmetric monoidal $\infty$-category of quasi-coherent crystals along $\mathcal{F}$. The underlying quasi-coherent complex is here $\mathrm{Sym}_A(\mathcal{N}_{\mathcal{F}}^*[2])$. The second mixed structure tells us that moreover this commutative algebra object in $\mathbf{QCoh}(\mathcal{F})$ is endowed with an internal graded mixed structure. We can therefore consider its de Rham cohomology of the second kind as in Definition A.8.0.2, and define

$$ \hat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}} := \hat{C}_{\mathbf{DR}}^{II}(\mathbf{DR}(\mathcal{F})/\mathbf{DR}(X)) = |\mathbf{DR}^{gr}(\mathbf{DR}(\mathcal{F})/\mathbf{DR}(X))|_2 $$

which is a complete filtered commutative algebra object in $\mathbf{QCoh}(\mathcal{F})$.

When $X$ is a more general derived Artin stack locally of finite presentation, and $\mathcal{F}$ is a perfect derived foliation on $X$, we obtain (by gluing the above constructions over all affines over $X$) a globally defined object $\hat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}$ which is a complete filtered commutative algebra object in $\mathbf{QCoh}(\mathcal{F})$.

**DEFINITION 3.4.0.1.** The sheaf of transversal formal functions along $\mathcal{F}$ is $\hat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}$ defined above. It is a complete filtered commutative algebra object in $\mathbf{QCoh}(\mathcal{F})$.

By construction, the underlying object of $\hat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}$ is a sheaf of complete filtered $\mathcal{O}_X$-linear cdga's on $X$ whose associated graded is equivalent to $\mathrm{Sym}_{\mathcal{O}_X}(\mathcal{N}_{\mathcal{F}}^*)$ where $\mathcal{N}_{\mathcal{F}}^*$ is of weight $-1$ (remind from (4) before proposition 1.3.2.2 that the associated graded of the realization $|E|$ of a graded mixed complexes is of the form $\oplus_p E^{-p}[-2p]$).

To conclude this part, we mention here that the object $\hat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}$ can be used in order to define a very general notion of algebraic holonomy of $\mathcal{F}$, by using the Tannakian formalism. This algebraic holonomy will be compared to the more classical notion of holonomy in the analytic setting later in §6.4. Note however that the definition below makes sense over any general bases (including non-zero characteristics bases, see §7). It provides a notion of algebraic holonomy defined over smaller fields, for instance number fields, if $\mathcal{F}$ is itself defined over smaller fields.

Let $\mathcal{F}$ be a perfect derived foliation over a derived Artin stack $X$ locally of finite presentation over $k$. Let $K$ be a $k$-field and $x \in X(K)$ be a global $K$-point. We can consider the fiber functor

$$ \omega_x : \mathrm{Perf}(\mathcal{F}) \to \mathrm{Perf}(K), $$

sending a perfect crystal along $\mathcal{F}$ to its fiber at $x$. We let $\langle \mathcal{N}_{\mathcal{F}}^* \rangle^{\otimes} \subset \mathrm{Perf}(\mathcal{F})$ be the smallest thick rigid tensor triangulated $\infty$-category containing the object $\mathcal{N}_{\mathcal{F}}^*$, or in other words the thick triangulated sub-$\infty$-category generated by all tensor powers of $\mathcal{N}_{\mathcal{F}}^*$ and its duals. We restrict the fiber functor to a symmetric monoidal $\infty$-functor

$$ \omega_x : \langle \mathcal{N}_{\mathcal{F}}^* \rangle^{\otimes} \to \mathrm{Perf}(K). $$
As explained in [Toe], we can form the stack of k-linear tensor automorphisms $aut^\otimes(\omega_x)$ of $\omega_x$. This is the (underived) group stack

$$ aut^\otimes(\omega_x) : \mathbf{Aff}_k^{op} \to Gp(\mathbb{T}), $$

sending a (genuine) commutative k-algebra A to the group-like simplicial monoid of self-equivalences of $A \otimes_k \omega_x(-)$ considered as an object in the $\infty$-category

$$ \mathrm{Fun}_k^{\infty, \otimes}(\langle \mathcal{N}_F^* \rangle^{\otimes}, \mathrm{Perf}(A)), $$

of k-linear symmetric monoidal $\infty$-functors from $\langle \mathcal{N}_F^* \rangle^{\otimes}$ to $\mathrm{Perf}(A)$. Its classifying stack $\mathrm{Baut}^\otimes(\omega_x)$ can be shown to be a schematic homotopy type over K in the sense of [Toe06]. For the sake of completeness we include a proof.

**LEMMA 3.4.0.2.** *The (underived) stack $\mathrm{Baut}^\otimes(\omega_x)$ above is a schematic homotopy type over the field K.*

*Proof of the lemma.* By Definition [Toe06, 3.1.2] we have to prove the following two assertions.

(1) The stack $auu^\otimes(\omega_x)$ is affine.

(2) The stack $\mathrm{Baut}^\otimes(\omega_x)$ is P-local.

As $aut^\otimes(\omega_x)$ coincide with $end^\otimes(\omega_x)$ (by the natural inclusion), because $\omega_x$ is a symmetric monoidal $\infty$-functor between rigid symmetric monoidal $\infty$-categories. But $end^\otimes(\omega_x)$ is always equivalent to a certain limit of stacks of the form $\mathrm{End}_K(E)$ for E a perfect complex over K, where $\mathrm{End}_K(E)$ is the stack of endomorphisms of E. Clearly this last stack is linear and thus affine, and explicitly given by the spectrum of $Sym_K(E \otimes_K E^\vee)$. As affine stacks are stable by limits the first assertion follows. Moreover, each stack $\mathrm{End}_K(E)$ is P-local in the sense of [Toe06, Def. 3.1.1], as it classifies endomorphisms of E. In other words, for any other K-stack F, we have

$$ \mathrm{Map}(F, \mathrm{End}_K(E)) \simeq \mathrm{Map}_{\mathrm{Perf}(F)}(p^*(E), p^*(E)) $$

for $p : F \to \mathrm{Spec} K$ the canonical projection. As P-local stacks are also stable by limits this finishes the proof of the lemma. $\square$

By the lemma 3.4.0.2, we can apply [Toe06, Thm. 3.2.9], and show that the sheaves $\pi_i(\mathrm{Baut}^\otimes(\omega_x)) \simeq \pi_{i-1}(aut^\otimes(\omega_x))$ are all representable by affine group schemes over K, which are furthermore products of additive groups when $i > 1$.

**DEFINITION 3.4.0.3.** *The algebraic holonomy type of $\mathcal{F}$ at $x$ is the schematic homotopy type $\mathrm{Baut}^\otimes(\omega_x)$ defined above, and is denoted by $\widehat{\mathrm{Hol}}_{\mathcal{F}}^{\mathrm{alg}}(x)$. Its homotopy groups are called the abstract algebraic holonomy groups of $\mathcal{F}$ at $x$.*

**REMARK 3.4.0.4.** (1) As a first comment, we have used the expression *abstract algebraic holonomy groups* in order to emphasize the existence of a more classical/topological version of algebraic holonomy in the analytic setting, which will be discussed in §6.2.
For $\pi_1$ a more concrete definition of algebraic holonomy will be also discussed below in Definition 3.4.0.5.

(2) The above definition depends on the choice of a global base point $x$ in $X(K)$. There exists a base-point free version by considering instead the whole stack of $k$-linear symmetric monoidal $\infty$-functors $\langle \mathcal{N}_{\mathcal{F}}^* \rangle \to \mathbf{Perf}$, to the stack of perfect complexes. This leads to a stack that deserves the name of foliated algebraic holonomy type of $\mathcal{F}$. The sheaf of connected components of this algebraic homotopy type is an algebraic version of an hypothetical *space of leaves* and can be very degenerate in general. A better behaved leaf space will be introduced in the analytic setting in §6.2

(3) Definition 3.4.0.3 can also be extended by considering instead the tensor automorphisms of the fiber functor defined now on the whole $\infty$-category $\mathbf{Perf}(\mathcal{F})$ rather than just on $\langle \mathcal{N}_{\mathcal{F}}^* \rangle$. This leads to a similar schematic homotopy type which is called the *algebraic homotopy (or monodromy) type of $\mathcal{F}$ at $x$, denoted by $\widehat{\mathbf{Mon}}_{\mathcal{F}}^{\text{alg}}(x)$. By restriction, there is a canonical projection morphism from the algebraic monodromy to the algebraic holonomy, which corresponds to some form of a maximal effective quotient, in the same spirit as the maximal effective quotient of a DM stack discussed in 6.2. When $X$ is a smooth variety over a field $k$, and $\mathcal{F} = *_X$ is the totological foliation, this algebraic holonomy is the differential-Galois homotopy types, and its fundamental group is the usual (affine part) of the differential Galois group of $X$. In this case the algebraic holonomy is easily seen to be trivial as $\mathcal{N}_{\mathcal{F}}^* = 0$. Finally, in the other extreme case where $\mathcal{F} = 0_X$, the algebraic holonomy is trivial because of the Tannakian reconstruction theorem (see for instance [BHL17, Iwa18]).

We want to discuss a more concrete realization of the abstract holonomy groups defined above, in particular for $\pi_1(\widehat{\mathrm{Hol}}_{\mathcal{F}}^{\mathrm{alg}}(x))$, in order to relate it to the more usual notion of holonomy as the action of fundamental groups of leaves on transversals. For this, we first notice that $\widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}$ can be considered as a complete filtered algebra objects in the $\infty$-category $\mathrm{Pro}(\langle \mathcal{N}_{\mathcal{F}}^* \rangle)$ of pro-objects. Indeed, as the filtration on $\widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}$ is concentrated in non-negative degrees, we can extra from it a pro-object

$$ \text{``} \lim_{i \le 0} \text{``} \widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}/F^i.$$

Each individual piece $\widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}/F^i$ is here the realization of the truncated internal de Rham algebra $\mathbf{DR}^{gr, \le i}(\mathbf{DR}(\mathcal{F})/\mathbf{DR}(X))$, which is itself considered as a graded mixed commutative algebra object in $\mathbf{QCoh}(\mathcal{F})$ as explained earlier in this section. Each $\widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}/F^i$ is endowed with its natural filtration (as being a realization of a graded mixed object) whose associated graded is $\mathrm{Sym}_{\widehat{\mathcal{O}}_X}^{\le i}(\mathcal{N}_{\mathcal{F}}^*)$. This implies that indeed each piece $\widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}/F^i$ is a commutative algebra object in $\langle \mathcal{N}_{\mathcal{F}}^* \rangle$, and thus defines $\widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}} = \text{``} \lim_{i \le 0} \text{``} \widehat{\mathcal{O}}_{\mathcal{N}_{\mathcal{F}}}/F^i$ as a pro-object in commutative algebras in $\langle \mathcal{N}_{\mathcal{F}}^* \rangle$.

The symmetric monoidal $\infty$-functor $\omega_x$ induces an $\infty$-functor on the corresponding $\infty$-categories of commutative algebra objects, and we can thus consider the action of the $\mathrm{aut}^\otimes(\omega_x)$
on $\omega_x(\hat{\mathcal{O}}_{\mathcal{N}_F})$ by automorphisms of complete cdga's. We get this way a natural morphism of group stacks

$$ aut^{\otimes}(\omega_x) \to aut_{\mathcal{cdga}^{fil}}(\hat{\mathcal{O}}_{\mathcal{N}_F, x}) $$

where $\hat{\mathcal{O}}_{\mathcal{N}_F, x}$ is the fiber of $\hat{\mathcal{O}}_{\mathcal{N}_F}$ at $x$, and $aut_{\mathcal{cdga}^{fil}}$ denotes the automorphism group stack of self-equivalences of filtered cdga's. Note that $\hat{\mathcal{O}}_{\mathcal{N}_F, x}$ can be morally identified here with the cdga of formal functions at $x$ along in the transversal direction. This defines an action of $aut^{\otimes}(\omega_x)$ on the formal transversal at $x$, which is closer to the usual notion of a holonomy action.

To go a tiny bit further, we assume now $\mathcal{F}$ is transversally smooth, or in other words that $\mathcal{N}_F^*$ is a vector bundle on $X$. the filtered cdga $\hat{\mathcal{O}}_{\mathcal{N}_F, x}$ is thus a discrete filtered cdga, as its associated graded of is $Sym_K(\mathcal{N}_F^*)$, the symmetric algebra over the fiber $\mathcal{N}_F^*$ of $\mathcal{N}_F$ at $x$. In this case, the group stack $aut_{\mathcal{cdga}^{fil}}(\hat{\mathcal{O}}_{\mathcal{N}_F, x})$ is a genuine sheaf of groups, which is moreover representable by an affine group scheme over $K$ which is abstractly isomorphic to the group scheme $\hat{G}^d$ of (origin preserving) automorphisms of a formal affine space $\hat{\mathbb{A}}^d$ where $d$ is the rank of $\mathcal{N}_F^*$. The morphism $aut^{\otimes}(\omega_x) \to aut_{\mathcal{cdga}^{fil}}(\hat{\mathcal{O}}_{\mathcal{N}_F, x})$ thus factors as

$$ \pi_1(\widehat{Hol}_{\mathcal{F}}^{alg}(x)) = \pi_0(aut^{\otimes}(\omega_x)) \to aut_{\mathcal{cdga}^{fil}}(\hat{\mathcal{O}}_{\mathcal{N}_F, x}) \simeq \hat{G}^d. $$

**DEFINITION 3.4.0.5.** With the notations and convention above, if $\mathcal{F}$ is a transversally smooth derived foliation on $X$, and $x \in X(K)$, the algebraic holonomy group at $x$ is the image of the morphism of affine group schemes over $K$

$$ \pi_1(\widehat{Hol}_{\mathcal{F}}^{alg}(x)) \to aut_{\mathcal{cdga}^{fil}}(\hat{\mathcal{O}}_{\mathcal{N}_F, x}). $$

It is denoted by $Hol_{\mathcal{F}}^{alg}(x)$.

Later in this book, the topological version of holonomy will be introduced in the complex analytic setting. We will show that $Hol_{\mathcal{F}}^{alg}(x)$ always contains the topological holonomy as a sub-group. When $X$ and $\mathcal{F}$ are defined over smaller fields, for instance number fields, this can provide interesting informations about the topological holonomy.
<figure><img src="image_24.png" /></figure>
# CHAPTER 4

## Operations on crystals

In this chapter we study furthermore crystals and their functorial properties. We present in particular a new description of crystals, as dg-modules over a certain sheaf of differential operators which is closer to the usual notion of $\mathcal{D}$-modules. We study the standard functorialities of crystals, pull-backs and push-forwards. We use the $\mathcal{D}$-module approach in order to introduce the notion of good filtration on crystals and use it to define characteristic cycles. The Grothendieck-Riemann-Roch theorem then states that characteristic cycles are compatible with push-forward along proper and quasi-smooth morphisms.

All along this section $k$ is a fixed $\mathbb{Q}$-algebra.

### 4.1. Sheaves of differential operators

In this Section we associate to a derived foliation $\mathcal{F}$ on a derived DM-stack $X$, a sheaf of filtered dg-algebras $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$ on $X$, called the sheaf of differential operators along $\mathcal{F}$, and prove that dg-modules over the underlying dg-algebra $\mathcal{D}_{\mathcal{F}} := (\mathcal{D}_{\mathcal{F}}^{\text{fil}})^u$ (i.e. $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$ with filtration forgotten) corresponds to quasi-coherent crystals along $\mathcal{F}$.

#### 4.1.1. Sheaf of differential operators.
Let $X = \text{Spec } A$ be a derived affine $k$-scheme, and $\mathcal{F}$ be a derived foliation on $X$. We have a corresponding graded mixed cdga $\text{DR}(\mathcal{F})$, and a canonical augmentation of graded mixed cdga's $\text{DR}(\mathcal{F}) \to A$. This makes $A$ into a graded mixed $\text{DR}(\mathcal{F})$-dg-module.

**REMARK 4.1.1.1.** The graded mixed $\text{DR}(\mathcal{F})$-dg-module $A$ is not a crystal along $\mathcal{F}$, as it is obviously not free as a graded $\text{DR}(\mathcal{F})$-module (unless $X$ is étale over $\text{Spec } k$). It should not be confused with the canonical crystal structure on the structure sheaf $\mathcal{O}_X$ in the sense of Definition 3.1.0.3, which corresponds to $\text{DR}(\mathcal{F})$ as a graded mixed dg-module over itself. In this section we will use at several places non-free graded mixed modules, which thus do not correspond to crystals along $\mathcal{F}$.

We consider $\text{RHom}_{\text{DR}(\mathcal{F})}(A, A)$, the internal Hom object of endomorphisms of $A \in \text{DR}(\mathcal{F})$ – $\epsilon - \text{dg}_X^{gr}$. This is a graded mixed associative algebra in the $\infty$-category $\text{DR}(\mathcal{F}) - \epsilon - \text{dg}_k^{gr}$ of graded mixed $\text{DR}(\mathcal{F})$-dg-modules

$$ (12) \qquad \text{RHom}_{\text{DR}(\mathcal{F})}(A, A) \in \text{Alg}(\epsilon - \text{dg}_k^{gr}). $$
Its underlying graded dg-algebra, obtained by forgetting the mixed structure, is explicitly given by

$$ \mathbb{R}\mathcal{H}om_{\mathbf{DR}(\mathcal{F})}(A, A)^{\ell} \simeq \mathbb{R}\mathcal{H}om_{Sym_A(\mathbb{L}_{\mathcal{F}}[1])}(A, A) \simeq \bigoplus_{i \ge 0} (Sym_A^i(\mathbb{L}_{\mathcal{F}}))^\vee [-2i], $$

where $(-)^\vee$ denotes the $\mathcal{O}_X$-linear dual. When $\mathcal{F}$ is a perfect derived foliation, $\mathbb{L}_{\mathcal{F}}$ is perfect and thus duality commutes with symmetric powers, and in this case we can write furthermore

$$ \mathbb{R}\mathcal{H}om_{\mathbf{DR}(\mathcal{F})}(A, A)^{\ell} \simeq Sym_A(\mathbb{T}_{\mathcal{F}}[-2]) $$

where the tangent complex $\mathbb{T}_{\mathcal{F}} = \mathbb{L}_{\mathcal{F}}^{\vee}$ is here of weight $-1$. Note that, in general, the mixed structure induced on the right hand side is non-trivial. For instance, it induces a morphism in $\mathbf{dg}_k$ from the piece of weight $-2$ to the (shift by $-1$ of the) piece of weight $-1$ $Sym_A^2(\mathbb{T}_{\mathcal{F}})[-4] \to \mathbb{T}_{\mathcal{F}}[-3]$, which defines an extension class

$$ Sym_A^2(\mathbb{T}_{\mathcal{F}}) \to \mathbb{T}_{\mathcal{F}}[1]. $$

The corresponding triangle

$$ \mathbb{T}_{\mathcal{F}} \longrightarrow \mathcal{D}_{\mathcal{F}}^{[1,2]} \longrightarrow Sym_A^2(\mathbb{T}_{\mathcal{F}}) $$

which defines the complex of $\mathcal{D}_{\mathcal{F}}^{[1,2]}$ differential operators of degree bigger than 1 less than 2 along $\mathcal{F}$.

**DEFINITION 4.1.1.2.** The filtered ring of differential operators of $\mathcal{F}$ is defined to be the realization

$$ \mathcal{D}_{\mathcal{F}}^{\text{fil}} := |\mathbb{R}\mathcal{H}om_{\mathbf{DR}(\mathcal{F})}(A, A)|^t \in \mathbf{Alg}(\mathbf{dg}_k^{\text{fil}}). $$

The ring of differential operators along $\mathcal{F}$ is the underlying $E_1$-algebra over $X$ obtained by forgetting the filtration and is denoted by

$$ \mathcal{D}_{\mathcal{F}} := (\mathcal{D}_{\mathcal{F}}^{\text{fil}})^u = \text{colim}_{i \ge 0} F^i \mathcal{D}_{\mathcal{F}}^{\text{fil}} \in \mathbf{Alg}(\mathbf{dg}_k). $$

**REMARK 4.1.1.3.** The filtered dg-algebra $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$ depends on the base ring $k$, and should rather be denoted by $\mathcal{D}_{\mathcal{F}/k}^{\text{fil}}$. We will keep the base ring implicit and continue to use the notation $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$ unless necessary.

By construction, $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$ is $k$-linear associative dg-algebras. As usual, we set

$$ \mathcal{D}_{\mathcal{F}}^{\le i} := F^i \mathcal{D}_{\mathcal{F}}, $$

and call $\mathcal{D}_{\mathcal{F}}^{\le i}$ the ring of differential operators along $\mathcal{F}$ of order $\le i$. Using the explicit description of the $\infty$-functor $|-|^t$ (see §1.3.2 (4)) it is straightforward to verify that, when $\mathcal{F}$ is perfect the associated graded $\text{Gr}(\mathcal{D}_{\mathcal{F}}^{\text{fil}})$ is naturally equivalent to $\text{Sym}_{\mathcal{O}_X}(\mathbb{T}_{\mathcal{F}})$, where $\mathbb{T}_{\mathcal{F}}$ has weight 1. When $\mathcal{F}$ is not perfect a more complicated formula holds

$$ \text{Gr}(\mathcal{D}_{\mathcal{F}}^{\text{fil}}) \simeq \bigoplus_{i \ge 0} (Sym_A^i(\mathbb{L}_{\mathcal{F}}))^\vee [-2i]. $$
Suppose that $\mathcal{F}$ is a smooth derived foliation, and $X$ is a non-derived scheme (e.g. a smooth variety), $\mathbb{T}_{\mathcal{F}}$ is a vector bundle on $X$, and thus $Gr^*(\mathcal{D}_{\mathcal{F}}^{\text{fil}})$ is automatically cohomologically concentrated degree 0. As it is a complete filtered object, this implies that $\mathcal{D}_{\mathcal{F}}^{\leq i}$ is also in degree 0 for all $i$. Therefore, in this case, $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$ is a genuine filtered $k$-algebra. For more general $X$, $\mathcal{D}_{\mathcal{F}}$ is a dg-algebra that might have non-trivial cohomologies in an infinite number of degrees. As $\mathcal{F}$ is assumed to be perfect, $\mathbb{T}_{\mathcal{F}}$ is a perfect $A$-module, as well as its symmetric powers $Sym_A^i(\mathbb{T}_{\mathcal{F}})$. As a result each $\mathcal{D}_{\mathcal{F}}^{\leq i}$ is a perfect $A$-module. It is moreover concentrated in non-negative degrees when $X$ is a non-derived scheme. When $\mathcal{F}$ and $X$ are both quasi-smooth (i.e. $\mathbb{L}_{\mathcal{F}}$ and $\mathbb{L}_X$ are perfect of tor-amplitude $[-1, 0]$), the dg-algebra $\mathcal{D}_{\mathcal{F}}$ is moreover cohomologically bounded. This can be checked, for instance, using the exact triangles

$$ \mathcal{D}_{\mathcal{F}}^{\leq i} \longrightarrow \mathcal{D}_{\mathcal{F}}^{\leq i+1} \longrightarrow Sym_{\mathcal{O}_X}^{i+1}(\mathbb{T}_{\mathcal{F}}) $$

and induction on $i$. More is true: if $X$ is a non-derived scheme and $\mathbb{T}_{\mathcal{F}}$ can be represented by a two term complex of vector bundles $V \to W$, then $\mathcal{D}_{\mathcal{F}}$ is cohomologically concentrated in degrees $[0, rk(W)]$.

Definition 4.1.1.2 can be generalized to non-affine objects by sheafification. However, we have to restrict to Deligne-Mumford stacks, as the construction of $\mathcal{D}_{\mathcal{F}}$ is not stable by pull-backs unless we restrict to étale morphisms. It is possible to define a non-quasi-coherent sheaf of dg-algebras over general derived Artin stacks, but we do not find it particularly helpful so will not discuss this here.

Let $f: X = \text{Spec } A \to Y = \text{Spec } B$ be a formally étale morphism between derived affine schemes over $k$, and $\mathcal{F} \in \text{Fol}^p(Y/k)$ be a perfect derived foliation on $Y$. We contemplate the commutative diagram of graded mixed cdga's

<figure><img src="image_25.png" /></figure>

and realize that the top square is cartesian by Definition of $f^*(\mathcal{F})$, and the bottom square is cartesian to because $f$ is formally étale. As a result, the exterior square is cartesian. Therefore, the base change $\infty$-functor on graded mixed dg-modules

$$ \mathrm{DR}(f^*(\mathcal{F})) \otimes_{\mathrm{DR}(\mathcal{F})} : \mathrm{DR}(\mathcal{F}) - \epsilon - \mathrm{dg}_k^{gr} \to \mathrm{DR}(f^*(\mathcal{F})) - \epsilon - \mathrm{dg}_k^{gr} $$

sends $B$ to $A$ via the canonical morphism $\mathrm{DR}(f^*(\mathcal{F})) \otimes_{\mathrm{DR}(\mathcal{F})} B \to A$. This implies that the formation of $\mathcal{D}_{\mathcal{F}}$ is functorial in $f$ and in particular it can be sheafified over the small étale site of any derived Deligne-Mumford stack.
For any derived Deligne-Mumford stack $X$ over $k$ and $\mathcal{F} \in \mathcal{F}ol(X)$, we first have a sheaf of graded mixed associative algebras $\mathbb{R}\mathcal{Hom}_{\mathbb{DR}_{\mathcal{F},et}}(\mathcal{O}_X, \mathcal{O}_X)$, and its realization

$$ \mathcal{D}_{\mathcal{F}}^{\text{fil}} := |\mathbb{R}\mathcal{Hom}_{\mathbb{DR}_{\mathcal{F},et}}(\mathcal{O}_X, \mathcal{O}_X)| \in \mathbf{Alg}(\epsilon - \mathbf{dg}_k(X)), $$

where $\epsilon - \mathbf{dg}_k(X)$ is the $\infty$-category of sheaves of graded mixed $k$-modules over $X_{\text{ét}}$ the small étale site of $X$.

**DEFINITION 4.1.1.4.** For a derived DM-stack $X$ over $k$, the filtered sheaf of differential operators of a derived foliation $\mathcal{F} \in \mathcal{F}ol(X/k)$ is

$$ \mathcal{D}_{\mathcal{F}}^{\text{fil}} := |\mathbb{R}\mathcal{Hom}_{\mathbb{DR}_{\mathcal{F},et}}(\mathcal{O}_X, \mathcal{O}_X)|^t \in \mathbf{Alg}(\mathbf{dg}_k^{\text{fil}}(X)) $$

defined above. The sheaf of differential operators along $\mathcal{F}$ is the underlying $E_1$-algebra over $X$ obtained by forgetting the filtration and is denoted by

$$ \mathcal{D}_{\mathcal{F}} := (\mathcal{D}_{\mathcal{F}}^{\text{fil}})^u = \text{colim}_{i \ge 0} F^i \mathcal{D}_{\mathcal{F}}^{\text{fil}} \in \mathbf{Alg}(\mathbf{dg}_k(X)). $$

When $X$ is a smooth DM-stack over $k$ and $\mathcal{F} = *_X$ is the tautological foliation, $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$ is a sheaf of filtered associative $k$-algebras whose associated graded in $\text{Sym}_{\mathcal{O}_X}(\mathbb{T}_{X/k})$. The following result identifies it with the usual ring of differential operators on $X$ relative to $k$.

**PROPOSITON 4.1.1.5.** Let $X$ be a smooth Deligne-Mumford stack over $\text{Spec } k$. There is a natural isomorphism of sheaf of filtered $k$-algebras on $X_{\text{ét}}$

$$ \text{Diff}_X \simeq \mathcal{D}_{*X}^{\text{fil}} $$

where $\text{Diff}_X$ is the usual sheaf of rings of differential operators (relative to $k$) filtered by order of differential operators.

*Proof.* We use corollary 1.3.4.4 in order to compute $|\mathbb{R}\mathcal{Hom}_{\mathbb{DR}_{\mathcal{F},et}}(\mathcal{O}_X, \mathcal{O}_X)|^t$. It can be identified as the internal derived endomorphism objects inside $\hat{C}_{DR}(-) - \hat{\mathbf{dg}}_k^{\text{fil}}(X)$, the $\infty$-category of sheaves of complete filtered complexes of $\hat{C}_{DR}(-)$-modules on $X_{\text{ét}}$

$$ |\mathbb{R}\mathcal{Hom}_{\mathbb{DR}_{\mathcal{F},et}}(\mathcal{O}_X, \mathcal{O}_X)|^t \simeq \mathbb{R}\mathcal{Hom}_{\hat{C}_{DR}(-)-\hat{\mathbf{dg}}_k^{\text{fil}}(X)}(\mathcal{O}_X, \mathcal{O}_X). $$

Here $\hat{C}_{DR}(-)$ is the sheaf of filtered dg-algebras $(U \to X) \mapsto \hat{C}_{DR}(U, \mathcal{O}_U)$, defined on the small étale site of $X$ and given by the Hodge-completed derived de Rham cohomology of Definition 1.4.2.1. As $X$ is smooth this is nothing else than the genuine de Rham complex $(\Omega_{X/k}^*, dR)$, with $\Omega_{X/k}^i$ in cohomological degrees $i$, and endowed with its Hodge filtration $\Omega_{X/k}^{\ge p} \subset \Omega_{X/k}^*$. The statement of the proposition is then that the usual ring of differential operators $\text{Diff}_X$ appears as

$$ \text{Diff}_X \simeq \mathbb{R}\mathcal{Hom}_{\hat{C}_{DR}(-)-\hat{\mathbf{dg}}_k^{\text{fil}}(X)}(\mathcal{O}_X, \mathcal{O}_X), $$

which is a consequence of the classical Koszul duality between the ring of differential operators and the de Rham complex. One manner to see this is to use the standard resolution of $\mathcal{O}_X$ by
locally free module over $Diff_X$ given by the Spencer complex $Diff_X \otimes_{\mathcal{O}_X} \wedge^* T_{X/k}$

$$0 \longrightarrow Diff_X \otimes_{\mathcal{O}_X} \omega_{X/k} \longrightarrow Diff_X \otimes_{\mathcal{O}_X} \wedge^{d-1} T_{X/k} \dots \longrightarrow Diff_X \otimes_{\mathcal{O}_X} T_{X/k} \longrightarrow Diff_X \longrightarrow \mathcal{O}_X$$

where $d$ is the rank of $T_{X/k}$ (we can assume $X$ connected). This resolution implies the existence of a natural quasi-isomorphism of sheaves of filtered dg-algebras

$$\mathbb{R}\mathcal{H}\mathit{om}_{Diff_X - \widehat{\mathbf{d}}_{\mathbf{g}_k}^{\mathit{fil}}(X)}(\mathcal{O}_X, \mathcal{O}_X) \simeq \hat{C}_{DR}(-).$$

By adjunction, this produces a canonical morphism of sheaves of complete filtered dg-algebras on $X$

$$\phi : Diff_X \longrightarrow \mathbb{R}\mathcal{H}\mathit{om}_{\widehat{C}_{DR}(-) - \widehat{\mathbf{d}}_{\mathbf{g}_k}^{\mathit{fil}}(X)}(\mathcal{O}_X, \mathcal{O}_X).$$

This last morphism can then be checked to be an equivalence by looking at the associated graded, which reduces to the standard equivalence of graded rings

$$Gr^*(\phi) : \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{T}_{X/k}) \simeq \mathbb{R}\mathcal{H}\mathit{om}_{\Omega_{X/k}^* - \mathbf{d}_{\mathbf{g}_k}^{gr}(X)}(\mathcal{O}_X, \mathcal{O}_X).$$

□

Here are some basic examples of Rings of differential operators.

EXAMPLE 4.1.1.6.

(1) When $\mathcal{F}$ is the zero foliation (i.e. $\mathbb{L}_{\mathcal{F}} = 0$) then $\mathcal{D}_{\mathcal{F}} = \mathcal{O}_X$ endowed with the trivial filtration.

(2) When $\mathcal{F}$ is the tautological foliation on a smooth variety $X$ (i.e. $\mathbf{DR}(\mathcal{F}) = \mathbf{DR}(X)$), then $\mathcal{D}_{\mathcal{F}} = \mathcal{D}_X$ is the usual ring of differential operators endowed with its usual filtration by the order of operators. This is the proposition 6.1.0.2 above.

(3) When $\mathcal{F}$ is the Dolbeault foliation on a smooth variety, that is $\mathbf{DR}(\mathcal{F}) = \mathrm{Sym}_{\mathcal{O}_X}(\Omega_X^1[1])$ with trivial mixed structure, then $\mathcal{D}_{\mathcal{F}} = \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{T}_X)$ with the split filtration. More generally, when $\mathbf{DR}(\mathcal{F}) = \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\mathcal{F}}[1])$ with trivial mixed structure (abelian derived foliation), then $\mathcal{D}_{\mathcal{F}} \simeq \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{T}_{\mathcal{F}})$.

(4) If the foliation $\mathcal{F}$ is smooth, i.e it is a Lie algebroid (see Theorem 2.2.0.7 or [TV23a, Section 1.3.1.])), induced by a smooth groupoid $G$ acting on $X$, then $\mathcal{D}_{\mathcal{F}}$ is the ring of distributions on $G$, i.e. the $\mathcal{O}_X$-linear dual of formal functions of $G$, endowed with the convolution product. This coincides with the universal enveloping algebra of the Lie algebroid.

(5) When $\mathcal{F}$ is globally integrable by a flat and generically smooth morphism of smooth varieties $f : X \longrightarrow Y$, $\mathcal{D}_{\mathcal{F}}$ is called the dg-algebra of relative differential operators. The reason for this name comes from the fact that $\mathbf{H}^0(\mathcal{D}_{\mathcal{F}})$ is indeed a subring of $\mathcal{D}_X$ consisting of differential operators stabilizing the fibers of $f$.

REMARK 4.1.1.7. (1) It is interesting to note the following basic example. Let $X = \mathrm{Spec}(k[u])$ with $u$ in degree $-2$, and $\mathcal{F} = *_X$ be the tautological (i.e. final) derived foliation on $X$. Then $\mathcal{D}_{\mathcal{F}}$ is here the Weyl dg-algebra over one generator $u$ in degree
-2. In other words, it is the dg-algebra freely generated by two cocycle $u$ and $\frac{\partial}{\partial u}$, respectively in degrees -2 and 2, with the usual commutation relation $[u, \frac{\partial}{\partial u}] = -1$. Note that $\mathcal{D}_F$ is not Morita equivalent to $k$, as opposed to the case when $\deg u$ is odd (see [Ber21, Proof of Cor. 4.3.13]). The reader will find more details about $\mathcal{D}_{*X}$ in [Ber21], even for more general $X$ than the ones considered here.

(2) The previous example shows that our notion of differential operators, and thus of crystals (see below) differs from the notion of $\mathcal{D}$-modules use in [GR14]. For more precise comparison we refer to [Buc].

For $X$ a fixed derived DM-stack, the construction $\mathcal{F} \mapsto \mathcal{D}_\mathcal{F}$ defines an $\infty$-functor $\mathcal{Fol}(X) \to \mathbf{Alg}(\mathrm{dg}^{\mathrm{fil}}(X))$: if $\mathcal{F} \to \mathcal{F}'$ is a morphism in $\mathcal{Fol}(X)$, given by a morphism $\varphi : \mathrm{DR}'_\mathcal{F} \to \mathrm{DR}_\mathcal{F}$ of sheaves of graded mixed cdga's over $X$, $\varphi$ induces a morphism on internal endomorphism objects $\mathbb{R}\mathrm{Hom}_{\mathrm{DR}_\mathcal{F}}(\mathcal{O}_X, \mathcal{O}_X) \to \mathbb{R}\mathrm{Hom}_{\mathrm{DR}'_\mathcal{F}}(\mathcal{O}_X, \mathcal{O}_X)$, and thus a morphism $\mathcal{D}^{\mathrm{fil}}_\mathcal{F} \to \mathcal{D}^{\mathrm{fil}}_{\mathcal{F}'}$ in $\mathbf{Alg}(\mathrm{dg}^{\mathrm{fil}}(X))$. In particular, the maps $0_X \to \mathcal{F} \to *_X$, from the initial and to the final foliations, provide maps of filtered dg-algebras over $X$

$$ \mathcal{O}_X \longrightarrow \mathcal{D}_\mathcal{F}^{\mathrm{fil}} \longrightarrow \mathcal{D}_X^{\mathrm{fil}}, $$

where $\mathcal{O}_X$ has the trivial filtration, and $\mathcal{D}_X^{\mathrm{fil}}$ is by definition the ring of differential operators on $X$ with its natural filtration by order of differential operators. Moreover, the first morphism exhibits $\mathcal{O}_X$ as $F^0\mathcal{D}_\mathcal{F}^{\mathrm{fil}}$. By induction on the filtration it is easy to see, when $\mathcal{F}$ is assumed to be perfect, that $\mathcal{D}_\mathcal{F}$ is a quasi-coherent complex on $X$, for the left $\mathcal{O}_X$-module structure given by the morphism above $\mathcal{O}_X \to \mathcal{D}_\mathcal{F}$

**4.1.2. Quasi coherent $\mathcal{F}$-crystals and $\mathcal{D}_\mathcal{F}$-modules.** As recalled in §1.3.4.4, the realization functor on $X$ provides an equivalence of symmetric monoidal $\infty$-categories

$$ |-|^t : \epsilon - \mathrm{dg}_k^{gr} \simeq \widehat{\mathrm{dg}}_k^{\mathrm{fil}}, $$

where the right hand side is endowed with the complete tensor product of complete filtered complexes. By passing to sheaves on the small étale sites, we have a symmetric monoidal equivalence

$$ |-|^t : \epsilon - \mathrm{dg}_k^{gr}(X) \simeq \widehat{\mathrm{dg}}_k^{\mathrm{fil}}(X), $$

for any derived DM-stack $X$ over $k$.

Let us fix a derived DM-stack $X$ and $\mathcal{F} \in \mathcal{Fol}(X)$ a derived foliation on $X$. We will construct an $\infty$-functor

$$ \theta_X : \mathrm{QCoh}(\mathcal{F}) \to \mathcal{D}_\mathcal{F} - \mathrm{dg}, $$

from quasi-coherent crystals along $\mathcal{F}$ to sheaves of (left) $\mathcal{D}_\mathcal{F}$-dg-modules on $X_{et}$. We start, as in formula (12) above, by letting

$$ S_{\mathcal{F}} := \mathbb{R}\mathrm{Hom}_{\mathrm{DR}_{\mathcal{F},et}}(\mathcal{O}_X, \mathcal{O}_X) \in \mathbf{Alg}(\epsilon - \mathrm{dg}^{gr}(X)), $$

the internal Hom object of endomorphisms of $\mathcal{O}_X$ as a sheaf of graded mixed $\mathrm{DR}_{\mathcal{F},et}$-modules over $X_{et}$.
The object $\mathcal{O}_X$ can thus be considered as a graded mixed bi-module with its right action by $\mathbf{DR}_{\mathcal{F},et}$ and left action by $S_{\mathcal{F}}$. Therefore it can be used in order to produce the following $\infty$-functor (between categories of left modules)

$$ \mathcal{O}_X \otimes_{\mathbf{DR}_{\mathcal{F},et}} : \mathbf{DR}_{\mathcal{F},et} - \epsilon - \mathbf{dg}^{gr} \to S_{\mathcal{F}} - \epsilon - \mathbf{dg}^{gr}. $$

Note that this $\infty$-functor obviously sends $\mathbf{DR}_{\mathcal{F},et}$-dg-modules on $X$ which are quasi-coherent on $X$ and graded free on weight zero, i.e. quasi-coherent crystals along $\mathcal{F}$, to graded mixed $S_{\mathcal{F}}$-dg-modules whose underlying graded modules are pure of weight 0 and $\mathcal{O}_X$-quasi-coherent. We thus get an induced $\infty$-functor

$$ \theta'_X : \mathbf{QCoh}(\mathcal{F}) \to S_{\mathcal{F}} - \epsilon - \mathbf{dg}_{qcoh, w=0}^{gr}, $$

where $S_{\mathcal{F}} - \epsilon - \mathbf{dg}_{qcoh, w=0}^{gr} \subset S_{\mathcal{F}} - \epsilon - \mathbf{dg}^{gr}$ is the full sub-$\infty$-category of objects whose underlying graded $S_{\mathcal{F}}$-module are pure of weight 0 and quasi-coherent over $\mathcal{O}_X$.

We now compose the previous $\infty$-functor with the Tate realization $| - |^t$ in order to get an $\infty$-functor to dg-modules over $\mathcal{D}_F$

$$ \theta_X : \mathbf{QCoh}(\mathcal{F}) \xrightarrow{\theta'_X} S_{\mathcal{F}} - \epsilon - \mathbf{dg}_{qcoh, w=0}^{gr} \xrightarrow{|-|^t} \mathcal{D}_F - \mathbf{dg}. $$

**THEOREM 4.1.2.1.** Let $X$ be a derived DM-stack over $k$ and $\mathcal{F} \in \mathcal{F}ol^p(X)$ a perfect derived foliation. The $\infty$-functor $\theta_X : \mathbf{QCoh}(\mathcal{F}) \to \mathcal{D}_F - \mathbf{dg}$ defined above is fully faithful. Its essential image, $\mathcal{D}_F - \mathbf{dg}_X^{qcoh}$, consists of all $\mathcal{D}_F$-modules over $X$ which are quasi-coherent as $\mathcal{O}_X$-modules

$$ \theta_X : \mathbf{QCoh}(\mathcal{F}) \simeq \mathcal{D}_F - \mathbf{dg}_X^{qcoh}. $$

*Proof.* The key lemma to prove the theorem is the following, which is a well known version of Koszul duality in the graded context.

**LEMMA 4.1.2.2.** Let $A$ be a cdga and $M$ a perfect $A$-dg-module. We let $C := \mathrm{Sym}_A(E)$. Then, the following holds.

(1) There exists a natural identification of graded dg-algebras

$$ \mathbb{R}\mathrm{Hom}_C(A, A) \simeq \mathrm{Sym}_A(E^{\vee}[-1]) := D. $$

(2) Using the identification above makes $A$ into a bi-dg-module over $(C, D)$, and the induced $\infty$-functor on $\infty$-categories of graded modules

$$ A \otimes_C : C - \mathbf{dg}^{gr} \to D - \mathbf{dg}^{gr} $$

induces an equivalence from the full sub-$\infty$-category of graded $C$-modules which are graded free over their part of weight 0, and the graded $D$-modules which are pure of weight 0 as graded modules over $k$ (i.e. comes from graded $A$-dg-modules via the augmentation $D \to A$).

*Proof of the lemma.* This is pretty standard. The $\infty$-functor $A \otimes_C$-graded modules of the form $C \otimes_A M(0)$ to $M(0)$ with the trivial $D$-module structure induced by the augmentation
$D \to A$. Conversely, the right adjoint to $A \otimes_C -$ is $\mathbb{R}\mathcal{H}om_D(A, -)$. We first prove that for any $A$-dg-module $M(0)$, considered as graded pure of weight 0, the adjunction morphism

$$C \otimes_A M(0) \to \mathbb{R}\mathcal{H}om_D(A, M(0))$$

is an equivalence. For this, we use the explicit cofibrant replacement $QA \to A$ in $D - \mathbf{dg}^{gr}$ given explicitly by $QA := \mathrm{Sym}_A(E^\vee \oplus E^\vee[-1])$ where the differential on $E^\vee \oplus E^\vee[-1]$ is the usual differential making it contractible $\begin{pmatrix} d & 0 \\ id & -d \end{pmatrix}$. This implies that $\mathbb{R}\mathcal{H}om_D(A, M(0))$ is equivalent to $\mathbb{R}\mathcal{H}om_A(\mathrm{Sym}_A(E^\vee), M(0))$, the graded hom complexes over $A$ of morphisms from $\mathrm{Sym}_A(E^\vee)$ to $M(0)$. But, as $E$ is perfect, and as these are computed in the graded modules $\infty$-categories, we have

$$\mathbb{R}\mathcal{H}om_A(\mathrm{Sym}_A(E^\vee), M(0)) \simeq \mathrm{Sym}(E) \otimes_A M(0)$$

as required. This shows that the adjunction $(A \otimes -, \mathbb{R}\mathcal{H}om_D(A, -))$ restricts to an equivalence on the $\infty$-categories stated in the lemma. $\square$

We now prove the theorem, and we prove first that the $\infty$-functor

$$\theta_X' : \mathbb{Q}\mathrm{Coh}(\mathcal{F}) \to S(\mathcal{F}) - \epsilon - \mathbf{dg}_{q\mathrm{coh},w=0}^{gr}$$

is an equivalence of $\infty$-categories, and then identify the essential image of $\theta_X$.

For this, we consider the $\infty$-functor that forgets the mixed structures and only retain the graded structures. We have obtained this way a commutative diagram of $\infty$-categories

$$\begin{tikzcd}
\mathbf{DR}_{\mathcal{F},et} - \epsilon - \mathbf{dg}^{gr} \arrow[r, "\theta_X'"] & S_{\mathcal{F}} - \epsilon - \mathbf{dg}^{gr} \\
\arrow[d] & \arrow[d]
\end{tikzcd}
\qquad
\begin{tikzcd}
\mathbf{DR}_{\mathcal{F},et} - \mathbf{dg}^{gr} \arrow[r] & S_{\mathcal{F}} - \mathbf{dg}^{gr} \\
\arrow[d] & \arrow[d]
\end{tikzcd}$$

where the vertical $\infty$-functors consists of forgetting the mixed structures. The horizontal bottom $\infty$-functor is the one of the lemma $\mathcal{O}_X \otimes_{\mathbf{DR}_{\mathcal{F},et}} -$ . Moreover, the above square is right adjointable in the sense of see [Lur22a, Def. 4.7.4.13], and the right adjoints are given by $\mathbb{R}\mathcal{H}om_{S_{\mathcal{F}}}(\mathcal{O}_X, -)$. The lemma 4.1.2.2 implies that $\theta_X'$ is an equivalence of $\infty$-categories, as the unit of conunit of the involved adjunction can be checked on the level of the underlying graded objects without mixed structures.

To finish the proof of the theorem, we now consider the realization

$$| - |^t : S_{\mathcal{F}} - \epsilon - \mathbf{dg}_X^{gr} \to \mathcal{D}_{\mathcal{F}}^{\mathrm{fil}} - \mathbf{dg}^{fil},$$

and observe that, by definition, $\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}} := |S_{\mathcal{F}}|^t$. We know that this $\infty$-functor is fully faithful, and its image consists of complete filtered $\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}$-modules over $X$. We restrict this to $S_{\mathcal{F}} - \epsilon - \mathbf{dg}_{q\mathrm{coh},w=0}^{gr}$, the full sub-$\infty$-category of graded mixed module which are quasi-coherent and of weight 0. Its image by $| - |^t$ is then easily checked to consist of all filtered $\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}$-modules $E$ satisfying the following two conditions
(1) The filtration on $E$ is tautological: $F^i(E) = E$ if $i \ge 0$ and $0$ if $i < 0$.

(2) The underlying $\mathcal{O}_X$-module of $E$ is quasi-coherent.

Now, these two conditions define a full sub-∞-category of $\mathcal{D}_F^{\text{fil}} - \text{dg}^{\text{fil}}$ which is equivalent, via the underlying object $\infty$-functor $(-)^u : \text{dg}_k^{\text{fil}}(X) \to \text{dg}_k(X)$, to the $\infty$-category $\mathcal{D}_F - \text{dg}_{q\text{coh}}$, of unfiltered $\mathcal{D}_F$-modules that are $\mathcal{O}_X$-quasi-coherent. $\square$

The following definition/notation will be used throughout the rest of the Chapter.

**DEFINITION 4.1.2.3.** Let $\mathcal{F} \in \mathcal{Fol}(X)$ be a perfect derived foliation over a derived scheme $X$. We denote by $\text{QCoh}^{\text{fil}}(\mathcal{F})$ the full sub-∞-category of $\mathcal{D}_F^{\text{fil}} - \text{dg}^{\text{fil}}$ consisting of all filtered modules which are quasi-coherent as filtered $\mathcal{O}_X$-modules via restriction of scalars along the natural morphism of filtered dg-algebras $\mathcal{O}_X \to \mathcal{D}_F^{\text{fil}}$. We will call $\text{QCoh}^{\text{fil}}(\mathcal{F})$ the $\infty$-category of filtered quasi-coherent crystals along $\mathcal{F}$.

**REMARK 4.1.2.4.** Without the perfect condition the theorem 4.1.2.1 does not hold. In fact, in general the sheaf $\mathcal{D}_F$ has no reasons of being quasi-coherent over $\mathcal{O}_X$, even if $\mathcal{F}$ is almost perfect.

**4.1.3. Inverse image and induction $\infty$-functors.** Let $u : \mathcal{F} \to \mathcal{G}$ be a morphism of derived foliations on a derived DM-stack $X$, which we assume to be perfect in order to be able to use our theorem 4.1.2.1. We have seen that it induces a morphism of sheaves of filtered dg-algebras over $X$

$$ \mathcal{D}_F^{\text{fil}} \to \mathcal{D}_G^{\text{fil}}. $$

Associated to this, we have the usual forgetful and base-change adjunction on the corresponding $\infty$-categories of sheaves of filtered modules

$$ (13) \qquad u_! := \mathcal{D}_G^{\text{fil}} \otimes_{\mathcal{D}_F^{\text{fil}}} - : \mathcal{D}_F^{\text{fil}} - \text{dg}^{\text{fil}} \rightleftharpoons \mathcal{D}_G^{\text{fil}} - \text{dg}^{\text{fil}} : u_! $$

The right adjoint $u_!$ is called the *inverse image $\infty$-functor*. The left adjoint $u_!$ is called the *induction along $u$* or *direct image $\infty$-functor*. By forgetting the filtrations, we have a corresponding non-filtered adjunction

$$ (14) \qquad u_! := \mathcal{D}_G \otimes_{\mathcal{D}_F} - : \mathcal{D}_F - \text{dg} \rightleftharpoons \mathcal{D}_G - \text{dg} : u_! $$

Both $\infty$-functors $u_!$ and $u_!$ preserve quasi-coherence, and thus induce an adjunction on quasi-coherent modules. By Theorem 4.1.2.1, this can also be interpreted as an adjunction on the $\infty$-category of quasi-coherent crystals

$$ u_! : \text{QCoh}(\mathcal{F}) \rightleftharpoons \text{QCoh}(\mathcal{G}) : u_! $$

where again, $u_!$ is called the inverse image $\infty$-functor, and $u_!$ the induction or direct image $\infty$-functor. Through Definition 4.1.2.3, the filtered adjunction (13) can be regarded as an adjunction on filtered crystals, as well

$$ u_! : \text{QCoh}^{\text{fil}}(\mathcal{F}) \rightleftharpoons \text{QCoh}^{\text{fil}}(\mathcal{G}) : u_! $$
DEFINITION 4.1.3.2. Let $\mathcal{F} \in \mathcal{F}ol(X)$ be a perfect derived foliation and $u : 0_X \to \mathcal{F}$ the canonical morphism. The induction $\infty$-functor for $\mathcal{F}$ is the $\infty$-functor

$$ \mathrm{Ind}_{\mathcal{F}} := u_{!} : \mathrm{QCoh}(X) \to \mathrm{QCoh}(\mathcal{F}). $$

The filtered induction $\infty$-functor for $\mathcal{F}$ is the $\infty$-functor

$$ \mathrm{Ind}_{\mathcal{F}}^{\mathrm{fil}} := u_{!} : \mathrm{QCoh}^{\mathrm{fil}}(X) \to \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}). $$

A direct consequence of the existence of the induction $\infty$-functor is the following important fact.

COROLLARY 4.1.3.3. Let $X$ be a derived DM-stack over $\mathrm{lk}$. If the $\infty$-category $\mathrm{QCoh}(X)$ admits a compact generator, then so are the $\infty$-categories $\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$ and $\mathrm{QCoh}(\mathcal{F})$.

Proof. Pick a family of compact generators $\{E_i\}_{i \in I}$ in $\mathrm{QCoh}(X)$. It is formal to check that the family of induced objects $\mathrm{Ind}_{\mathcal{F}}(E_i) \in \mathrm{QCoh}(\mathcal{F})$ is a family of compact generators. In the filtered case, we first consider the family of objects $(i_n)_{!}(E_i) \in \mathrm{QCoh}^{\mathrm{fil}}(X)$, for $\in \mathbb{Z}$ and $i \in I$. Here $(i_n)_{!}(E_i)$ is the filtered object in $\mathrm{QCoh}(X)$ whose filtration layers are all zero below $n$ and are equal to $E_i$ in layers above $n$. In other words, we have

$$ \mathrm{Map}_{\mathrm{QCoh}^{\mathrm{fil}}(X)}((i_n)_{!}(E_i), E') \simeq \mathrm{Map}_{\mathrm{QCoh}(X)}(E_i, F^n E'). $$

The family of objects $\{(i_n)_{!}E_i\}_{n,i}$ is a family of compact generators of $\mathrm{QCoh}^{\mathrm{fil}}(X)$, and it is formal to check that the induced family $\mathrm{Ind}_{\mathcal{F}}^{\mathrm{fil}}(\{(i_n)_{!}E_i\})_{n,i}$ is again a family of compact generators of $\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$. $\square$

COROLLARY 4.1.3.4. Let $X$ be either a quasi-compact and quasi-separated derived $k$-scheme, or a separated and quasi-compact derived Deligne-Mumford $k$-stack, and let $\mathcal{F} \in \mathcal{F}ol^p(X/k)$ a perfect derived foliation on $X$. Then the $\infty$-categories $\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$ and $\mathrm{QCoh}(\mathcal{F})$ are both compactly generated.

Proof. If $X$ is a quasi-compact and quasi-separated derived scheme then $\mathrm{QCoh}(X)$ is compactly generated by [Toë12, Thm. 3.7]. When $X$ is a separated and quasi-compact derived DM-stack it possesses a coarse moduli space $\pi : X \to M$ where $M$ is a separated and quasi-compact derived algebraic space (see for instance [AHPS23, Thm. 3.12]). We can write $\mathrm{QCoh}(X) \simeq \Gamma(M_{et}, \pi_*\mathrm{QCoh}))$, the global sections of the quasi-coherent stack $\pi_*\mathrm{QCoh}$) on $M_{et}$, the small étale site of $M$. Locally on $M_{et}$ the derived stack $X$ is equivalent to a quotient stack $[V/G]$ with $V$ affine and $G$ a finite group, and thus $\pi_*\mathrm{QCoh}$) is locally compactly generated as a quasi-coherent stack on $M_{et}$. We can thus apply [Lur22b, Thm. 10.3.2.1], which implies that $\Gamma(M_{et}, \pi_*\mathrm{QCoh})) \simeq \mathrm{QCoh}(X)$ is compactly generated.

We thus have seen that in both cases we can apply the corollary 4.1.3.3 and therefore have proven the result. $\square$
REMARK 4.1.3.5. Slightly more is true for corollary 4.1.3.4, as we also know that the compact objects in $\mathbf{QCoh}(X)$ are the perfect complexes, are both results [Toë12, Thm. 3.7] and [Lur22b, Thm. 10.3.2.1] identify the compact objects in the global section $\infty$-categories as the objects being compact locally.

The other interesting special case of (14) is that of the unique morphism $u : \mathcal{F} \to *_X$ to the final foliation $*_X$. By definition $\mathcal{D}_{*_X} = \mathcal{D}_X$ is the sheaf of differential operators on $X$, which coincides with the usual sheaf of differential operators when $X$ is smooth (see proposition 4.1.1.5). The induction $\infty$-functor, in this situation, produces a $\infty$-functor

$$u_! : \mathbf{QCoh}(\mathcal{F}) \to \mathcal{D}_X - \mathrm{dg}^{q\mathrm{coh}},$$

from quasi-coherent crystals along $\mathcal{F}$ to quasi-coherent $\mathcal{D}_X$-modules on $X$. The $\mathcal{D}_X$-modules of the form $u_!(E)$ will be called induced from the foliation $\mathcal{F}$. One important example is the induced $\mathcal{D}_X$-module $u_!(\mathcal{O}_X)$. This is a canonical $\mathcal{D}_X$-module on $X$ associated to the derived foliation $\mathcal{F}$ which contains interesting informations about $\mathcal{F}$. For instance, when $\mathcal{F}$ is smooth, then $u_!(\mathcal{O}_X)$ is a coherent $\mathcal{D}_X$-module. However, this is not true anymore for non-smooth derived foliations $\mathcal{F}$. Measuring the defect of coherence of $u_!(\mathcal{D}_X)$ is a very interesting question related to invariants of singularities of derived foliations, generalizing classical invariants such as Milnor numbers. We will touch the general study of singularities of derived foliations in this book and details will appear in a future work.

## 4.2. Direct and inverse images of filtered crystals

In this Section we define (filtered and unfiltered) direct image functors between quasi-coherent crystals, along proper and quasi-smooth maps. We also prove the important result that filtered direct and filtered inverse images commutes both with the underlying and the associated graded objects functors.

**4.2.1. Direct images.** We have seen in §3.1 that $(X, \mathcal{F}) \mapsto \mathbf{QCoh}(\mathcal{F})$ is a contravariant $\infty$-functor by using !-pull-backs. This functoriality can be extended to the filtered case as follows. Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism of pairs consisting of derived DM-stacks endowed with perfect derived foliations. Thus $f$ is given by a morphism $g : X \to Y$ of derived stacks, and a morphism $u : \mathcal{F} \to g^*(\mathcal{G})$ of derived foliations on $X$ (i.e. a morphism $\mathbf{DR}_{g^*(\mathcal{G})} \to \mathbf{DR}_{\mathcal{F}}$ of graded mixed cdga's over $X$). This pair induces a morphism on the corresponding sheaves of differential operators

$$\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}} \to \mathcal{D}_{g^*(\mathcal{G})}^{\mathrm{fil}}.$$

We consider the sheaf of filtered $k$-modules on $X$ defined by

$$\mathcal{D}_f^{\mathrm{fil}} := |\mathbb{R}\mathrm{Hom}_{g^{-1}(\mathbf{DR}_\mathcal{G}) - \epsilon - \mathrm{dg}^{gr}}(g^{-1}(\mathcal{O}_Y), \mathcal{O}_X)|^t.$$

This obviously carries a right filtered $g^{-1}(\mathcal{D}_\mathcal{G}^{\mathrm{fil}})$-module structure by via the canonical morphism of sheaves filtered dg-algebras

$$g^{-1}(\mathcal{D}_\mathcal{G}^{\mathrm{fil}}) = g^{-1}|(\mathbb{R}\mathrm{Hom}_{\mathbf{DR}_\mathcal{G}-\epsilon-\mathrm{dg}^{gr}}(\mathcal{O}_Y, \mathcal{O}_Y)|^t \to |\mathbb{R}\mathrm{Hom}_{g^{-1}(\mathbf{DR}_\mathcal{G})-\epsilon-\mathrm{dg}^{gr}}(g^{-1}(\mathcal{O}_Y), g^{-1}(\mathcal{O}_Y))|^t.$$
The filtered complex $\mathcal{D}_f^{\text{fil}}$ also carries a left filtered by $\mathcal{D}_{g^*(G)}^{\text{fil}}$ by means of the natural morphism of sheaves of filtered dg-algebras

$$ \mathcal{D}_{g^*(G)}^{\text{fil}} = |\mathbb{R}\operatorname{Hom}_{(\mathbb{D}\mathbf{R}_X \otimes_{g^{-1}(\mathbb{D}\mathbf{R}_Y)} g^{-1}(\mathbb{D}\mathbf{R}_G)) - \epsilon - \mathbf{d}\mathbf{g}^{gr}}(\mathcal{O}_X, \mathcal{O}_X)|^t \to |\mathbb{R}\operatorname{Hom}_{(g^{-1}(\mathbb{D}\mathbf{R}_G)) - \epsilon - \mathbf{d}\mathbf{g}^{gr}}(\mathcal{O}_X, \mathcal{O}_X)|^t. $$

The filtered bi $(\mathcal{D}_{g^*(G)}^{\text{fil}}, g^{-1}(\mathcal{D}_G^{\text{fil}})$-module $\mathcal{D}_f^{\text{fil}}$ defines a pull-back $\infty$-functor on filtered modules

$$ \phi : \mathcal{D}_G^{\text{fil}} - \mathbf{d}\mathbf{g}^{\text{fil}} \to \mathcal{D}_{g^*(G)}^{\text{fil}} - \mathbf{d}\mathbf{g}^{\text{fil}} $$

by the standard formula

$$ \phi(E) := \mathcal{D}_f^{\text{fil}} \otimes_{g^{-1}(\mathcal{D}_G^{\text{fil}})} g^{-1}(E). $$

Finally, we can compose this with the forgetful $\infty$-functor for the morphism of filtered dg-algebras $u : \mathcal{D}_{g^*(G)}^{\text{fil}} \to \mathcal{D}_F^{\text{fil}}$ in order to get the requires pull-back $\infty$-functor

$$ f^1 : \mathcal{D}_G^{\text{fil}} - \mathbf{d}\mathbf{g}^{\text{fil}} \to \mathcal{D}_F^{\text{fil}} - \mathbf{d}\mathbf{g}^{\text{fil}}. $$

On the underlying $\mathcal{O}$-modules, the $\infty$-functor $f^1$ acts as the usual pull-back, as shown by the following lemma.

**LEMMA 4.2.1.1.** With the notations and assumptions above, we have a natural equivalence of filtered $\mathcal{O}_X$-modules

$$ \mathcal{D}_f^{\text{fil}} \simeq \mathcal{O}_X \otimes_{g^{-1}(\mathcal{O}_Y)} g^{-1}(\mathcal{D}_G^{\text{fil}}). $$

In particular, the underlying $\mathcal{O}_X$-module of $f^1(E)$ is naturally equivalent to $f^*(E)$ and is therefore quasi-coherent. As a consequence we have a commutative square of $\infty$-functors

$$ \begin{tikzcd} \mathbb{Q}\mathrm{Coh}^{\text{fil}}(\mathcal{G}) \arrow[r, "f^!"] & \mathbb{Q}\mathrm{Coh}^{\text{fil}}(\mathcal{F}) \arrow[d] \\ \mathbb{Q}\mathrm{Coh}^{\text{fil}}(Y) \arrow[r, "g^*"'] & \mathbb{Q}\mathrm{Coh}^{\text{fil}}(X), \arrow[d] \end{tikzcd} $$

where the vertical $\infty$-functors are the forgetful $\infty$-functors to filtered quasi-coherent complexes.

*Proof of the lemma.* This computations have already been seen in the lemma 4.1.2.2, because $\mathcal{G}$ is perfect we have an equivalence of sheaves of graded mixed complexes

$$ \mathbb{R}\operatorname{Hom}_{g^{-1}(\mathbb{D}\mathbf{R}_G) - \epsilon - \mathbf{d}\mathbf{g}^{gr}}(g^{-1}(\mathcal{O}_Y), \mathcal{O}_X) \simeq \mathbb{R}\operatorname{Hom}_{g^{-1}(\mathbb{D}\mathbf{R}_G) - \epsilon - \mathbf{d}\mathbf{g}^{gr}}(g^{-1}(\mathcal{O}_Y), g^{-1}(\mathcal{O}_Y)) \otimes_{g^{-1}(\mathcal{O}_Y)} \mathcal{O}_X. $$

Passing to the realization provides the wanted result

$$ \mathcal{D}_f^{\text{fil}} \simeq \mathcal{O}_X \otimes_{g^{-1}(\mathcal{O}_Y)} g^{-1}(\mathcal{D}_G^{\text{fil}}). $$

The rest of the lemma follows directly from the first assumption as the quasi-coherent pull-back is given by the standard formula

$$ g^*(E) \simeq g^{-1}(E) \otimes_{g^{-1}(\mathcal{O}_Y)} \mathcal{O}_X. $$

□
According to the previous lemma we do have constructed the pull-back of filtered crystals

$$f^! : \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}) \to \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}).$$

**DEFINITION 4.2.1.2.** The $\infty$-functor constructed above is called the filtered pull-back of filtered crystals.

A first important property is the existence of a left adjoint to the $!$-pull back, given by the $!$-pushforward. It will exists under sufficient properness and quasi-smoothness conditions to insure that the standard direct image of quasi-coherent sheaves preserves perfect complexes.

**PROPOSITON 4.2.1.3.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism between derived DM-stacks endowed with perfect derived foliations. We assume that $g : X \to Y$ is proper and quasi-smooth, and that $Y$ is either a quasi-compact and quasi-separated derived scheme, or a separated and quasi-compact derived DM-stack. Then, the $\infty$-functor

$$f^! : \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}) \to \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$$

admits a left adjoint

$$f_! : \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}) \to \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}).$$

*Proof.* We already know from Corollary 4.1.3.3 that both $\infty$-categories are compactly generated. For the existence of $f_!$ it is thus enough to check that $f_!^!$ commutes with limits. For this, we use that $f_!^!$ is compatible with the usual pull-backs of filtered $\mathcal{O}$-modules along $g$ (see Lemma 4.2.1.1)

$$\begin{tikzcd}
\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}) \arrow[r] \arrow[d, "f^!"'] & \mathrm{QCoh}^{\mathrm{fil}}(Y) \arrow[d, "(g^*)^{\mathrm{fil}}"'] \\
\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}) \arrow[r] & \mathrm{QCoh}^{\mathrm{fil}}(X).
\end{tikzcd}$$

The horizontal $\infty$-functors are clearly conservative and commute with limits and colimits. Therefore, to check that $f_!^!$ preserves limits, it is enough to show that $(g^*)^{\mathrm{fil}}$ does so. Again, as $(g^*)^{\mathrm{fil}}$ is obtained by extending the usual pull-back $g^* : \mathrm{QCoh}(Y) \to \mathrm{QCoh}(X)$ to filtered objects, we are reduced to show that $g^*$ commutes with limits.

This last step follows easily from the assumption that $g$ is proper and quasi-smooth. Indeed, let $E \in \mathrm{QCoh}(X)$ be a compact generator (thus a perfect complex on $X$), and $\{F_i\}_{i \in I}$ a diagram in $\mathrm{QCoh}(Y)$. Using that $E$ is dualizable and the projection formula, we have

$$\mathrm{Map}(E, g^*(\lim_i F_i)) \simeq \mathrm{Map}(\mathcal{O}_X, E^\vee \otimes_{\mathcal{O}_X} g^*(\lim_i F_i)) \simeq \mathrm{Map}(\mathcal{O}_Y, g_*(E^\vee) \otimes_{\mathcal{O}_Y} (\lim_i F_i)).$$

Now we use that $g$ is proper and quasi-smooth, so that $g_*(E^\vee)$ is again perfect and thus dualizable, and therefore the functor $g_*(E^\vee) \otimes_{\mathcal{O}_Y}$ commutes with limits. So we have

$$\mathrm{Map}(\mathcal{O}_Y, g_*(E^\vee) \otimes_{\mathcal{O}_Y} (\lim_i F_i)) \simeq \lim_i \mathrm{Map}(\mathcal{O}_Y, g_*(E^\vee) \otimes_{\mathcal{O}_Y} F_i) \simeq \lim_i \mathrm{Map}(E, g^*(F_i)).$$
This shows that the canonical map

$$ Map(E, g^*(\lim_i F_i)) \to \lim_i Map(E, g^*(F_i)) \simeq Map(E, \lim_i g^*(F_i)) $$

is indeed in equivalence, so that $g^*$ preserves limits (since $E$ is a generator of $\mathbf{QCoh}(Y)$). $\square$

The proposition 4.2.1.3 allows us to give the following definition.

**DEFINITION 4.2.1.4.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism with $g : X \to Y$ proper and quasi-smooth and satisfying the condition of the proposition 4.2.1.3. The filtered direct image is the $\infty$-functor

$$ f_! : \mathbf{QCoh}^{\text{fil}}(\mathcal{F}) \to \mathbf{QCoh}^{\text{fil}}(\mathcal{G}), $$

left adjoint to the pull-back $\infty$-functor $f^!$.

Note that for $f = (\text{id}_X, u : 0_X \to \mathcal{F})$, we clear have $f_! = \text{Ind}_{\mathcal{F}}^{\text{fil}}$ (see Definition 4.1.3.2).

**REMARK 4.2.1.5.** Note that, although it is possible, we do not try to define direct images for non-proper or non-quasi-smooth morphisms (this would require working with Ind-coherent crystals rather than quasi-coherent ones). In the rest of the book, we will only need direct images for proper and quasi-smooth morphisms.

We also define the unfiltered direct image

$$ f_! : \mathbf{QCoh}(\mathcal{F}) \to \mathbf{QCoh}(\mathcal{G}) $$

as being the left adjoint to the unfiltered version of $f^!$ whose existence is proved similarly.

The direct image functors satisfy the usual pseudo-functoriality properties, $(f f')_! \simeq f_! f^!_!$ (for composeable, proper and quasi-smooth $f$ and $f'$). An important consequence of this property is the following result.

**PROPOSITON 4.2.1.6.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism with $g$ proper and quasi-smooth, $\mathcal{F}$ and $\mathcal{G}$ perfect, and $Y$ either a $qcqs$ derived scheme or a separated and quasi-compact derived DM-stack. Then, the following diagram naturally commutes

$$ \begin{tikzcd} \mathbf{QCoh}^{\text{fil}}(X) \arrow[r, "\text{Ind}_{\mathcal{F}}^{\text{fil}}"] & \mathbf{QCoh}^{\text{fil}}(\mathcal{F}) \\ \mathbf{QCoh}^{\text{fil}}(Y) \arrow[r, "\text{Ind}_{\mathcal{G}}^{\text{fil}}"] & \mathbf{QCoh}^{\text{fil}}(\mathcal{G}) \arrow[down, "g*(\omega_{X/Y}\otimes-) [d]"'] \arrow[down, "f!"'] \end{tikzcd} $$

where $\omega_{X/Y}$ is the relative canonical line bundle of $X$ over $Y$, and $d$ the relative dimension of $X$ over $Y$.
*Proof.* We have a commutative diagram of pairs

<figure><img src="image_26.png" /></figure>

where $u$ and $v$ are the unique morphisms from the initial foliation. We get from this a natural isomorphism of $\infty$-functors

$$f_! u_! \simeq v_! g_!.$$

The proposition then follows from the fact that, by definition $u_!$ and $v_!$ are the induction $\infty$-functors, and from the explicit formula for $g_! : \mathrm{QCoh}(X) \to \mathrm{QCoh}(Y)$ in terms of relative Serre duality. $\square$

An interesting application of proposition 4.2.1.6 is to the direct image of $\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}$ itself. Keeping the same notations we obtain

$$f_!(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}) \simeq \mathcal{D}_{\mathcal{G}}^{\mathrm{fil}} \otimes_{\mathcal{O}_Y} g_*(\omega_{X/Y})[d].$$

When $X$ is proper and quasi-smooth over $k$, and $f : (X, \mathcal{F}) \to (\mathrm{Spec} k, 0)$ is the projection to the point (endowed with its trivial foliation), we see in particular that $f_!(\mathcal{D}_{\mathcal{F}})$ computes $H^{*+d}(X, \omega_X)$, that is coherent homology of $X$ with coefficients in $\mathcal{O}_X$.

The following result is a direct consequence of the fact that $f_!$ commutes with colimits (and can also be deduced by using proposition 4.2.1.6 and the proof of corollary 4.1.3.3).

**COROLLARY 4.2.1.7.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism of derived schemes endowed with perfect derived foliations, with $g$ proper and quasi-smooth and $Y$ being either a $qcqs$ derived scheme or a separated quasi-compact derived DM-stack. Then, the direct image $\infty$-functors

$$f_! : \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}) \to \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}) \qquad f_! : \mathrm{QCoh}(\mathcal{F}) \to \mathrm{QCoh}(\mathcal{G})$$

preserve compact objects.

**4.2.2. Compatibility with underlying and associated graded objects.** We conclude this section with the important result that filtered direct images commute with both taking underlying and associated graded objects. For this, let us consider a morphism $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ with $g$ proper and quasi-smooth, and assume the usual assumptions of proposition 4.2.1.3. We have the corresponding adjunction on filtered crystals

$$f_!^{\mathrm{fil}} : \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}) \rightleftharpoons \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}) : f_!^{\mathrm{fil}},$$

and its unfiltered version

$$f_! : \mathrm{QCoh}(\mathcal{F}) \rightleftharpoons \mathrm{QCoh}(\mathcal{G}) : f_!.$$
We may also consider $\mathcal{F}^{\epsilon=0}$ which is the derived foliation whose underlying graded mixed cdga is $\mathbf{DR}(\mathcal{F})^{\epsilon=0}$ (i.e. with trivial mixed structure $\epsilon = 0$), and the same for $\mathcal{G}^{\epsilon=0}$. For such derived foliations, we have now a graded push-forward

$$f_{!}^{\mathrm{gr}} : \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{F}^{\epsilon=0}) \to \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{G}^{\epsilon=0})$$

defined as the left adjoint to the graded pull-back $f_{\mathrm{gr}}^{!} : \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{G}^{\epsilon=0}) \to \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{F}^{\epsilon=0})$. Note that the $\infty$-categories $\mathrm{QCoh}^{\mathrm{gr}}(\mathcal{F}^{\epsilon=0})$ and $\mathrm{QCoh}^{\mathrm{gr}}(\mathcal{G}^{\epsilon=0})$ can also be identified with the categories of graded $\mathrm{Sym}_{\mathcal{O}_X}(\mathbb{T}_{\mathcal{F}})$-modules over $X$ and of graded $\mathrm{Sym}_{\mathcal{O}_Y}(\mathbb{T}_{\mathcal{G}})$-modules over $Y$, respectively (where $\mathbb{T}_{\mathcal{F}}$ and $\mathbb{T}_{\mathcal{G}}$ both sit in weight 1). We leave to the reader the precise construction of these graded functorialities, which are obtained using similarly as the filtered case.

By putting all these functors together, we may write the following diagram of vertical adjunctions

$$(17) \qquad \begin{tikzcd} \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{F}^{\epsilon=0}) \arrow[r, "\mathrm{Gr}"] \arrow[d, "f_{!}^{\mathrm{gr}}"] & \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}) \arrow[r, "(-)^u"] \arrow[d, "f_{!}^{\mathrm{fil}}"] & \mathrm{QCoh}(\mathcal{F}) \arrow[d, "f_{!}"] \\ \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{G}^{\epsilon=0}) \arrow[r, "\mathrm{Gr}"] \arrow[d, "f_{!}^{\mathrm{fil}}"] & \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}) \arrow[r, "(-)^u"] \arrow[d, "f_{!}^{\mathrm{fil}}"] & \mathrm{QCoh}(\mathcal{G}) \arrow[d, "f_{!}^{!}"] \end{tikzcd}$$

We already know that this diagram, when restricted to the inverse images only, naturally commutes. This implies that the diagram restricted to direct images is naturally lax commutative. In fact, the induced natural transformations

$$f_{!}^{\mathrm{gr}} \circ \mathrm{Gr} \Rightarrow \mathrm{Gr} \circ f_{!}^{\mathrm{fil}} \qquad f_{!} \circ (-)^u \Rightarrow (-)^u \circ f_{!}^{\mathrm{fil}}$$

turn out to be equivalences. Indeed, as all $\infty$-functors involved commute with colimits, it is enough to check this property on compact generators, and we can thus use proposition 4.2.1.6 and the proof of corollary 4.1.3.3 to conclude. As a consequence of the commutativity of (17), we are allowed to (and will from now on) simply write $f_!$ and $f_!^!$ without any decorations $(-)^{\mathrm{fil}}$ or $(-)^{\mathrm{gr}}$. Because of its importance, and for later reference, we state this result in the following corollary.

**COROLLARY 4.2.2.1.** With the notations and assumptions above, the filtered direct and filtered inverse images of quasi-coherent crystals commute with taking the underlying or the associated graded objects, i.e. either considering its $f_!^!$ part or its $f_!$ part, the following diagram naturally commutes

$$(18) \qquad \begin{tikzcd} \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{F}^{\epsilon=0}) \arrow[r, "\mathrm{Gr}"] \arrow[d, "f_{!}^{\mathrm{gr}}"] & \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F}) \arrow[r, "(-)^u"] \arrow[d, "f_{!}^{\mathrm{fil}}"] & \mathrm{QCoh}(\mathcal{F}) \arrow[d, "f_{!}^{!}"] \\ \mathrm{QCoh}^{\mathrm{gr}}(\mathcal{G}^{\epsilon=0}) \arrow[r, "\mathrm{Gr}"] \arrow[d, "f_{!}^{\mathrm{fil}}"] & \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{G}) \arrow[r, "(-)^u"] \arrow[d, "f_{!}^{\mathrm{fil}}"] & \mathrm{QCoh}(\mathcal{G}) \arrow[d, "f_{!}^{!}"] \end{tikzcd}$$
Here is an equivalent way of looking at the leftmost adjunction $f_!^{gr} : \mathbf{QCoh}^{gr}(\mathcal{F}^{\epsilon=0}) \stackrel{\sim}{\to} \mathbf{QCoh}^{gr}(\mathcal{G}^{\epsilon=0}) : f^{gr}!$ of (17) that avoids introducing the auxiliary foliations $\mathcal{F}^{\epsilon=0}$ and $\mathcal{G}^{\epsilon=0}$. First of all, we have the associated graded object $\infty$-functor

$$ \mathrm{Gr} : \mathbf{QCoh}^{\mathrm{fil}}(\mathcal{F}) = \mathcal{D}_{\mathcal{F}}^{\mathrm{fil}} - \mathbf{dg}_{qcoh}^{\mathrm{fil}} \to \mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}) - \mathbf{dg}_{qcoh}^{\mathrm{gr}}, $$

from $\mathcal{O}_X$-quasi-coherent sheaves of filtered $\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}$-dg-modules on $X$ to $\mathcal{O}_X$-quasi-coherent sheaves of graded $\mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}})$-dg-modules on $X$ (and similarly for $\mathcal{G}$ and $Y$). Note that $\mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}) \simeq \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{T}_{\mathcal{F}})$ and $\mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}}) \simeq \mathrm{Sym}_{\mathcal{O}_Y}(\mathbb{T}_{\mathcal{G}})$ in the $\infty$-category of sheaves of graded associative dg-algebras $\mathrm{Alg}(\mathbf{dg}^{gr}(X))$ and $\mathrm{Alg}(\mathbf{dg}^{gr}(Y))$ (with $\mathbb{T}_{\mathcal{F}} = \mathbb{L}_{\mathcal{F}}^{\vee}$ and $\mathbb{T}_{\mathcal{G}} = \mathbb{L}_{\mathcal{G}}^{\vee}$ of pure weight 1). Now, we proceed as in §4.2.1 and consider

$$ \mathrm{Gr}(\mathcal{D}_{f}^{\mathrm{fil}}) \simeq g^{*}(\mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}})) = \mathcal{O}_{X} \otimes_{g^{-1}(\mathcal{O}_{Y})} g^{-1}(\mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}})). $$

which is a graded $(\mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}), g^{-1}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}}))$-bi-module over $X$, and as such, it defines a pull-back $\infty$-functor on graded modules

$$ f_{\mathrm{gr}}^{!} : \mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}}) - \mathbf{dg}^{gr} \to \mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}) - \mathbf{dg}^{gr} $$

by

$$ f_{\mathrm{gr}}^{!}(E) := \mathrm{Gr}(\mathcal{D}_{f}^{\mathrm{fil}}) \otimes_{g^{-1}(\mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}}))} g^{-1}(E). $$

This $\infty$-functor respects the property of being quasi-coherent over $X$ and $Y$, so it induces a $\infty$-functor

$$ f_{\mathrm{gr}}^{!} : \mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}}) - \mathbf{dg}_{qcoh}^{gr} \to \mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}) - \mathbf{dg}_{qcoh}^{gr} $$

which has a left adjoint

$$ f_!^{\mathrm{gr}} : \mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}) - \mathbf{dg}_{qcoh}^{\mathrm{gr}} \stackrel{\sim}{\to} \mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}}) - \mathbf{dg}_{qcoh}^{\mathrm{gr}} : f_{\mathrm{gr}}^! $$

that coincides with the leftmost adjunction of (17), via the natural equivalences $\mathbf{QCoh}^{gr}(\mathcal{F}^{\epsilon=0}) \simeq \mathrm{Gr}(\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}) - \mathbf{dg}_{qcoh}^{gr}$, and $\mathbf{QCoh}^{gr}(\mathcal{G}^{\epsilon=0}) \simeq \mathrm{Gr}(\mathcal{D}_{\mathcal{G}}^{\mathrm{fil}}) - \mathbf{dg}_{qcoh}^{gr}$.

### 4.3. Characteristic cycles

In this section we introduce the notion of characteristic cycle of a quasi-coherent crystal (or of a $\mathcal{D}_{\mathcal{F}}$-module thanks to the equivalence of Theorem 4.1.2.1) along a derived foliation $\mathcal{F}$. For this we first introduce the global cotangent stack $T^*\mathcal{F}$ of a derived foliation $\mathcal{F}$, which is a derived Artin $n$-stack, where $n$ is the tor-amplitude of the perfect complex $\mathbb{L}_{\mathcal{F}}$. We will then discuss the notion of bounded coherent crystals and of good filtrations on them. By definition, the associated graded to a good filtration will be a $\mathcal{G}_m$-equivariant perfect complex on $T^*\mathcal{F}$, that will be used to define characteristic cycles. We investigate the existence of good filtrations and prove some independence (of good filtrations) results for characteristic cycles.