$\mathbb{L}_{\mathcal{F}}^{\text{big}}$ called the *big cotangent complex of* $\mathcal{F}$. Its associated quasi-coherent complex (see §A.4) is called *the cotangent complex of* $\mathcal{F}$, and is denoted by

$$ \mathbb{L}_{\mathcal{F}} := (\mathbb{L}_{\mathcal{F}}^{\text{big}})^{qcoh} \in \mathbf{QCoh}(X). $$

A derived foliation $\mathcal{F}$ on $X$ is called *perfect* if its cotangent complex $\mathbb{L}_{\mathcal{F}}$ is perfect.

For any derived stack $X$ over $k$, we have a final object $*_{X} \in \mathcal{Fol}(X/k)$ which is given by $\mathbb{V}(\mathcal{F}) = \hat{\mathcal{L}}^{gr}(X/k)$. Similarly, when $X$ is a derived Artin stack locally of finite presentation over $k$, the $\infty$-category $\mathcal{Fol}(X/k)$ has an initial object $0_X \in \mathcal{Fol}(X/k)$. It can be described as an explicit derived stack $\mathbb{V}(\mathcal{F}) \to \hat{\mathcal{L}}^{gr}(X)$ derived stack as follows. We write $X$ as a colimit of derived affine opschemes $\text{colim}_i U_i$, so that $\hat{\mathcal{L}}^{gr}(X) \simeq \text{colim}_i \mathcal{L}^{gr}(U_i)$ ($\hat{\mathcal{L}}^{gr}$ commutes with colimits has it is defined by a left Kan extension). For each index $i$, we form the relative graded loop stack

$$ \hat{\mathcal{L}}^{gr}(U_i/X) := \mathcal{L}^{gr}(U_i) \times_{\hat{\mathcal{L}}^{gr}(X)} X \to \mathcal{L}^{gr}(U_i), $$

which comes equipped with its natural $\mathcal{H}$-action. The $\mathcal{H}$-equivariant derived stacks $\hat{\mathcal{L}}^{gr}(U_i/X)$ certainly glue to a global object over $\hat{\mathcal{L}}^{gr}(X)$, however in general $\hat{\mathcal{L}}^{gr}(U_i/X)$ is not a derived affine stack over $U_i$. We thus replace $\hat{\mathcal{L}}^{gr}(U_i/F)$ by its graded affinization (see §A.9.2)

$$ \hat{\mathcal{L}}^{gr}(U_i/F)^{aff} := \mathbb{R}\mathbf{Spec}^{\Delta,gr}(\mathcal{O}(\hat{\mathcal{L}}^{gr}(U_i/F)^{aff})) \to U_i. $$

The group stack $\mathcal{H}$ being affine over $\mathbb{BG}_m$, it continues to act on $\hat{\mathcal{L}}^{gr}(U_i/F)^{aff}$. Finally, by our descent result for cotangent complexes (see §A.5) we have that $\hat{\mathcal{L}}^{gr}(U_i/F)^{aff} \simeq \mathbb{V}(\mathbb{L}_{U_i/F}[1])$ as graded derived affine stacks. The family of objects $\hat{\mathcal{L}}^{gr}(U_i/F)^{aff}$ then glue as a global $\mathcal{H}$-equivariant derived stack $\mathbb{V}(\mathcal{F}) \to \hat{\mathcal{L}}^{gr}(F)$, which thus defines an object in $\mathcal{Fol}(F)$. It can be checked, as in §2.1.5.1, that this defines an initial object in $\mathcal{Fol}(F)$.

We finish this general part by stating that integrable sub-bundles of the tangent bundle of a smooth scheme canonically define derived foliations in the sense of Definition 7.2.0.1. This result is taken from [Mon22], whose proof is reproduced below.

**PROPOSITON 7.2.0.3.** Let $f: X \to S$ be a smooth morphism of derived Deligne-Mumford stacks. Let $E \subset \mathbb{T}_{X/S}$ be a sub-bundle of the relative tangent bundle of $X$ over $S$ such that $E$ is stable by the Lie bracket of relative vector fields. Then, there exists a unique action of $\mathcal{H}$ on $\mathbb{V}(E^{\vee}[1])$, covering the $\mathcal{H}$-action on $\mathbb{V}(\mathbb{L}_{X/S}[1]) \simeq \mathcal{L}^{gr}(X/S)$.

*Proof.* The purpose of this proof is to show that the space of $\mathcal{H}$-action on $\mathbb{V}(E^{\vee}[-1])$ compatible, via the map $\mathbb{V}(\mathbb{L}_{X/S}[1]) \simeq \mathcal{L}^{gr}(X/S) = \mathbb{V}(\mathbb{L}_X[-1])$, with the $\mathcal{H}$-action on $\mathbb{V}(\mathbb{L}_X[1])$, is contractible. This is a local statement on the small étale site of $X$ and thus we can assume that $X$ is an affine smooth scheme over $k$.

We first consider $\mathbb{V}(E^{\vee}[1])$ as a $\mathbb{G}_m$-equivariant derived stack, and we study the $\mathbb{G}_m$-equivariant group stack $G$ of self equivalence of $\mathbb{V}(E^{\vee}[1])$. More precisely, $G$ is the group
stack over $BG_m$ of self-equivalences of $\mathcal{V}(E^\vee[1]) \to BG_m$. The group stack $G$ a priory comes equipped with a natural derived structure, but we will only be interested in the underlying underived stack $G \to BG_m$. Also, we will not need the full stack $G$ but only $G_0 \subset G$ the full sub-stack of the connected component of the identity, defined as the pull-back of stacks relative to $BG_m$

<figure><img src="image_47.png" /></figure>

The group stack $G_0$ is connected, $\pi_0(G_0) \simeq *$ as a sheaf over $BG_m$.

Because $\mathcal{V}(E^\vee[1]) = \mathbf{Spec} \, \mathrm{Sym}_A(E^\vee[1])$, it is easy to compute the homotopy sheaves $\pi_i(G_0)$, as sheaves over $BG_m$, or equivalently as $G_m$-equivariant sheaves of groups on $\mathbf{Spec} \, k$. We have short exact sequence of sheaves on the big fpqc site of affine $k$-schemes

$$ 0 \longrightarrow \mathrm{Hom}_A(E^\vee, \wedge^i(E^\vee)) \xrightarrow{\sim} \pi_i(G_0) \longrightarrow \mathrm{Der}_k(A, \wedge^{i-1}E^\vee) \xrightarrow{\sim} 0. $$

The left hand side is the quasi-coherent sheaf on $\mathbf{Spec} \, k$ associated with the $k$-module of $A$-linear maps $E^\vee \to \wedge^i(E^\vee)$, whereas the right hand side is the quasi-coherent sheaf on $\mathbf{Spec} \, k$ associated with the $k$-module of $k$-linear derivations from $A$ to the $A$-module $\wedge^{i-1}E^\vee$. The $G_m$-action on both of these quasi-coherent sheaves is induced by the canonical homothetic action on $E^\vee$ and is thus pure of weight $i-1$ on both side.

Now, an $\mathcal{H}$-action on $\mathcal{V}(E^\vee[-1])$, compatible with the already existing $G_m$-action, is given by a $G_m$-equivariant morphism

$$ \rho : \mathcal{BH} \to BG_0. $$

The stack $\mathcal{BH}$ is an affine stack (see Appendix A.9) of the form $\mathbf{Spec}^\Delta \mathbb{Z}[u]$, where $u$ is a free variable in degree 2, and $\mathbb{Z}[u]$ is the free commutative cosimplicial-simplicial ring freely generated by $u$, as this follows easily from the fact that $\mathcal{H} = \mathbf{Spec}^\Delta H^*(S^1)$ (see 7.1.0.3 and [MRT22, Thm. 3.4.17]). Moreover, the $G_m$-action on $\mathcal{BH}$ is such that $u$ is of weight 1. We can thus understand the space of $G_m$-equivariant morphisms $\rho : \mathcal{BH} \to BG_0$ by Postnikov induction on $BG_0$, with in mind the above description of the homotopy sheaves $\pi_i(BG_0) \simeq \pi_{i-1}(G_0)$. For this we use the standard cellular decomposition, inside the $\infty$-category of the $G_m$-equivariant affine stacks, $\mathcal{BH} \simeq \mathrm{colim}_k(\mathcal{BH})_{\le 2k}$

<figure><img src="image_48.png" /></figure>

where $S_{gr}^p(q)$ denotes the *affine graded sphere* $\mathbf{Spec}^\Delta \mathbb{Z} \oplus \mathbb{Z}[-p](q)$, and $h_k$ is induced by the graded versions of the standard Hopf maps $S_{gr}^{2k+1}(k+1) \to \mathcal{BH}_{\le 2k}$.
Using this cellular decomposition, and by weight considerations, we see in particular that the projection on the first non-trivial Postnikov layer

$$BG_0 \to \tau_{\le 2}(BG_0) = K(\pi_2, 2)$$

induces an equivalences of mapping spaces between $\mathbb{G}_m$-equivariant stacks

$$\mathbf{Map}_{\mathbb{G}_m-\mathbf{st}}(B\mathcal{H}, BG_0) \simeq \mathbf{Map}_{-\mathbb{G}_m-\mathbf{st}}(B\mathcal{H}, K(\pi_2, 2)) \simeq \mathrm{Hom}_{\mathbb{G}_m-G_p}(\mathbb{G}_a^\sharp, \pi_2),$$

where $\pi_2 = \pi_1(G_0)$. Using the fact that group morphisms $\mathbb{G}_a^\sharp \to \mathbb{G}_a$ are in one-to-one correspondence with elements in $k$ (because $H^1(B\mathbb{G}_a^\sharp, \mathcal{O}) \simeq k$, see Proposition 7.1.0.3), we see that $\mathrm{Hom}_{\mathbb{G}_m-G_p}(\mathbb{G}_a^\sharp, \pi_2) \simeq \pi_2(\mathbf{Spec} k)$ can be identified with the global sections of the sheaf $\pi_2$.

We have thus seen that the space of $\mathbb{G}_m$-compatible actions on $\mathcal{V}(E^\vee[1])$ is discrete and equivalent to the underlying set of $\pi_2(\mathbf{Spec} k)$, which sits in a short exact sequence

$$0 \to \mathrm{Hom}_A(E^{\vee}, \wedge^2(E^{\vee})) \to \pi_2(\mathbf{Spec} k) \to \mathrm{Der}_k(A, E^{\vee}) \to 0.$$

The elements in the set $\pi_2(\mathbf{Spec} k)$ can be described as pairs $(d, u)$, where $d : A \to E^\vee$ is a $k$-linear derivation, and $u : E^\vee \to \wedge^2(E^\vee)$ is a $d$-semi-linear morphism

$$u(am) = d(a) \wedge m + a.u(m).$$

We can therefore conclude the proof of the proposition, as we can apply this result on both $\mathcal{V}(\mathbb{L}_X[-1])$ and $\mathcal{V}(E^\vee[1])$ and see that the only possible compatible $\mathcal{H}$-action on $\mathcal{V}(E^\vee[1])$ must be induced by the de Rham differential on the quotient

$$\mathrm{Sym}_A(\Omega_A^1[1]) \to \mathrm{Sym}_A(E^\vee)$$

and is therefore given by the pair $(d, u)$, where $d : A \to E^\vee$ is the dual of the anchor map $a : E \to \mathbb{T}_X$, and $u$ is dual to the restriction of the Lie bracket on $E$ (i.e. the Chevalley Eilenberg differential)

$$u(\phi)(x \wedge y) = a(x)(\phi(y)) - a(y)(\phi(x)) - \phi([a(x), a(y)])$$

for $\phi : E \to A$ an $A$-linear form and $x, y \in E$. $\square$

<figure><img src="image_49.png" /></figure>

### 7.3. Foliated de Rham theory

Recall that any morphism of derived stacks $f : X \to Y$ induces and adjunction given by push-forward and pull-back on quasi-coherent sheaves

$$f^* : \mathrm{QCoh}(X) \hookrightarrow \mathrm{QCoh}(Y) : f_*.$$

The left adjoint $p^*$ is a symmetric monoidal $\infty$-functor, and therefore this adjunction induces an adjunction on the corresponding $\infty$-categories of commutative algebras in the sense of [Lur22a, Def. 2.1.3.1]

$$p^* : \mathrm{Alg}(X) \hookrightarrow \mathrm{Alg}(Y) : p_*,$$

and this adjunction covers the adjunction on quasi-coherent sheaves (i.e. $p_*$ and $p^*$ commute with the forgetful $\infty$-functor forgetting the commutative algebra structure). To void confusions
objects in $Alg(QCoh(X)))$ will be called quasi-coherent $E_\infty$-algebras on $X$, and we will use the general notation

$$E_\infty - \mathrm{Alg}(\mathcal{C}) := \mathrm{Alg}(\mathcal{C}),$$

for any symmetric monoidal $\infty$-category $\mathcal{C}$. In characteristic zero these coincide with quasi-coherent sheaves of $\mathcal{O}_X$-cdga's over $X$.

Let $X \in \mathrm{dSt}_k$ be a derived Artin stack over $k$, and $\mathcal{F} \in \mathrm{Fol}(X/k)$ be a derived foliation (relative to $k$). By definition, $\mathcal{F}$ is given by an $\mathcal{H}$-equivariant derived stack $\mathcal{V}(\mathcal{F})$ together with an $\mathcal{H}$-equivariant morphism

$$\mathcal{V}(\mathcal{F}) \to \hat{\mathcal{L}}^{gr}(X).$$

We can pass to the corresponding quotient stacks, and obtain this way a morphism of derived stacks over $B\mathcal{H}$

<figure><img src="image_50.png" /></figure>

By considering the adjunctions induced by this morphisms we get a canonical morphism of quasi-coherent $E_\infty$-algebras $q_*(\mathcal{O}) \to p_*(\mathcal{O})$ on $B\mathcal{H}$. We can compose with the symmetric monoidal equivalence given by the Tate realization (see Corollary 1.3.4.4)

$$| - |^t : QCoh(B\mathcal{H}) \simeq \widehat{\mathrm{dg}}_k^{\mathrm{fil}},$$

and get this way a morphism of filtered and complete $E_\infty$-algebras $|q_*(\mathcal{O})|^t \to |p_*(\mathcal{O})|^t$.

**Definition 7.3.0.1.** Let $X \in \mathrm{dSt}_k$, and $\mathcal{F} \in \mathrm{Fol}(X/k)$. With the above notations, we define

(1) the completed derived de Rham cohomology of $X$ (relative to $k$) is the complete filtered $E_\infty$-algebra

$$\hat{C}_{DR}(X, \mathcal{O}) := |q_*(\mathcal{O})|^t \in E_\infty - \mathrm{Alg}(\widehat{\mathrm{dg}}_k^{\mathrm{fil}}).$$

(2) The completed foliated derived de Rham cohomology of the derived foliation $\mathcal{F}$ (relative to $k$) is the complete filtered $E_\infty$-algebra

$$\hat{C}_{DR}(\mathcal{F}, \mathcal{O}) := |p_*(\mathcal{O})|^t \in E_\infty - \mathrm{Alg}(\widehat{\mathrm{dg}}_k^{\mathrm{fil}}).$$

The corresponding cohomology groups will be denoted by

$$\hat{H}_{DR}^*(X) := H^i(\hat{C}_{DR}^*(X, \mathcal{O})) \qquad \hat{H}_{DR}^*(\mathcal{F}) := H^i(\hat{C}_{DR}^*(\mathcal{F}, \mathcal{O})).$$

We also have a definition with coefficients in crystals, as in Definition 3.2.0.1. To get this, we introduce the $\infty$-category of (quasi-coherent) crystals along $\mathcal{F}$ as a certain full sub-$\infty$-category $QCoh(\mathcal{F})$ of $QCoh([\mathcal{V}(\mathcal{F})/\mathcal{H}])$. To explain the definition of $QCoh(\mathcal{F})$, we start with a general derived stack $X$ with an action of $\mathbb{G}_m$ together with a $\mathbb{G}_m$-equivariant morphism $X \to Y$,
considered as a morphism $p : [X/\mathbb{G}_m] \to Y$. A quasi-coherent complex $E \in \mathbf{QCoh}([X/\mathbb{G}_m])$ is said to be *graded free relative to* $Y$, if the canonical morphism

$$p^*(p_*(E)) \to E$$

is an equivalence in $\mathbf{QCoh}([X/\mathbb{G}_m])$. With this notion, by definition an object $E \in \mathbf{QCoh}([\mathbb{V}(\mathcal{F})/\mathcal{H}])$ lies in $\mathbf{QCoh}(\mathcal{F})$ if its pull-back to $\mathbf{QCoh}([\mathbb{V}(\mathcal{F})/\mathbb{G}_m])$ is graded free with respect to the $\mathbb{G}_m$-equivariant projection $\mathbb{V}(\mathcal{F}) \to X$.

For any crystal $E \in \mathbf{QCoh}(\mathcal{F})$ on $\mathcal{F}$, we thus have a *complete foliated derived de Rham cohomology with coefficients in E*, defined by

$$\hat{C}_{DR}(\mathcal{F}, E) := |p_*(E)|^t.$$

The object $\hat{C}_{DR}(\mathcal{F}, E)$ is canonically a complete filtered module over the complete filtered $E_\infty$-algebra $\hat{C}_{DR}(\mathcal{F}, \mathcal{O})$.

One can prove, using the descent results of §A.5, that the underlying graded complex of $p_*(\mathcal{O})$ is equivalent to

$$p_*(\mathcal{O}) \simeq \bigoplus_p \mathbb{H}(F, \mathbb{L}\Lambda^p\mathbb{L}_F)[p].$$

The Hodge filtration discussed in §3.3 also makes sense in the present setting. The only difference is here that the *left derived functors* of the exterior product must be used. We gather the results in the following Proposition, and leave the details to the reader.

**PROPOSITON 7.3.0.2.** For a perfect derived foliation $\mathcal{F}$ on a derived Artin stack $X$, there exists a canonical filtration $F_F^*$ on the $E_\infty$-algebra $\hat{C}_{DR}(X/k)$ of completed de Rham cohomology on $X$, with the following properties.

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

(2) The associated graded, as a graded $E_\infty$-algebra, is given by

$$Gr_{F_F^*}^i(\hat{C}_{DR}(X/k)) \simeq \hat{C}_{DR}(\mathcal{F}, \mathbb{L}\Lambda^{-i}\mathcal{N}_F^*)([i]).$$

In the above proposition, $\mathbb{L}\Lambda^i\mathcal{N}_F^*$ is the $i$-th derived exterior product of $\mathcal{N}_F^*$ (defined as usual as being the fiber of $\mathbb{L}_{X/k} \to \mathbb{L}_{\mathcal{F}/k}$), equipped with its natural structure of a perfect crystal along $\mathcal{F}$ (see Proposition 3.3.0.2).

## 7.4. Characteristic classes

In this section we construct *characteristic classes* of derived foliations as cohomology classes in the foliated de Rham cohomology introduced in the previous section. When the base is a perfect field, we then use the comparison between derived de Rham cohomology and crystalline cohomology to prove the existence of residues for foliations, in the sense of [BB72b], in the positive characteristic setting.
We start by general results and constructions concerning the de Rham cohomology of the classifying stacks $BGl_n$. The completed derived de Rham complex $\hat{C}_{DR}^*(BGl_n, \mathcal{O})$ is a complete filtered $E_\infty$-algebra whose associated graded is equivalent to

$$ \bigoplus_p H^*(BGl_n, S^p(\mathfrak{g}^*))[-2p], $$

where $\mathfrak{g} = \mathrm{Lie}(Gl_n)$ is the Lie algebra of the group scheme $Gl_n$ and $S^p(\mathfrak{g}^*)$ denotes the (underyived) $p$-th symmetric power of its linear dual (as a $k$-module). The group $Gl_n$ acts by the adjoint action on $\mathfrak{g}$ and thus on $S^p(\mathfrak{g}^*)$, this action defines $S^p(\mathfrak{g}^*)$ as a quasi-coherent sheaf on the stack $BGl_n$.

By construction the complete filtered $E_\infty$-algebra $\hat{C}_{DR}^*(BGl_n)$ can be written as $|q_*(\mathcal{O})|^t$ where $q_*(\mathcal{O})$ is a quasi-coherent $E_\infty$-algebra over $BH$. The symmetric monoidal $\infty$-category QCoh($BH$) can be endowed with its canonical t-structure, for which the pull-back along the canonical point $\mathrm{Spec} k \to QCoh(BH)$ is t-exact and conservative. Therefore, $q_*(\mathcal{O})$ possesses a connective cover $B \to q_*(\mathcal{O})$ inside $E_\infty - \mathrm{Alg}(QCoh(BH))$, which is simply given by the non-derived direct image of $\mathcal{O}$ by $q$. The underlying graded complex of this connective cover $B$ is concentrated in degree 0 and is isomorphic to $\bigoplus_p(S^p(\mathfrak{g}^*))^{Gl_n}$. As $B$ is concentrated in cohomological degree 0 the $BH$-action is automatically trivial, and thus the induced morphism on the Tate realizations provides a morphism of complete filtered $E_\infty$-algebras We thus obtain a natural morphism of filtered $E_\infty$-algebras

$$ |B|^t := \mathcal{A}_n \to \hat{C}_{DR}^*(BGl_n, \mathcal{O}) = |q_*(\mathcal{O})|^t. $$

By the definition of the Tate realization and because the mixed structure is trivial on $B$, we have

$$ \mathcal{A}_n \simeq \bigoplus_p S^p(\mathfrak{g}^*)^{Gl_n} [-2p] $$

where the filtration is the split filtration associated with the graduation for which $(\mathfrak{g}^*)^{Gl_n} [-2]$ sits in weight $-1$.

The inclusion of the maximal torus in $Gl_n$ induces, by restrictions, a morphism of graded $E_\infty$-algebras

$$ \mathcal{A}_n \to k[x_1, \dots, x_n]^{\Sigma_n} $$

which is an isomorphism (and each $x_i$ sits in degree 2 and weight $-1$). We thus have an isomorphism of graded $E_\infty$-algebras

$$ k[c_1, \dots, c_n] \simeq \mathcal{A}_n $$

where $c_i$ corresponds to the $i$-th elementary symmetric polynomial in the $x_1, \dots, x_n$, and is therefore of cohomological degree $2i$ and weight $-i$. Via the map $\mathcal{A}_n \to \hat{C}_{DR}^*(BGl_n, \mathcal{O})$ these classes defines natural morphism of complexes of $k$-modules

$$ c_i : k \to F^{-i} \hat{C}_{DR}^*(BGl_n)[2i]. $$
REMARK 7.4.0.1. In [Tot18, Thm. 10.2] it is proven that the higher cohomology groups of $\mathbb{H}(B\mathbf{G}\mathbf{l}_n, S^p(\mathbf{g}^*))$ all vanish, and therefore the natural morphism induces an isomorphism

$$ \mathcal{A}_n \to \hat{C}_{DR}^*(B\mathbf{G}\mathbf{l}_n) $$

However, the construction of the Chern classes $c_i$ in $\hat{C}_{DR}^*(B\mathbf{G}\mathbf{l}_n)$ does not require the results of [Tot18], as the mere existence of the morphism from $\mathcal{A}_n$ to $\hat{C}_{DR}^*(B\mathbf{G}\mathbf{l}_n)$ is sufficient.

We now extend the universal Chern class to a class in derived completed de Rham cohomology of the derived stack of perfect complexes, using the standard methods of [Gil81].

Each Chern class $c_i$ can be considered as a morphism of underived stacks on the big etale site of affine $k$-schemes

$$ c_i : \mathbb{Z} \times B\mathbf{G}\mathbf{l}_{\infty} \to F^{-i}\hat{C}_{DR}^*(-, \mathcal{O})[2i], $$

where $\mathbf{G}\mathbf{l}_{\infty}$ is defined as the colimit of the $\mathbf{G}\mathbf{l}_n$ for the natural inclusion $\mathbf{G}\mathbf{l}_n \hookrightarrow \mathbf{G}\mathbf{l}_{n+1}$ given by the direct sum wuth a trivial line bundle. By the universal property of Quillen $+$ construction", this factors as a morphism of stacks $c_i : (\mathbb{Z} \times B\mathbf{G}\mathbf{l}_{\infty})^+ \to F^{-i}\hat{C}_{DR}^*(-, \mathcal{O})[2i]$, or in other words as a morphism

$$ c_i : \underline{K}^{\text{cl}} \to F^{-i}\hat{C}_{DR}^*(-, \mathcal{O})[2i] $$

where $\underline{K}^{\text{cl}}$ is the $K$-theory stack (i.e. the étale stackification of the $K$-theory functor sending a commutative $k$-algebra $A$ to its $K$-theory space $K(A)$). If we denote by $i^* : \mathbf{d}\mathbf{St}_k \to \mathbf{St}_k$ the restriction functor from the $\infty$-category of derived stacks to the $\infty$-category of stacks, the above morphism can be written as

$$ c_i : i^*(\underline{K}) \to i^*(F^{-i}\hat{C}_{DR}^*(-, \mathcal{O})[2i]) $$

where now $\underline{K}$ is the derived stack sending a simplicial commutative $k$-algebra $A$ to the Waldhausen $K$-theory space of the $\infty$-category of perfect $A$-modules. The functor $i^*$ has a left adjoint $i_! : \mathbf{St}_k \to \mathbf{d}\mathbf{St}_k$ obtained by the left Kan extension induced by the embedding of $k$-algebras into simplicial $k$-algebras. By adjunction, the morphism $c_i$ induces a morphism of derived stacks

$$ c_i : i_!i^*(\underline{K}) \to F^{-i}\hat{C}_{DR}^*(-, \mathcal{O})[2i], $$

However, it is well known that the adjunction morphism $i_!i^*(\underline{K}) \to \underline{K}$ is an equivalence, and this can be seen easily from the fact that the derived stack of vector bundles $\mathbf{Vect}$ is such that the adjunction morphism $i_!^*(\mathbf{Vect}) \to \mathbf{Vect}$ is an equivalence (simply because $B\mathbf{G}\mathbf{l}_n$ is a smooth underived stack). We therefore obtain a morphism of derived stacks

$$ c_i : \underline{K} \to F^{-i}\hat{C}_{DR}^*(-, \mathcal{O})[2i]. $$

Composed with the canonical morphism $\mathbf{Perf} \to \underline{K}$ we obtain the desired extensions

$$ c_i : \mathbf{Perf} \to F^{-i}\hat{C}_{DR}^*(-, \mathcal{O})[2i], $$

and thus canonical classes

$$ c_i \in H^{2i}(F^{-i}\hat{C}_{DR}^*(\mathbf{Perf}, \mathcal{O})). $$
DEFINITION 7.4.0.2. The universal i-th Chern class is

$$c_i \in H^{2i}(F^{-i} \hat{C}_{DR}^*(\mathrm{Perf}, \mathcal{O}))$$

constructed above.

We are now ready to define characteristic classes of perfect crystals over derived foliations. Let $X$ be a derived Artin stack (locally of finite presentation over $k$) and $\mathcal{F} \in \mathcal{F}ol(X/k)$ a derived foliation on $X$ (relative to $k$).

Let $E \in \mathrm{Perf}(\mathcal{F})$ be a perfect crystal along $\mathcal{F}$. The object $E$ can be represented by a morphism between pairs, consisting of derived stacks with derived foliations

$$\phi_E : (X, \mathcal{F}) \to (\mathrm{Perf}, \mathcal{O}_{\mathrm{Perf}})$$

where $\mathcal{O}_{\mathrm{Perf}}$ is the initial foliation on $\mathrm{Perf}$. Therefore, the morphism $\phi_E$ induces a morphism on completed derived de Rham cohomologies

$$\hat{C}_{DR}(\mathrm{Perf}, \mathcal{O}) \to \hat{C}_{DR}(X, \mathcal{O})$$

which is a morphism of filtered $E_\infty$-algebras for the Hodge filtrations on both sides (Proposition 7.3.0.2). The Hodge filtration $F_{\mathcal{O}_{\mathrm{Perf}}}^*$ on the left hand side is the usual Hodge filtration $F$ as the derived foliation on $\mathrm{Perf}$ is here the initial foliation. We therefore obtain for any integer $i$ an induced morphism

$$F^i \hat{C}_{DR}^*(\mathrm{Perf}, \mathcal{O}) \to F_{\mathcal{F}}^i \hat{C}_{DR}^*(X, \mathcal{O}).$$

As a consequence, we see that the image of the universal Chern classes $c_i$ in $\hat{H}_{DR}^{2i}(X)$ has natural lifts through $\hat{H}^{2i}(F_{\mathcal{F}}^{-i} \hat{C}_{DR}^*(X, \mathcal{O})) \to \hat{H}_{DR}^{2i}(X)$.

A first direct consequence of the previous considerations is the following vanishing result of Chern classes for perfect crystals along derived foliations.

COROLLARY 7.4.0.3. Let $X$ be a derived Artin stack locally of finite presentation over $k$, and $\mathcal{F} \in \mathcal{F}ol(X/k)$ be a perfect derived foliation. Then, for any $E \in \mathrm{Perf}(\mathcal{F})$, the chern classes $c_i(E) \in \hat{H}_{DR}^{2i}(X)$ of the underlying perfect complex $E$ on $X$ possess canonical lifts to $H^{2i}(F_{\mathcal{F}}^{-i} \hat{C}_{DR}^*(X, \mathcal{O}))$. It particular, the image of $c_i(E)$ by the natural moprism

$$H_{DR}^{2i}(X) \to H_{DR}^{2i}(\mathcal{F}, \mathcal{O}) \simeq H^{2i}(Gr_{F_{\mathcal{F}}}^0 \hat{C}_{DR}^*(X, \mathcal{O}))$$

is zero as soon as $i > 0$.

A slightly more sophisticated consequence is the following result on characteristic classes of the conormal complex of a derived foliation. Recall from Proposition 7.3.0.2 that, for $X$ and $\mathcal{F}$ as above, the Hodge filtration $F_{\mathcal{F}}^*$ is a filtration on $\hat{C}_{DR}^*(X, \mathcal{O})$ whose associated graded is given, as a graded $E_\infty$-algebra, by foliated de Rham cohomology with coefficients in the derived external products of the conormal complex

$$Gr_{F_{\mathcal{F}}}^i \hat{C}_{DR}^*(X, \mathcal{O}) \simeq \hat{C}_{DR}^*(\mathcal{F}, \mathbb{L}\Lambda^{-i}\mathcal{N}_{\mathcal{F}}^*)(i]$$
where $\mathbb{L}\Lambda^i\mathcal{N}_{\mathcal{F}}^*$ is the $i$-th derived exterior product of $\mathcal{N}_{\mathcal{F}}^*$, equipped with its natural structure of a perfect crystal along $\mathcal{F}$. In particular, when $\mathcal{N}_{\mathcal{F}}$ is a vector bundle of rank $r$, $\mathbb{L}\Lambda^i\mathcal{N}_{\mathcal{F}}^* \simeq \Lambda^i\mathcal{N}_{\mathcal{F}}^* \simeq 0$ for all $i > r$. As a consequence, using the multiplicative nature of the Hodge filtration, we obtain the following important corollary.

**COROLLARY 7.4.0.4.** Let $X$ be a derived Artin stack locally of finite presentation over $k$, and $\mathcal{F} \in \mathcal{Fol}(X)$ be a perfect derived foliation whose conormal complex $\mathcal{N}_{\mathcal{F}}^*$ is a vector bundle of rank $r$. Then, for any homogenuous polynomial $P(x_1, \dots, x_r)$, with $\deg(x_i) = -2i$, of total degree $\deg(P) = q < -r$, the element

$$P(c_1(\mathcal{N}_{\mathcal{F}}^*), \dots, c_r(\mathcal{N}_{\mathcal{F}}^*)) : k \to \hat{C}_{DR}^*(X, \mathcal{O})$$

is canonically homotopic to zero.

*Proof.* The vector bundle $\mathcal{N}_{\mathcal{F}}^*$ has canonical structure of a crystal along $\mathcal{F}$, and thus, by Corollary 7.4.0.3, all the morphisms of dg-modules over $k$

$$c_i(\mathcal{N}_{\mathcal{F}}^*) : k \to \hat{C}_{DR}^*(X, \mathcal{O})$$

possess canonical factorisations

$$c_i(\mathcal{N}_{\mathcal{F}}^*) \to F_{\mathcal{F}}^{-i} \hat{C}_{DR}^*(X, \mathcal{O}).$$

As the Hodge filtration is a filtration of $E_{\infty}$-algebras, we deduce that the induced morphism $P(c_1(\mathcal{N}_{\mathcal{F}}^*), \dots, c_r(\mathcal{N}_{\mathcal{F}}^*))$ has a canonical factorisation as

$$P(c_1(\mathcal{N}_{\mathcal{F}}^*), \dots, c_r(\mathcal{N}_{\mathcal{F}}^*)) : k \to F_{\mathcal{F}}^{-r-1} \hat{C}_{DR}^*(X, \mathcal{O}) \to \hat{C}_{DR}^*(X, \mathcal{O}).$$

But, by assumption $\mathbb{L}\Lambda^i\mathcal{N}_{\mathcal{F}}^* = 0$ for $i > r$, and thus $F_{\mathcal{F}}^{-r-1}\hat{C}_{DR}^*(X, \mathcal{O}) \simeq 0$. $\square$

**7.5. Residues of singular foliations in positive characteristics**

In this section we exploit the results on characteristic classes defined and studied in the previous section, in the specific case of a smooth variety over a perfect field of characteristic $p \neq 2$. Our objective is to prove a crystalline version of the Baum-Bott residue theorem of [BB72a].

We let $k$ be a perfect field of characteristic $p > 0$, and we assume $p \neq 2$. We start by explaining the well-known relations between the crystalline cohomology over $W(k)$ and derived de Rham cohomology. We let $S_n = \text{Spec} W_n(k)$ be the spectrum of the ring of Witt vectors of length $n$ over $k$, $W(k) = \text{lim}_n W_n(k)$ the corresponding filtered complete algebra, and $S := \text{Spf} W(k) = \text{colim}_n S_n$ its formal spectrum. In the sequel, in order to avoid confusions with our notation $\mathbf{DR}(A/W_n(k))$, we will write $dR(A/W_n(k))$ to denote the usual (underived) de Rham complex of a commutative $W_n(k)$-algebra $A$

$$dR(A/W_n(k)) := (A \to \Omega_{A/W_n(k)}^1 \to \Omega_{A/W_n(k)}^2 \to \dots)$$

where $A$ sits in degree 0.
For a commutative simplicial $W_n(k)$-algebra $A$, which we assume to be a finite cell $W_n(k)$-algebra (so each $A_m$ is a polynomial algebra over $W_n(k)$), we consider the projection $A \to \pi_0(A)$ and the corresponding augmentation simplicial ideal $I \subset A$. We let $D(I)$ be the divided power envelope of $A$ along $I$ relative to $W_n(k)$ (endowed with its canonical divided power structure), so we have a commutative diagram of commutative simplicial $W_n(k)$-algebras

<figure><img src="image_51.png" /></figure>

The commutative $A$-algebra $D(I)$ carries a canonical flat connection, and thus we can form its usual de Rham complex, which is the simplicial object in $dg_k^{fil}$

$$[m] \mapsto D(I) \otimes_{A_m} dR(A_m/W_n(k)),$$

where the filtrations are here given by the PD-Hodge filtration of [Ber74, §V-2.3.1] (convolution of the standard Hodge filtration with the PD-adic filtration). We remind that the $(-i)$-layer of this filtration is the subcomplex (for a fixed given $m$)

$$I_m^{[i]} \to I_m^{[i-1]} \otimes \Omega_{A_m/W_n(k)}^1 \to I_m^{[i-2]} \otimes \Omega_{A_m/W_n(k)}^2 \to \dots,$$

where $I_m^{[q]}$ denotes the $q$-th PD-powers of the ideal $I_m$ in $D(I_m)$. By taking the geometric realization of this simplicial object (i.e. its colimit in the $\infty$-category $dg_k^{fil}$) we obtain a well defined object $|dR(D(I)/W_n(k))| \in dg_{WW(k)}^{fil}$ which is functorial in $A$. It defines an $\infty$-functor $A \mapsto |dR(D(I))|$, which by [Ber74, §V-2.3.2] is naturally equivalent to the functor sending $A$ to the complex of crystalline cohomology of the $W_n(k)$-algebra $\pi_0(A)$.

For any $A$ as above, the morphism $A \to D(I)$ induces a morphism on the de Rham complexes $|dR(A)| \to |dR(D(I))|$ where the left hand side is the geometric realization of the simplicial object $[m] \mapsto dR(A_m/W_n(k))$. We can complete this morphism with respect to the filtrations on both sides, and thus get an induced morphism

$$|d\widehat{R(A)}| \to |d\widehat{R(D(I))}|,$$

which is functorial in $A$. The left hand side is naturally equivalent to the complex $\widehat{C}_{DR}^*(X/S_n, \mathcal{O})$ computing derived de Rham cohomology of $X = \text{Spec } A$ relative to $W_n(k)$, whereas the right hand side is the PD-completed crystalline cohomology of $t_0(X) = \text{Spec } \pi_0(A)$ relative to $W_n(k)$ denoted by

$$\widehat{C}_{crys}^*(t_0(X)/W_n(k)) := |d\widehat{R(D(I))}|.$$

We thus have constructed a functorial diagram of objects in $dg_{W(k)}^{fil}$

$$\widehat{C}_{DR}^*(X/S_n, \mathcal{O}) \longrightarrow \widehat{C}_{crys}^*(t_0(X)/S_n) \longleftarrow C_{crys}^*(t_0(X)/S_n)$$
where the map on the right hand side is the completion morphism with respect to the PD-Hodge filtration. This diagram of morphisms is not only functorial in $X$, but also functorial with respect to the natural projections $W_{n+1}(k) \to W_n(k)$, so that we can pass to the limit over $n$

$$ \hat{C}_{DR}^*(X/S) \xrightarrow{\phi} \hat{C}_{crys}^*(t_0(X)/S) \leftarrow \hat{C}_{crys}^*(t_0(X)/S). $$

Finally, the functoriality in $X$ can be used to glue these constructions to get a global version valid for any derived stack $X$ which is locally of finite presentation over $S$ (i.e. a compatible system of derived Artin stacks $X_n$ of finite presentation over $S_n$). Moreover, we could also keep track of the multiplicative structures involved and contemplate the natural lift of the previous diagram to the $\infty$-category of filtered $E_\infty$-algebras over $W(k)$. Note also that, as $\hat{C}_{crys}^*(s/S) \simeq W(k)$, the morphism $\phi$ enters in a commutative diagram of complete filtered $E_\infty$-algebras

$$ \begin{array}{ccc} \hat{C}_{DR}^*(X/S) & \longrightarrow & \hat{C}_{crys}^*(X/S) \\ \uparrow & & \uparrow \\ \hat{C}_{DR}^*(s/S) & \longrightarrow & W(k) \end{array} $$

and thus produces a morphism

$$ \hat{C}_{DR}^*(X/S) \otimes_{\hat{C}_{DR}^*(s/S)} W(k) \to \hat{C}_{crys}^*(X/S) $$

(here we have noted $s = \text{Spec } k$).

**PROPOSITON 7.5.0.1.** Let $X$ be a smooth variety over $k$, considered as a derived scheme of finite presentation over $S$ via the natural morphism $s = \text{Spec } k \to S$.

(1) The completion morphism with respect to the PD-Hodge filtration

$$ \mathcal{C}_{crys}^*(X/S) \to \hat{\mathcal{C}}_{crys}^*(X/S) $$

is an equivalence.

(2) The morphism $\phi$ constructed above

$$ \hat{C}_{DR}^*(X/S) \otimes_{\hat{C}_{DR}^*(s/S)} W(k) \to \hat{C}_{crys}^*(X/S) $$

is an equivalence.

*Proof.* Assertion (1) follows from the fact that the PD-Hodge filtration is already complete (in fact it is finite) when $X$ is smooth over $k$. Indeed, by descent we can assume that $X$ is affine to prove (1). We can then compute the crystalline cohomology of $X$ over $S_n$ by choosing a smooth lift $X_n$ over $S_n$. The scheme $X_n$ is the PD-hull of $X$ relative to $S_n$, and the PD-adic filtration is then finite as this can be seen using [Ber74, §I-3.2.4] (we use here that $p \neq 2$). We thus have that the PD-Hodge filtration is finite and thus complete.
It remains to prove (2). By construction

$$ \hat{C}_{DR}^*(X/S) \otimes_{\hat{C}_{DR}^*(s/S)} \mathcal{W}(k) \to \hat{C}_{crys}^*(X/S) $$

is a morphism of complete $\mathcal{W}(k)$-dg-modules, and thus to check that it is an equivalence it is enough to check that it is so after base change along the augmentation morphism $\mathcal{W}(k) \to k$. After such a base change, the morphism $\psi$ can be written as

$$ \hat{C}_{DR}^*(X \times_s s \times_S s/s) \otimes_{\hat{C}_{DR}^*(s \times_S s/s)} k \to \hat{C}_{DR}^*(X/s). $$

By Künneth, this is also written as the canonical morphism

$$ \hat{C}_{DR}^*(X/s) \otimes_k \hat{C}_{DR}^*(s \times_S s/s) \otimes_{\hat{C}_{DR}^*(s \times_S s/s)} k \to \hat{C}_{DR}^*(X/s) $$

which is obviously an equivalence. $\square$

By combining Corollary 7.4.0.4 and Proposition 7.5.0.1 we finally find the following corollary, which is our version of the Baum-Bott's residue theorem in characteristic $p > 0$. We let $X$ be a smooth variety over $k$. For a derived foliation $\mathcal{F}$ on $X$ relative to $s = \text{Spec } k$, we will call an $S$-structure on $\mathcal{F}$ the data of a derived foliation $\mathcal{F}' \in \text{Fol}(X/S)$ lifting $\mathcal{F}$ via the pull-back $i : \text{Fol}(X/S) \to \text{Fol}(X/s)$ along $i : s \hookrightarrow S$.

**COROLLARY 7.5.0.2.** Let $X$ be a smooth variety over $k$, and $D \subset \Omega_{X/k}^1$ be a coherent subsheaf which is a differential ideal: $\text{d}R : D \to \Omega_{X/k}^2$ factors through the image of $D \otimes \Omega_{X/k}^1 \to \Omega_{X/k}^2$. Let $U \subset X$ be a dense open on which $D$ is a sub-bundle of rank $q$ of $\Omega_{X/k}^1$, and $Z$ its reduced closed complement. Let us fix an $S$-structure on the derived foliation defined by $D|_U \subset \Omega_{U/k}^1$ (by Proposition 7.2.0.3).

Then, for any homogeneous polynomial $P \in \mathbb{Z}[c_1, \dots, c_q]$ of degree $d < -q$ (with $\deg(c_i) = -2i$), there exists a canonical element in local cohomology supported on $Z$

$$ \mathrm{Res}_{D,P} \in H_{Z,\mathrm{crys}}^{-d}(X/S) $$

whose image in $H_{crys}^{-d}(X/S)$ coincide with $P(c_1(D), \dots, c_q(D))$.

Note that the residue $\mathrm{Res}_{D,P}$ does depend on the choice of the $S$-structure on the foliation defined by the differential ideal $D|_U$. Note also that when $U = X$, that is when $D$ is a sub-bundle of rank $q$ in $\Omega_{X/k}^1$ then we find that $P(c_1(D), \dots, c_q(D))$ vanishes in crystalline cohomology $H_{crys}^d(X/S)$. Without the condition on the existence of an $S$-structure, we would only have that this class is divisible by $p$.

Finally, we note that if both $X$ and $D$ lifts to characteristic zero, then the existence of the $S$-structure is automatic (and the lift provides a canonical one). However, $S$-structures can exist without $X$ being liftable to $S$. For instance, the tautological foliation $\ast_{X/S} \in \text{Fol}(X/s)$ always possesses a canonical lift, namely $\ast_{X/S}$, even if $X$ does not itself lift to $S$. In a way, the existence of an $S$-structure should be thought as the existence of a lift of an hypothetical leaf space of the foliation from $s$ to $S$.
## 7.6. Infinitesimal derived foliations

In this section we present an alternative notion of derived foliations in arbitrary characteristics. In characteristic zero this notion is equivalent to our original notion of derived foliations, by means of the famous red shift self equivalence of the $\infty$-category of graded mixed complexes. However, outside of characteristic zero this notion is different from the one presented previously in this chapter (Definition 7.2.0.1). We call this alternative notion infinitesimal derived foliations, as its relation with infinitesimal cohomology is similar to the relation between derived foliation and de Rham cohomology. It is also very closely related to the notion of partition Lie algebras and Lie algebras and we refer to [Fu12] for a precise comparison statement.

We will show that any infinitesimal derived foliation possesses an underlying derived foliation in the sense of 7.2.0.1. It is important to understand that infinitesimal derived foliations are somehow more restricted and could be thought as derived foliations together with some extra structures/properties. For instance, infinitesimal derived foliations should also be closely related to p-restricted foliations of [Eke87].

We start by considering a derived group stack $\mathcal{H}_\pi$, which will be playing a similar role than the group stack $\mathcal{H}$ considered in Section 7.1. For this we consider $\mathbb{G}_a[-1] := \Omega_0\mathbb{G}_a$, the derived group scheme (over $k$) defined as the derived loop space of the additive group $\mathbb{G}_a$ taken at 0. By definition, we have

$$ \mathbb{G}_a[-1] \simeq \mathbf{Spec} k \times_{\mathbb{G}_a} \mathbf{Spec} k \simeq \mathbf{Spec} k \oplus k[1], $$

where $k \oplus k[1]$ is the simplicial commutative ring obtained as the trivial square zero extension of $k$ by $k[1]$. The natural $\mathbb{G}_m$-action on $\mathbb{G}_a$, which by our convention is of weight $-1$ (see right before Definition 1.5.0.1), induces an action of $\mathbb{G}_m$ on the derived group scheme $\mathbb{G}_a[-1]$. Algebraically, this $\mathbb{G}_m$ action on $\mathcal{O}(\mathbb{G}_a[-1]) = k \oplus k[1]$ is such that $k[1]$ is pure of weight 1.

**DEFINITION 7.6.0.1.** The derived group scheme $\mathcal{H}_\pi$ is defined to be the semi-direct product of $\mathbb{G}_a[-1]$ by $\mathbb{G}_m$

$$ \mathcal{H}_\pi := \mathbb{G}_m \times_{\mathbb{G}_a} \mathbb{G}_a[-1]. $$

According to the above definition, the underlying derived scheme of $\mathcal{H}_\pi$ is $\mathbf{Spec}(k \oplus k[1])[t, t^{-1}]$, i.e. the spectrum of the Laurent polynomials ring with coefficients in $k \oplus k[1]$. The group structure on $\mathcal{H}_\pi$ provides $(k \oplus k[1])[t, t^{-1}]$ with a structure of a cogroup object inside the $\infty$-category of simplicial commutative $k$-algebras, or in other words as a $k$-linear commutative Hopf simplicial algebra. If we write $k \oplus k[1]$ as $k[\epsilon]$, with $\epsilon$ a free generator in homotopical degree 1 (so cohomological degree $-1$), this makes $k[\epsilon][t, t^{-1}]$ into a simplicial commutative Hopf algebra for the usual twisted comultiplication

$$ \Delta(\epsilon) = \epsilon \otimes 1 + t \otimes \epsilon \qquad \Delta(t) = t \otimes t. $$

Similarly to Proposition 7.1.0.10, we have the following result, identifying graded mixed complexes with representations of the group $\mathcal{H}_\pi$.
**PROPOSITON 7.6.0.2.** There exists a natural equivalence of k-linear symmetric monoidal $\infty$-categories

$$ \epsilon - \mathrm{dg}_k^{gr} \simeq \mathrm{QCoh}(B\mathcal{H}_\pi). $$

*Proof.* The proof is similar to [MRT22, Prop. 4.2.3 (ii)], and in fact easier as the group $\mathcal{H}_\pi$ is an affine derived group scheme, which simplifies some of the arguments. We start by writing $B\mathcal{H}_\pi$ as the standard geometric realization of the simplicial object $[n] \mapsto \mathcal{H}_\pi^n$. By descent we have a canonical equivalence of symmetric monoidal $\infty$-categories

$$ \mathrm{QCoh}(B\mathcal{H}_\pi) \simeq \lim_n \mathcal{O}(\mathcal{H}_\pi)^{\otimes n} - \mathrm{dg}_k. $$

We observe that the simplicial object $[n] \to \mathcal{H}_\pi^n$ is the nerve of the zero section $\mathrm{Spec} k \to [\mathbb{A}^1/\mathbb{G}_m]$, so we have, by pull-back, a symmetric monoidal $\infty$-functor

$$ \mathrm{QCoh}([\mathbb{A}^1/\mathbb{G}_m]) \to \lim_n \mathcal{O}(\mathcal{H}_\pi)^{\otimes n} - \mathrm{dg}_k \simeq \mathrm{QCoh}(B\mathcal{H}_\pi). $$

By the comparison between graded mixed complexes and complete filtered complexes (see Corollary 1.3.4.4) the Tate realization provides a lax symmetric monoidal full embedding

$$ \epsilon - \mathrm{dg}_k^{gr} \hookrightarrow \mathrm{QCoh}([\mathbb{A}^1/\mathbb{G}_m]) $$

and thus by composition a lax symmetric monoidal $\infty$-functor

$$ \psi : \epsilon - \mathrm{dg}_k^{gr} \to \mathrm{QCoh}(B\mathcal{H}_\pi). $$

We now prove that $\psi$ is a symmetric equivalence. For this, we start by noticing that the lax symmetric monoidal structure of $\psi$ is in fact strict. For this we consider the natural projection $\mathcal{H}_\pi \to \mathbb{G}_m$, and the induced morphism $p : B\mathcal{H}_\pi \to B\mathbb{G}_m$ on classifying derived stacks. The composition of $\psi$ by the push-forward $p_* : \epsilon - \mathrm{dg}_k^{gr} \to \mathrm{QCoh}(B\mathbb{G}_m) \simeq \mathrm{dg}_k^{gr}$, is easily seen to be equivalent to the $\infty$-functor sending a graded mixed objects $E$ to the graded object $E^{(*)}[-2*]$. In particular, its lax symmetric monoidal structure is strict. Moreover, as $p_* : \mathrm{QCoh}(B\mathcal{H}_\pi) \to \mathrm{QCoh}(B\mathbb{G}_m)$ is clearly conservative (because $p$ is induced by an affine morphism), we deduce that the lax symmetric monoidal structure on $\psi$ is a symmetric monoidal structure.

It remains to show that the $\infty$-functor underlying $\psi$ is an equivalence of $\infty$-categories. For this we come back to the situation

$$ \epsilon - \mathrm{dg}_k^{gr} \hookrightarrow \mathrm{QCoh}([\mathbb{A}^1/\mathbb{G}_m]) \to \mathrm{QCoh}(B\mathcal{H}_\pi). $$

By Corollary 1.3.4.4, the image of the first full embedding is the full sub-$\infty$-category of objects which are complete at 0. We thus have to show that $\mathrm{QCoh}([\mathbb{A}^1/\mathbb{G}_m]) \to \mathrm{QCoh}(B\mathcal{H}_\pi)$ identifies the full sub-$\infty$-category of complete modules with $\mathrm{QCoh}(B\mathcal{H}_\pi)$. This follows easily from the fact that $B\mathcal{H}_\pi$ is the realization of the nerve of the zero section $B\mathbb{G}_m \hookrightarrow [\mathbb{A}^1/\mathbb{G}_m]$ as shown by the following lemma.
LEMMA 7.6.0.3. Let $A$ be a $k$-algebra and $f \in A$ be a non-zero divisor. Let $B^*$ be the cosimplicial $E_\infty$-algebra given by the derived conerve of $A \to A/f$

$$B^n := A/f \otimes_A A/f \otimes_A \cdots \otimes_A A/f.$$

Then the descent $\infty$-functor

$$\mathbf{dg}_A \to \lim_n \mathbf{dg}_{B^n}$$

induces an equivalence

$$\hat{\mathbf{dg}}_{A,f} \simeq \lim_n \mathbf{dg}_{B^n}$$

where $\hat{\mathbf{dg}}_A, f$ is the $\infty$-category of $f$-complete $A$-dg-modules (E such that $E \simeq \lim_n(E/f^n)$).

*Proof of the lemma.* The descent $\infty$-functor $\phi: \mathbf{dg}_A \to \lim_n \mathbf{dg}_{B^n}$ sends an $A$-dg-module to the pull-back $E \otimes_A B^*$. It has a right adjoint $\psi$ sending $E^* \in \lim_n \mathbf{dg}_{B^n}$ to $\lim_n E^n \in \mathbf{dg}_A$. We can form the cocartesian square of commutative cosimplicial $E_\infty$-algebras

<figure><img src="image_52.png" /></figure>

Then, $A/f \to C^*$ is the derived conerve of the morphism $A/f \to A/f \otimes_A A/f$ which possesses a retraction given by the multiplication in $A/f$. Therefore, the cosimplicial object $C^*$ is simplicially contractible. This implies that the descent $\infty$-functor

$$\mathbf{dg}_{A/f} \to \lim_n (\mathbf{dg}_{C^n})$$

is in fact an equivalence of $\infty$-categories. We consider now the commutative diagram of $\infty$-categories obtained by base change from the above diagram

<figure><img src="image_53.png" /></figure>

In this diagram the vertical $\infty$-functors are conservative and commutes with limits. Indeed, $A/f$ is a perfect $A$-dg-module and thus $(A/f) \otimes_A -$ commutes with limits. Similarly, each $C^n$ is perfect over $B^n$. As a result, the diagram is right adjointable in the sense of [Lur22a, Def. 4.7.4.13], because of the explicit description of the right adjoints of the horizontal $\infty$-functors in terms of limits. As the bottom horizontal $\infty$-functor is an equivalence, so is the top horizontal $\infty$-functor, because the unit of counit of the top adjunction can be checked to be equivalences after applying the vertical conservative $\infty$-functors (so that they become the unit and counit of the bottom adjunction). $\square$
We can apply the previous lemma to the natural simplicial objects representing $BG_m \hookrightarrow [A^1/G_m]$. This finishes the proof of the proposition. □

In parallel to the notion of graded loop space we introduce the infinitesimal graded loop space as follows.

**DEFINITION 7.6.0.4.** Let $X$ be a derived stack. The infinitesimal graded loop space of $X$ is defined by

$$ \mathcal{L}_{\pi}^{gr}(X) := \underline{\mathrm{Map}}(\mathbb{G}_a[-1], X) \in \mathrm{dSt}. $$

If $X \to Y$ is a morphism of derived stacks, the relative infinitesimal graded loop space of $X$ over $Y$ is defined by

$$ \mathcal{L}_{\pi}^{gr}(X/Y) := \mathcal{L}_{\pi}^{gr}(X) \times_{\mathcal{L}_{\pi}^{gr}(Y)} Y, $$

for the natural morphism $Y \to \mathcal{L}_{\pi}^{gr}(Y)$ corresponding to the projection $Y \times \mathbb{G}_a[-1] \to Y$.

The group $G_m$ acts on $\mathbb{G}_a[-1]$, and the group $\mathbb{G}_a[-1]$ acts on itself by translation. Therefore we get a natural action of $\mathcal{H}_{\pi} = G_m \times G_a[-1]$ on the stack $\mathbb{G}_a[-1]$, and thus on the infinitesimal graded loop space of any derived stack. The object $\mathcal{L}_{\pi}^{gr}(X)$ will always be considered as an object in the $\infty$-topos $\mathrm{dSt}^{\mathcal{H}_{\pi}}$ of derived stacks endowed with a $\mathcal{H}_{\pi}$-action (and $\mathcal{H}_{\pi}$-equivariant maps).

The following proposition identifies $\mathcal{L}_{\pi}^{gr}(X)$ with its $G_m$-action when $X$ is a derived Artin stack. For this we consider the canonical point $\ast \to \mathbb{G}_a[-1]$ given by the unit, and its induced projection $\mathcal{L}_{\pi}^{gr}(X) \to X$. Note that this projection is $G_m$-equivariant, but is not $\mathbb{G}_a[-1]$ equivariant as the $\mathbb{G}_a[-1]$-action by translation does not fix the base point. The derived stack $\mathcal{L}_{\pi}^{gr}(X) \to X$ thus lives in the $\infty$-topos $\mathrm{dSt}_{\mathbb{X}}^{\mathbb{G}_m}$ of $G_m$-equivariant objects over $X$. Recall from [Mon21] that this category contains as a full sub-$\infty$-category the $\infty$-category of derived linear stacks over $X$, which is equivalent to $\mathrm{QCoh}^{-}(X)$ of eventually coconnective quasi-coherent complexes on $X$. The full-embedding $\mathrm{QCoh}^{-}(X) \hookrightarrow \mathrm{dSt}_{\mathbb{X}}^{\mathbb{G}_m}$ is given by the construction $\mathbb{V}$, sending $E \in \mathrm{QCoh}^{-}(X)$ to $\mathbb{V}(E) \to X$ to the $\infty$-functor sending an affine $u : \mathrm{Spec} A \to X$ to the space $\mathrm{Map}_{\mathbb{dG}_A}(u^*(E), A)$ (see Definition A.9.2.4).

**PROPOSITON 7.6.0.5.** Let $X$ be a derived Artin stack. The $G_m$-equivariant derived stack $\mathcal{L}_{\pi}^{gr}(X) \to X$ is canonically equivalent to the linear stack $\mathbb{V}(\mathbb{L}_X[-1]) \to X$, associated to the quasi-coherent complex $\mathbb{L}_X[-1]$.

*Proof.* As $\mathbb{G}_a[-1] = \mathrm{Spec} k \oplus k[1]$, this is simply the definition of $\mathbb{L}_X$ in terms of points with values in square zero extensions (see [TV08, Def. 1.4.1.5]). □

The terminology infinitesimal graded loop space has been chosen because of the theorem below, expressing that $\mathcal{H}_{\pi}$-equivariant functions on $\mathcal{L}_{\pi}^{gr}(X)$ computes infinitesimal cohomology of $X$, at least in the smooth case. Recall that if $X \to S = \mathrm{Spec} k$ is a smooth morphism of schemes, we can define $(X/S)_{\mathrm{inf}}$ as the functor sending an affine $\mathrm{Spec} A \to S$ to $X(A_{\mathrm{red}})$,
where $A_{red}$ is defined to be the discrete ring $\pi_0(A)_{red}$. The functor $(X/S)_{inf}$ is a derived stack over $S$, and by definition infinitesimal cohomology of $X$ relative to $S$ is defined by

$$C_{inf}^{*}(X/S) := C^{*}((X/S)_{inf}, \mathcal{O}),$$

which will be considered as a k-linear $E_{\infty}$-algebra. Because $X$ is smooth over $S$, the canonical morphism $X \to (X/S)_{inf}$ is an epimorphism of derived stacks (because of the infinitesimal lifting property). Therefore, $H_{inf}^{*}(X/S)$ can also be computed as the limit

$$C_{inf}^{*}(X/S) \simeq \lim_{n}(C^{*}(X^{(n)}, \mathcal{O})),$$

where $X^{(*)}$ is the nerve of $X$ over $(X/S)_{inf}$, which can also be described as the nerve of the formal groupoid $X^{(2)} \simeq X$ where $X^{(2)}$ is the formal completion of $X \times_S X$ along the diagonal.

In order to state the theorem below, let us consider the quotient map

$$p : [\mathcal{L}_{\pi}^{gr}(X/S)/\mathcal{H}_{\pi}] \to S$$

The direct image of the structure sheaf $p_{*}(\mathcal{O})$ is by definition the $\mathcal{H}_{\pi}$-equivariant cohomology of $\mathcal{L}_{\pi}^{gr}(X/S)$ with coefficients in the structure sheaf $\mathcal{O}$. It will be symbolically denoted by $C^{*}(\mathcal{L}_{\pi}^{gr}(X/S))^{\mathcal{H}_{\pi}}$.

We produce a natural morphism $C_{inf}^{*}(X/S) \to C^{*}(\mathcal{L}_{\pi}^{gr}(X/S))^{\mathcal{H}_{\pi}}$ as follows. The natural morphism $X \to (X/S)_{inf}$ induces a morphism on infinitesimal graded loop spaces $\mathcal{L}_{\pi}^{gr}(X/S) \to \mathcal{L}_{\pi}^{gr}((X/S)_{inf}/S)$. By the definition of $(X/S)_{inf}$, and from the fact that the reduced group $\mathcal{H}_{\pi}$ is trivial, it is easy to see that the constant map morphism $(X/S)_{inf} \to \mathcal{L}_{\pi}^{gr}((X/S)_{inf}/S)$ is an equivalence. In particular, we get this way a natural morphism of derived stacks with $\mathcal{H}_{\pi}$-actions $\mathcal{L}_{\pi}^{gr}(X/S) \to (X/S)_{inf}$. By considering cohomology with coefficients in $\mathcal{O}$ we finally get the required morphism

$$\phi_{X/S} : C_{inf}^{*}(X/S) \to C^{*}(\mathcal{L}_{\pi}^{gr}(X/S))^{\mathcal{H}_{\pi}}.$$

By construction it is a morphism of $E_{\infty}$-algebras functorial in $X/S$. Note that the above morphism always exists, for a general morphism of derived stacks $X \to S$ without assuming any extra conditions.

**THEOREM 7.6.0.6.** Let $X \to S = \text{Spec } k$ be a smooth scheme. Then, the above morphism

$$\phi_{X/S} : C_{inf}^{*}(X/S) \simeq C^{*}(\mathcal{L}_{\pi}^{gr}(X/S))^{\mathcal{H}_{\pi}}$$

is an equivalence.

*Proof.* We consider the commutative diagram of derived stacks with $\mathcal{H}_{\pi}$-actions

<figure><img src="image_54.png" /></figure>
This diagram induces a morphism of simplicial objects on the corresponding nerves of the vertical morphisms. The nerve of the morphism $X \to \mathcal{L}_{\pi}^{gr}(X/S)$ can be described by the simplicial object $[n] \mapsto \mathcal{L}_{\pi}^{gr}(X/X^n)$, where $X$ sits inside $X^n$ by the diagonal embedding. At the simplicial level $n$, the morpshim induced on the nerve is thus an $\mathcal{H}_{\pi}$-equivariant morphism of stacks over $S$

$$ \psi_{X,n} : \mathcal{L}_{\pi}^{gr}(X/X^n) \to X^{(n)}, $$

where $X^{(n)}$ is the formal completion of $X^n$ along its diagonal.

**LEMMA 7.6.0.7.** *The morphism $\psi_{X,n}$ induces by pull-backs an equivalence of $E_{\infty}$-algebras*

$$ C^*(X^{(n)}, \mathcal{O}) \simeq C^*(\mathcal{L}_{\pi}^{gr}(X/X^n))^{\mathcal{H}_{\pi}}. $$

*Proof Lemma 7.6.0.7.* By descent we can easily restrict to the case where $X = \mathbf{Spec} A$ is a smooth affine scheme over $S = \mathbf{Spec} k$. In this case, $C^*(X^{(n)}, \mathcal{O})$ identifies canonically with $\hat{A}^{\otimes k^n}$, the formal completion of $A^{\otimes k^n}$ along the canonical augmentation $A^{\otimes k^n} \to A$. Similarly, $\mathcal{L}_{\pi}^{gr}(X/X^n)$ is canonically identified with the smooth affine scheme $\mathbf{Spec} \mathrm{Sym}_A((\Omega_{A/k}^1)^{n-1})$. Therefore, the morphism $\psi_{X,n}$ is a morphism of discrete commutative algebras

$$ \psi_{A,n} : \hat{A}^{\otimes k^n} \to |\mathrm{Sym}_A((\Omega_{A/k}^1)^{n-1})|^t, $$

where the right hand side is the Tate realization of the graded mixed object $\mathrm{Sym}_A((\Omega_{A/k}^1)^{n-1})$ induced by the $\mathcal{H}_{\pi}$-action on $\mathcal{L}_{\pi}^{gr}(X/X^n)$. The morphism $\psi_{A,n}$ is compatible with the canonical augmentations to $A$, and thus is a morphism of complete filtered commutative algebras. Therefore, in order to prove it is an isomorphism it is enough to show that it induces an isomorphism on the associated graded objects. This induced morphism is a morphism

$$ \mathrm{gr}(\psi_{A,n}) : \mathrm{Gr}^*(\hat{A}^{\otimes k^n}) \to \mathrm{Sym}_A((\Omega_{A/k}^1)^{n-1}). $$

Moreover, because $A$ is a smooth $k$-algebra, the left hand side is canonically isomorphic to $\mathrm{Sym}_A(I_n/I_n^2)$, where $I_n$ is the augmentation ideal. Finally, we have the usual natural isomorphism of $A$-modules $I_n/I_n^2 \simeq (\Omega_{A/k}^1)^{n-1}$, and thus the morphism $\mathrm{gr}\psi_{A,n}$ can be identified with a graded endomorphism of the graded algebra $\mathrm{Sym}_A((\Omega_{A/k}^1)^{n-1})$. As it is compatible with augmentations to $A$ it is thus determined by an $A$-linear endomorphism of $(\Omega_{A/k}^1)^{n-1}$. By functoriality in $A$ and in the simplicial direction $n$, we see moreover that it is determined by a functorial endomorphism the $A$-module $\Omega_{A/k}^1$, and thus must be the multiplication by an element $\lambda \in k$. To conclude the lemma, we have to show that $\lambda = 1$.

By functoriality, it is easy to reduce to the case $A = k[T]$ is a polynomial ring, and by base change we can even assume that $k = \mathbb{Z}$. In order to check that $\lambda = 1$ we can base change from $\mathbb{Z}$ to $\mathbb{Q}$, and thus reduce to the case $A = \mathbb{Q}[T]$ and $k = \mathbb{Q}$. As we are now in characteristic zero, $C^*(\mathcal{L}_{\pi}^{gr}(X/X^n))^{\mathcal{H}_{\pi}}$ identifies canonically with the completed de Rham complex $\hat{C}_{DR}^*(X/X^n)$. Indeed, the red shift equivalence sends the graded mixed cdga $\mathcal{O}(\mathcal{L}_{\pi}^{gr}(X/X^n))$ to $\mathcal{O}(\mathcal{L}^{gr}(X/X^n))$. Finally, the morphism $\psi_{X,n}$ is now the natural isomorphism relating derived de Rham cohomology of a closed embedding with functions on the formal completion (see for
instance Proposition 1.4.2.3).

The lemma implies the theorem by descent along the epimorphisms $X \to \mathcal{L}_{\pi}^{gr}(X/S)$ and $X \to (X/S)_{inf}$. $\square$

**7.6.1. Infinitesimal derived foliations.** As before, we fix $k$ a base simplicial commutative ring.

**DEFINITION 7.6.1.1.** Let $X = \mathbf{Spec} A$ be a derived affine scheme over $k$. $A$ (perfect) infinitesimal derived foliation $\mathcal{F}$ on $X$ (relative to $k$) consists of a $\mathcal{H}_{\pi}$-equivariant derived stack $\mathcal{V}(\mathcal{F})$ together with a $\mathcal{H}_{\pi}$-equivariant morphism $p : \mathcal{V}(\mathcal{F}) \to \mathcal{L}_{\pi}^{gr}(X/k)$, such that, as a $\mathbb{G}_{m}$-equivariant stack over $X$, $\mathcal{V}(\mathcal{F})$ is a perfect linear derived affine stack (i.e. of the form $\mathcal{V}(E)$ for $E$ a perfect complex on $X$).

The infinitesimal derived foliations on $X$ (relative to $k$) form an $\infty$-category $\mathcal{Fol}_{\pi}(X/k)$, which by definition is a full sub-$\infty$-category of $(\mathbf{dSt}_{k}^{H_{\pi}})/\mathcal{L}_{\pi}^{gr}(X/k)$, the $\mathcal{H}_{\pi}$-equivariant derived stacks over $\mathcal{L}_{\pi}^{gr}(X/k)$. The construction $X \mapsto \mathcal{Fol}_{\pi}(X/k)$ can be promoted to an $\infty$-functor $\mathcal{Fol}_{\pi}(-/k) : \mathbf{dAff}_{k}^{op} \to \mathbf{Cat}_{\infty}$, by defining pull-backs as follows: for a morphism $f : X \to Y$ of affine derived $k$-schemes, we define $f^{*}$ to be induced by the base change

$$ (\mathbf{dSt}_{k}^{H_{\pi}})/\mathcal{L}_{\pi}^{gr}(Y/k) \to (\mathbf{dSt}_{k}^{H_{\pi}})/\mathcal{L}_{\pi}^{gr}(X/k), $$

along the induced morphism $\mathcal{L}_{\pi}^{gr}(f/k) : \mathcal{L}_{\pi}^{gr}(X/k) \to \mathcal{L}_{\pi}^{gr}(Y/k)$. As the $\infty$-functor $\mathcal{Fol}_{\pi}(-/k)$ clearly is a hyperstack for the fpqc topology, it can be uniquely extended to a colimit preserving $\infty$-functor

$$ \mathcal{Fol}_{\pi}(-/k) : \mathbf{dSt}_{k} \to \mathbf{Cat}_{\infty}^{op}. $$

Let $X$ be a derived affine $k$-scheme, and $\mathcal{F} \in \mathcal{Fol}_{\pi}(X/k)$ be an infinitesimal derived foliation on $X$. By [Mon21], the linear stack $\mathcal{V}(\mathcal{F}) \to X$ is of the form $\mathcal{V}(\mathbb{L}_{\mathcal{F}/k}[-1])$, for some uniquely defined perfect complex $\mathbb{L}_{\mathcal{F}/k}$ on $X$.

**DEFINITION 7.6.1.2.** With the notations above, the complex $\mathbb{L}_{\mathcal{F}/k}$ is called the cotangent complex of $\mathcal{F}$ (relative to $k$). It is also simply denoted by $\mathbb{L}_{\mathcal{F}}$ if $k$ is clear from the context.

When $X$ is a general derived stack and $\mathcal{F} \in \mathcal{Fol}_{\pi}(X/k)$, it is also possible to define the cotangent complex $\mathbb{L}_{\mathcal{F}/k}$, as a quasi-coherent complex over $X$ following the same lines as Definition 2.1.3.4. We have first a big cotangent complex $\mathbb{L}_{\mathcal{F}}^{big}$, which is an $\mathcal{O}_{X}$-dg-module over the big site of derived affine schemes over $X$: it sends $u : \mathbf{Spec} A \to X$ to the $A$-dg-module $\mathbb{L}_{u^{*}(\mathcal{F})/k}$. This $\mathcal{O}_{X}$-dg-module is not quasi-coherent and we thus apply the quasi-coherator construction of §A.4 to get the desired cotangent complex

$$ \mathbb{L}_{\mathcal{F}/k} := (\mathbb{L}_{\mathcal{F}/k}^{big})^{qcoh}. $$
The special case where $X$ is a derived Deligne-Mumford stack simplifies, as the cotangent complexes of infinitesimal foliations are stable by étale pull-backs, and thus will glue globally on $X$. This applies in particular to the case where $X$ is any derived scheme. In these cases, we have the natural morphism $\mathbb{L}_{\mathcal{F}/k}^{\text{big}} \to \mathbb{L}_{\mathcal{F}/k}$ restricts to an equivalence on the small étale site of $X$.

Let $X$ be a derived affine scheme of finite presentation over $k$. The $\infty$-category $\mathcal{F}ol_{\pi}(X/k)$ admits a final object, namely the identity morphism $\mathcal{L}_{\pi}^{gr}(X/k) \to \mathcal{L}_{\pi}^{gr}(X/k)$. It is denoted by $*_X/k$, and its cotangent complex is naturally equivalent to $\mathbb{L}_{X/k}$, the cotangent complex of $X$. Indeed, it is easy to see, simply by contemplating the functors of points and using the very definition of the cotangent complex, that $\mathbf{Map}(G_0, X) \simeq \mathbf{V}(\mathbb{L}_{X/k}[-1])$. Similarly, the initial object in $\mathcal{F}ol_{\pi}(X/k)$ exists, is denoted by $0_{X/k}$, and its cotangent complex is 0. It corresponds to the constant-map morphism $\mathbf{V}(0_{X/k}) = X \to \mathcal{L}_{\pi}^{gr}(X/k)$.

**7.6.2. Infinitesimal cohomology of infinitesimal derived foliations.** Let $X$ be a derived Artin stack of finite presentation over $k$, and $\mathcal{F} \in \mathcal{F}ol_{\pi}(X/k)$ be an infinitesimal derived foliation on $X$. We define the $\infty$-category of quasi-coherent complexes along $\mathcal{F}$ as follows. We first consider $\mathbf{QCoh}([\mathbf{V}(\mathcal{F})/\mathcal{H}_{\pi}])$, the $\infty$-category of quasi-coherent complexes on the quotient stack $[\mathbf{V}(\mathcal{F})/\mathcal{H}_{\pi}]$ (which is also, by definition, the $\infty$-category of $\mathcal{H}_{\pi}$-equivariant quasi-coherent complexes on $\mathbf{V}(\mathcal{F})$). The $\infty$-category $\mathbf{QCoh}(\mathcal{F})$ is defined as the full sub-$\infty$-category of $\mathbf{QCoh}([\mathbf{V}(\mathcal{F})/\mathcal{H}_{\pi}])$ whose objects $E$ are graded free: the pull-back of $E$ on $[\mathbf{V}(\mathcal{F})/\mathcal{G}_m]$ is of the form $p^*(E_0)$ where $p: [\mathbf{V}(\mathcal{F})/\mathcal{G}_m] \to X$ is the natural projection.

In the definition above we use the identification $\mathbf{QCoh}(B\mathcal{H}_{\pi})$ with the $\infty$-category of graded mixed complexes (see Proposition 7.6.0.2).

**DEFINITION 7.6.2.1.** With the notations above, the $\infty$-category of quasi-coherent complexes along $\mathcal{F}$ is $\mathbf{QCoh}(\mathcal{F})$. For $E \in \mathbf{QCoh}^{\mathcal{F}}(X)$, the (derived and Hodge completed) the infinitesimal cohomology of $X$ along $\mathcal{F}$ with coefficients in $E$ is defined to be

$$ \hat{C}_{\text{inf}}^{*}(\mathcal{F}, E) := |q_{*}(E)|^{t} \in \hat{\mathbf{d}}_{k}^{\text{fil}} $$
where $q: [\mathbf{V}(\mathcal{F})/\mathcal{H}_{\pi}] \to B\mathcal{H}_{\pi}$ is the structural map.

The definition above is justified by our Theorem 7.6.0.6, stating that when $X$ is smooth over $k$, and $\mathcal{F} =*_X/k$ and $E = \mathcal{O}_X$, $\hat{C}_{\text{inf}}^*(\mathcal{F}, E)$ coincides with infinitesimal cohomology defined using the usual infinitesimal site. For a general derived Artin stack $X$, $\hat{C}_{\text{inf}}(*_{X/k}, \mathcal{O})$ computes the completed derived infinitesimal cohomology of $X$ (relative to $k$). Indeed, it is easy to see the $\infty$-functor $X \mapsto \mathcal{O}(\mathcal{L}_{\pi}^{gr}(X/k))$, from affine derived schemes to graded mixed complexes, is equivalent to its extension by sifted colimits from smooth $X$'s. For $A \in \mathbf{sCR}$, where each $A_n$ is a smooth algebra over $k$, we thus have

$$ \hat{C}_{\text{inf}}(*_X, \mathcal{O}_X) \simeq \text{colim}_{[n]} C^*((\text{Spec } A_n)_{\text{inf}}, \mathcal{O}), $$

where the colimit is computed in the $\infty$-category of complete filtered complexes. The right hand side of this equivalence is, by definition, the completed derived infinitesimal cohomology
of $X$, and is denoted by $\mathbb{R}\hat{C}^*(X_{\inf}, \mathcal{O})$. This notion is extended to any derived Artin stack locally of finite presentation over $k$ by the usual gluing formula

$$ \mathbb{R}\hat{C}^*(X_{\inf}, \mathcal{O}) := \lim_{U \to X} \mathbb{R}\hat{C}^*(U_{\inf}, \mathcal{O}) $$

where the limit runs over affine $U$ locally of finite presentation.

**COROLLARY 7.6.2.2.** For a general derived Artin stack $X$ of finite presentation over $k$, $\hat{C}_{\inf}(*_{X/k}, \mathcal{O})$ recovers the Hodge completed derived infinitesimal cohomology of $X$ relative to $k$

$$ \hat{C}_{\inf}(*_{X/k}, \mathcal{O}) \simeq \mathbb{R}\hat{C}^*(X_{\inf}, \mathcal{O}). $$

**7.6.3. Integrability by formal groupoids.** As opposed to derived foliations, infinitesimal derived foliations turn out to be formally integrable in a very strong sense. The most general integrability statements can be found in [BMN25]. Below, we prove a special easy case by elementary means, showing that the $\infty$-category of smooth infinitesimal derived foliations over a $k$-scheme $X$ is in fact equivalent to the category of formally smooth formal groupoids over $X$.

Let $X$ be a smooth $k$-scheme and $\mathcal{F} \in \mathcal{F}ol_{\pi}(X/k)$. We say that $\mathcal{F}$ is smooth if $\mathbb{L}_{\mathcal{F}}$ is a vector bundle over $X$. In this case, we consider $X \times_{\mathbb{V}(\mathcal{F})} X$, endowed with its natural $\mathcal{H}_{\pi}$-action and with its first projection to $X$. Note that, as a graded derived stack, $X \times_{\mathbb{V}(\mathcal{F})} X$ is of the form $\mathbb{V}(\mathbb{L}_{\mathcal{F}})$, and thus is a the total space of the tangent bundle $\mathbb{T}_{\mathcal{F}}$ of $\mathcal{F}$. Also, the natural projection $X \times_{\mathbb{V}(\mathcal{F})} X \to X$ is $\mathcal{H}_{\pi}$-equivariant. As a result, $X \times_{\mathbb{V}(\mathcal{F})} X$ is a new object in $\mathcal{F}ol_{\pi}(X)$, that will be simply denoted by $\Omega_X \mathcal{F}$, and called the loop space of $\mathcal{F}$. Being the fiber product of the canonical morphism $0_X \to \mathcal{F}$, the loop space $\Omega_X \mathcal{F}$ comes equipped with a natural structure of a groupoid object in $\mathcal{F}ol_{\pi}(X)$, whose object of objects is $0_X \in \mathcal{F}ol_{\pi}(X/k)$. Note that as $0_X$ is not the final object $\Omega_X \mathcal{F}$ is not a group object and merely a groupoid.

**PROPOSITON 7.6.3.1.** The full sub-$\infty$-category of $\mathcal{F}ol_{\pi}(X/k)$ formed by objects $\mathcal{F}$ whose cotangent complex is of the form $V[1]$ for $V$ a vector bundle on $X$, is naturally equivalent to the category of formal schemes $Z$ with an identification $Z_{red} \simeq X$, which are locally equivalent on $X$ to $X \times \hat{\mathbb{A}}^d$ where $d = \mathrm{rank}(V)$.

*Proof.* To each $\mathcal{F}$ as in the Proposition, we associate the sheaf of infinitesimal cohomology $\hat{C}_{\inf}(\mathcal{F})$ on $X$. By assumptions on the cotangent complex of $\mathcal{F}$, $\hat{C}_{\inf}(\mathcal{F})$ is a sheaf of discrete complete filtered commutative algebras, augmented to $\mathcal{O}_X$ and its associated graded object is isomorphic to $\mathcal{O}_X[[t_1, \dots, t_d]]$. Therefore, $Z_{\mathcal{F}} := \mathrm{Spf}(\hat{C}_{\inf}(\mathcal{F}))$ is a formal scheme with a canonical identification $Z_{red} \simeq X$ which is locally equivalent to $X \times \hat{\mathbb{A}}^d$.

The fact that the construction $\mathcal{F} \mapsto Z_{\mathcal{F}}$ induces the equivalence of the Proposition simply follows from the fact that the Tate realization induces an equivalence between graded mixed complexes and complete filtered complexes (see [TV20, Prop. 1.3.1]). $\square$
By Proposition A.9.2.1, for any smooth infinitesimal derived foliation $\mathcal{F}$, its loop space $\Omega_X \mathcal{F}$ can be realized as a groupoid object in formal schemes acting on $X$

$$Z_{\Omega_X \mathcal{F}} \to X \times X.$$

The construction $\mathcal{F} \mapsto Z_{\Omega_X \mathcal{F}}$ provides a functor from the category of smooth infinitesimal derived foliations on $X$, and the category of formally smooth formal groupoids over $X$. We can produce a functor in the other direction, as follows. Starting from a formally smooth formal groupoid $G \to X \times X$, we can consider its nerve $G_* \to X$, which is a simplicial object in formal schemes. Taking functions on $G_*$ provides a cosimplicial complete filtered commutative algebra $\mathcal{O}(G_*)$ whose associated graded is the cosimplicial algebra of functions on the simplicial scheme $BV$, where $V = e^*(\Omega_{G/X}^1)$ is the vector bundle of invariant relative forms on $G$. Using the equivalence between graded mixed complexes and complete filtered complexes of [TV20, Prop. 1.3.1], we find that $\mathcal{O}(G_*)$ can be realized as a cosimplicial object in the category of graded commutative algebras endowed with a compatible action on $G_0$. Passing then to $\text{Spec}$, we get a simplicial diagram of affine schemes endowed with a $\mathcal{H}_\pi$-action, whose underlying simplicial diagram (obtained by forgetting the action) is the usual simplicial scheme $BV$. Taking geometric realization we get a $\mathcal{H}_\pi$-action on the derived stack $\mathcal{V}(V[1])$, which defines a smooth infinitesimal derived foliations $\mathcal{F}$ on $X$.

We can then subsume the previous discussion in the following corollary.

**COROLLARY 7.6.3.2.** *The category of smooth infinitesimal derived foliations on $X$ is equivalent to the category of formally smooth formal groupoids on $X$.*

We note that the above corollary marks a major difference between derived foliations of Section §7.2 and infinitesimal derived foliations. Indeed, any finite dimensional Lie algebra $L$ over a field $k$, defines a smooth derived foliation over $k$, by considering $\text{Sym}(L^\vee[1])$ endowed with the mixed structure coming from the Chevalley-Eilenberg differential. However, it is not true that any Lie algebra can be integrated to a smooth formal scheme in general. On the contrary, the above corollary, when applied to $X = \text{Spec } k$ with $k$ a field, states that smooth infinitesimal derived foliations over $k$ form a category equivalent to smooth formal groups over $k$. Any such smooth infinitesimal derived foliation is of the form $\mathcal{V}(V[-1])$ for $V$ a finite dimensional $k$-vector space $V$. Therefore, the data of an infinitesimal derived foliation structure on $\mathcal{V}(V[-1])$ seems to be related to that of a *partition Lie algebra structure* in the sense of [BCN21]. The precise comparison is out of the scope of this note, but has been carried out in [Fu12].

**7.6.4. Comparison between infinitesimal derived foliations and derived foliations: the "de Rhamification" functor.** In this § we explain how to construct a derived foliation from any infinitesimal derived foliation. This construction will be called the *de Rhamification* functor, as it is related to the natural "forgetful" morphism from (Hodge-completed and derived) infinitesimal cohomology to de Rham cohomology.
We start by $X = \mathbf{Spec} A$ a derived affine scheme of finite presentation over $k$ and $\mathcal{F} \in \mathcal{Fol}_\pi(X)$ an infinitesimal derived foliation. The natural projection $\mathcal{V}(\mathcal{F}) \to X$ possesses a zero section $s: X \to \mathcal{F}$ which is moreover naturally $\mathcal{H}_\pi$-equivariant. We can then consider the relative derived loop space of $s \hat{\mathcal{L}}^{gr}(X/\mathcal{V}(\mathcal{F}))$, which fits in a Cartesian square of $\mathcal{H}_\pi$-equivariant derived stacks

<figure><img src="image_55.png" /></figure>

The group stack $\mathcal{H}$ acts by its standard action on $\hat{\mathcal{L}}^{gr}(X/\mathcal{V}(\mathcal{F}))$, and this action is itself $\mathcal{H}_\pi$-equivariant, or in other words is combined into a natural action of $\mathcal{H} \times \mathcal{H}_\pi$. The derived stack $\hat{\mathcal{L}}^{gr}(X/\mathcal{V}(\mathcal{F}))$ is a priori not affine, but we can consider its affinization (see §A.9)

$$ \hat{\mathcal{L}}^{gr}(X/\mathcal{V}(\mathcal{F}))^{aff} := \mathbb{R}\mathbf{Spec}^{\Delta}(\mathcal{O}(\mathcal{L}^{gr}(X/\mathcal{V}(\mathcal{F}))). $$

Because the group stack $\mathcal{H} \times \mathcal{H}_\pi$ is affine and $\mathcal{O}(\mathcal{H} \times \mathcal{H}_\pi)$ is perfect as a $k$-dg-module, it is easy to check that $\mathcal{H} \times \mathcal{H}_\pi$ continues to act on $\hat{\mathcal{L}}^{gr}(X/\mathcal{V}(\mathcal{F}))^{aff}$ in a natural manner (because the affinization preserves taking products with $\mathcal{H} \times \mathcal{H}_\pi$). Moreover, using the descent theorem 2.1.3.8 we have a canonical identification

$$ \hat{\mathcal{L}}^{gr}(X/\mathcal{V}(\mathcal{F}))^{aff} \simeq \mathbb{R}\mathbf{Spec}^{\Delta}(\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])) = \mathcal{V}(\mathbb{L}_{\mathcal{F}}[1]). $$

We have therefore constructed a natural action of $\mathcal{H} \times \mathcal{H}_\pi$ on $\mathcal{V}(\mathbb{L}_{\mathcal{F}}[1])$. By construction, the induced action of $B\mathbb{G}_m \times B\mathbb{G}_m$ is the diagonal action, making $\mathbb{L}_{\mathcal{F}}[1]$ pure of weight $(1, 1)$.

**LEMMA 7.6.4.1.** *With the above notations, the action of $\mathbb{G}_a[-1]$ on $\mathcal{V}(\mathbb{L}_{\mathcal{F}}[1])$ is canonically trivial.*

*Proof of the lemma.* This is formally deduced from the fact that the action of $\mathbb{G}_a[-1]$, or more generally of $\mathcal{H}_\pi$, respects the action of $\mathcal{H}$. In particular, $\mathbb{G}_a[-1]$ acts on $\mathcal{V}(\mathbb{L}_{\mathcal{F}}[1])$ by respecting the grading. Using [Mon21] this action is then determined by a linear action of $\mathbb{G}_a[-1]$ on the perfect complex $\mathbb{L}_{\mathcal{F}}$, compatible with the gradings, and thus by a graded mixed structure on $\mathbb{L}_{\mathcal{F}}$, which is pure of weight 1. Such a graded mixed $A$-dg-module is represented by an object $E \in \epsilon - \mathrm{dg}_k^{gr}$ with a fixed quasi-isomorphism $E^{(1)} \to \mathbb{L}_{\mathcal{F}}$ and $E^{(i)}$ being acyclic for all $i \neq 1$. Such an object is equivalent $E'$ with $(E')^{(1)} = \mathbb{L}_{\mathcal{F}}$ and $(E')^{(i)} = 0$ for $i \neq 1$ (and thus $\epsilon = 0$). $\square$

We now form the morphism $K := B\mathcal{H}_\pi \times_{B\mathbb{G}_m} B\mathcal{H} \to B\mathcal{H}_\pi \times B\mathcal{H}$, the corresponding quotient derived stack $[\mathcal{V}(\mathbb{L}_{\mathcal{F}}[1])/K] \to B\mathcal{H}$, and its affinization

$$ F := [\mathcal{V}(\mathbb{L}_{\mathcal{F}}[1])/K]^{aff} \to B\mathcal{H}. $$
Lemma 7.6.4.1 implies that $F \times_{BH} B\mathbb{G}_m$ is naturally equivalent to $\mathcal{V}(\mathbb{L}_F[1])$, and the natural projection $\mathcal{V}(\mathbb{L}_F[1]) \to X$ makes $F$ into a derived foliation on $X$ whose cotangent complex is identified with $\mathbb{L}_F$.

The above construction is clearly functorial in $\mathcal{F}$ and $X$, and thus defines for any derived Artin stack $X$ a well defined $\infty$-functor

$$ (-)^{DR} : \mathcal{F}ol_{\pi}(X/k) \to \mathcal{F}ol(X/k), $$

sending an infinitesimal derived foliation $\mathcal{F}$ to its de Rhamification $\mathcal{F}^{DR}$.

**DEFINITION 7.6.4.2.** The above $\infty$-functor $(-)^{DR}$ is called the de Rhamification construction. For an infinitesimal derived foliation $\mathcal{F} \in \mathcal{F}ol_{\pi}(X/k)$, the derived foliation $\mathcal{F}^{DR}$ is called the underlying derived foliation of $\mathcal{F}$.

We note in particular that the cotangent complexes are preserved by de Rhamification

$$ \mathbb{L}_{\mathcal{F}} \simeq \mathbb{L}_{\mathcal{F}^{DR}}. $$

The behaviour of the de Rhamification construction is rather subtle. For instance, already the question of which derived foliations arise as the de Rhamification of infinitesimal derived foliations is a non-trivial question. Derived foliations are not formally integrable, so our formal integrability result Corollary 7.6.3.2 provides a first obstruction for a derived foliation to come from an infinitesimal derived foliation. This can be seen already at the level of classical smooth foliations, as formally integrable foliations on a smooth varieties in characteristic $p > 0$ must always be stable by the $p$-th power operation on the tangent sheaf (see [Eke87]). The precise relation between infinitesimal derived foliations and an hypothetical notion $p$-restricted derived foliations is also not totally obvious. By the results of [Fu12] the question translate, for derived foliations over a perfect field, to the comparison between $p$-restricted dg-Lie algebras and partition Lie algebras of [BCN21]. As far as we know this comparison has not been clarified yet.

Nevertheless the existence of $\mathcal{F} \mapsto \mathcal{F}^{DR}$ is already helpful in the following sense. Starting with an infinitesimal derived foliation $\mathcal{F}$ we can exploit its formal integrability, and use $\mathcal{F}^{DR}$ when we want to study its de Rham cohomology, which is much better behaved than infinitesimal cohomology. For instance, we believe that the $\infty$-category of crystals on $\mathcal{F}^{DR}$ is more interesting for the study of cohomological properties than the $\infty$-category of crystals on $\mathcal{F}$ (defined naturally using $\mathcal{H}_{\pi}$-equivariant quasi-coherent complexes on $\mathcal{V}(\mathcal{F})$).
<figure><img src="image_56.png" /></figure>
# APPENDIX A

## A.1. Model categories and $\infty$-categories

**A.1.1. Model categories.** The notion of model category, due to D. Quillen [Qui67], is a very convenient axiomatization of a homotopy theoretical context. Modern standard references include [Hov99, Hir03].

**DEFINITION A.1.1.1.** Let $\mathcal{C}$ be a category. A model structure on $\mathcal{C}$ consists of the choice of three classes $W(\mathcal{C})$ (weak equivalences), $\mathrm{Fib}(\mathcal{C})$ (fibrations), and $\mathrm{Cofib}(\mathcal{C})$ (cofibrations) of morphisms in $\mathcal{C}$, subject to the following properties.

(1) If $(f, g)$ is a pair of composeable morphisms in $\mathcal{C}$ (i.e. $f \circ g$ exists in $\mathcal{C}$), if two among the morphisms $f, g, f \circ g$ is in $W(\mathcal{C})$, then the same is true for the remaining one.

(2) Each class $W(\mathcal{C})$, $\mathrm{Fib}(\mathcal{C})$, and $\mathrm{Cofib}(\mathcal{C})$ is stable under composition and retracts<fnref n="1" /> in $\mathcal{C}$.

(3) Let

$$ \begin{array}{ccc} X & \xrightarrow{f} & Y \\ \downarrow_i & & \downarrow_p \\ Z & \xrightarrow{g} & T \end{array} $$

be a commutative diagram in $\mathcal{C}$, with $i \in \mathrm{Cofib}(\mathcal{C})$ and $p \in \mathrm{Fib}(\mathcal{C})$. If either $i$ or $p$ also belongs to $W(\mathcal{C})$, then there exists a morphism $h : Z \to Y$ such that $p \circ h = g$ and $h \circ i = f$.

(4) Any morphism $f : X \to Y$ in $\mathcal{C}$ can be factored as $f = p \circ i$ and as $f = q \circ j$ where $i \in \mathrm{Cofib}(\mathcal{C})$, $p \in \mathrm{Fib}(\mathcal{C}) \cap W(\mathcal{C})$, $j \in \mathrm{Cofib}(\mathcal{C}) \cap W(\mathcal{C})$, $q \in \mathrm{Fib}(\mathcal{C})$. Moreover, these factorizations are both required to be functorial in $f$.

A model category consists of a complete and cocomplete category $\mathcal{C}$ endowed with a model structure.

**Model categories’ vocabulary.** Let $(\mathcal{C}, W(\mathcal{C}), \mathrm{Fib}(\mathcal{C}), \mathrm{Cofib}(\mathcal{C}))$ be a model category. Morphisms in $\mathrm{Fib}(\mathcal{C}) \cap W(\mathcal{C})$ are called *trivial fibrations*, while morphisms in $\mathrm{Cofib}(\mathcal{C}) \cap W(\mathcal{C})$ are called *trivial cofibrations*. An object $P$ (respectively, $Q$) in $\mathcal{C}$ is said to be *fibrant* (respectively, *cofibrant*) if the canonical map from $P$ to the final object $*_{\mathcal{C}}$ (resp., from the initial object $\emptyset_{\mathcal{C}}$

<footnote n="1">¹See [Lur09, Def. A.2.1.1 (3)] for a precise statement of stability under retracts</footnote>
to $\mathcal{Q}$) is a fibration (resp., a cofibration). By (4), any object $x \in \mathcal{C}$ admits a diagram, which is moreover *functorial* in $x$,

$$Qx \xrightarrow{p} x \xrightarrow{i} Rx$$

where $Qx$ is cofrant, $p$ is a trivial fibration (by applying (4) to $\emptyset_C \to x$), $Rx$ is fibrant and $i$ is a trivial cofibration (by applying (4) to $x \to *_C$). The *homotopy category ho($\mathcal{C}$)* of the model category $\mathcal{C}$ is the Gabriel-Zisman localization $\mathcal{C}[W(\mathcal{C})^{-1}]$ of $\mathcal{C}$ at weak equivalences. It admits also an equivalent description as the category whose objects are those objects in $\mathcal{C}$ that are both fibrant and cofibrant, with morphisms sets given by homotopy classes of morphisms between such objects ( [Hov99, §1.2]).

If $(\mathcal{C}, W(\mathcal{C}), \mathrm{Fib}(\mathcal{C}), \mathrm{Cofib}(\mathcal{C}))$ and $(\mathcal{D}, W(\mathcal{D}), \mathrm{Fib}(\mathcal{D}), \mathrm{Cofib}(\mathcal{D}))$ are model categories, an adjoint pair $(F, G)$, with left adjoint $F: \mathcal{C} \to \mathcal{D}$ is called a *Quillen adjunction* or *Quillen pair* (and $F$, respectively $G$, is called a *left*, resp. *right*, *Quillen functor*) if $F$ preserves cofibrations and trivial cofibrations or if, equivalently, $G$ preserves fibrations and trivial fibrations. A Quillen adjunction induces an adjunction at the level of the corresponding homotopy categories ( [Hov99, §1.3]), and it is said to be a Quillen equivalence if the induced adjoints are (mutually inverse) equivalences of categories.

**A.1.2. $\infty$-categories.** There are several, a priori different, definitions of $\infty$-categories: $\infty$-categories as simplicial (i.e. categories enriched over the category of simplicial sets) categories ( [Lur09, Definition 1.1.4.1]), $\infty$-categories as quasi-categories ( [Lur09, Definition 1.1.2.4]), $\infty$-categories as Segal categories ( [Sim11]), $\infty$-categories as complete Segal spaces ( [Rez01, JT07]), $\infty$-categories as Segal spaces ( [MN24]), $\infty$-categories as topological (i.e. categories enriched in the category of compactly generated weakly Hausdorff topological spaces) ( [Lur09, Definition 1.1.1.6]). The common idea behind these definitions is that a $\infty$-category should be a category (maybe weakly) enriched in topological spaces, or in any equivalent homotopical model for topological spaces (e.g. simplicial sets). For a precise comparison between these different models, the interested reader may consult [Lur09], [Ber10], and [Ber18].

For example, any usual 1-category gives rise to an $\infty$-category, to which it is usually identified. If for $\infty$-categories we take the model of quasi-categories, then the $\infty$-category associated to a 1-category $\mathcal{C}$ is its *nerve N($\mathcal{C}$)*, i.e. the simplicial set such that

$$\mathcal{N}(\mathcal{C})_n := \mathrm{Fun}([n], \mathcal{C}).$$

All the standard concepts in usual (1-)category theory have appropriate extensions to $\infty$-categories. For example, we have a notion of $\infty$-functor $\mathbf{F}: \mathbf{C} \to \mathbf{D}$: in the model of $\infty$-categories given by quasi-categories (respectively, simplicial categories), an $\infty$-functor is just a morphism between the corresponding simplicial sets (respectively, a simplicially enriched functor between the corresponding simplicially enriched categories). Given two $\infty$-categories $\mathbf{C}$ and $\mathbf{D}$, there is, actually, an $\infty$-category $\mathrm{Fun}_\infty(\mathbf{C}, \mathbf{D})$ of $\infty$-functors from $\mathbf{C}$ to $\mathbf{D}$ (see [Lur09,
1.2.7.2, 1.2.7.3] for quasi-categories). There is a notion of adjoint $\infty$-functors between $\infty$-categories (see [Lur09, Definition 5.2.2.1] for quasi-categories), notions of limit and colimit for diagrams in an $\infty$-category (see [Lur09, §1.2.13] for these concepts inside a quasi-category), Kan extensions (see [Lur09, §4.3] for quasi-categories), etc. For any pair $(x, y)$ of objects inside an $\infty$-category $\mathbf{C}$, there is a mapping space $Map_{\mathbf{C}}(x, y)$ between them, that can be identified with a simplicial set, and is defined up to homotopy (i.e. it is well defined in $ho(\mathbf{sSets}))$. The homotopy category of an $\infty$-category $\mathbf{C}$, is the 1-category $ho(\mathbf{C})$ with the same objects as $\mathbf{C}$ and morphism from $x$ to $y$ given by $\pi_0(Map_{\mathbf{C}}(x, y))$. There is an obvious $\infty$-functor $\mathbf{C} \to ho(\mathbf{C})$, and a morphism in $\mathbf{C}$ will be called an equivalence if it becomes an isomorphism in $ho(\mathbf{C})$. Moreover, (small) $\infty$-categories and $\infty$-functors between them, organize themselves<fnref n="2" /> again into a $\infty$-category $\mathbf{Cat}_{\infty}$ (see [Lur09, Chapter 3] for quasi-categories). More is true: $\mathbf{Cat}_{\infty}$ is moreover a $(\infty, 2)$-category, i.e. an $\infty$-categorical version of a bicategory, but we will mainly view it simply as an $\infty$-category. The nerve construction from 1-categories to $\infty$-categories, recalled above, provides a fully faithful functor.

For an $\infty$-category $\mathbf{C}$ and a set $\mathbf{W}$ of morphisms, one can construct another $\infty$-category denoted by either $L_{\mathbf{W}}(\mathbf{C})$ or $\mathbf{C}[\mathbf{W}^{-1}]$ where, informally speaking, the morphisms in $\mathbf{W}$ has been inverted. The construction also provides a canonical $\infty$-functor $L: \mathbf{C} \to L_{\mathbf{W}}(\mathbf{C})$ such that, for any $\infty$-category $\mathbf{D}$, the induced $\infty$-functor

$$ \mathbf{Fun}_{\infty}(L_{\mathbf{W}}(\mathbf{C}), \mathbf{D}) \to \mathbf{Fun}_{\infty}(\mathbf{C}, \mathbf{D}) $$

is fully faithful with essential image consisting of those $\infty$-functors $\mathbf{C} \to \mathbf{D}$ sending morphisms in $\mathbf{W}$ into equivalences in $\mathbf{D}$. The $\infty$-category $L_{\mathbf{W}}(\mathbf{C})$ is called the localization of $\mathbf{C}$ at (or along) $\mathbf{W}$.

**DEFINITION A.1.2.1.** Given a model category $(\mathbf{C}, W(\mathbf{C}), \mathrm{Fib}(\mathbf{C}), \mathrm{Cofib}(\mathbf{C}))$, its underlying or associated $\infty$-category will be the localization $L_{W(\mathbf{C})}(\mathbf{N}(\mathbf{C}^{\mathrm{cof}}))$, where $\mathbf{C}^{\mathrm{cof}}$ denotes the full sub-category of $\mathbf{C}$ consisting of cofibrant objects.

**REMARK A.1.2.2.** In the main text, if $\mathbf{C}$ is a model category, we have simply denoted by $\mathbf{C}$ its underlying $\infty$-category.

A Quillen adjunction between model categories (A.1.1) induces an adjunction between $\infty$-functors on the underlying $\infty$-categories ( [Lur09, §5.2.4]), which is an equivalence of $\infty$-categories if the Quillen pair is a Quillen equivalence.

## A.2. Derived schemes and stacks: basic facts and notations

In this § we will mainly fix the notations used in the main text, we do not mean to give a short introduction to derived algebraic geometry, for which we refer to [Toë09, Toe14, PV21].

<footnote n="2">²With appropriate set-theoretic or universes's caveats.</footnote>
Let $k$ be a base discrete commutative ring (e.g. $k = \mathbb{Z}$), we denote by $\mathbf{sAlg}_k$ the model category$^3$ of simplicial commutative $k$-algebras (where both weak equivalences and fibrations are detected by the forgetful functor $\mathbf{sAlg}_k \to \mathbf{sSets}$), and by $\mathbf{sAlg}_k$ the underlying $\infty$-category (see A.1.1). Note that, for $A \in \mathbf{sAlg}_k$, $\pi_*(A) = \bigoplus_{i \ge 0} \pi_i(A)$ has a natural induced structure of a commutative graded $k$-algebra. In particular, $\pi_0(A)$ has an induced structure of usual commutative $k$-algebra, and each $\pi_i(A)$ is a $\pi_0(A)$-module. We say that $A$ is *Noetherian* if $\pi_0(A)$ is a Noetherian ring, and each $\pi_i(A)$ is a finitely generated (discrete) $\pi_0(A)$-module. We will say that $A$ is *discrete* if $\pi_i(A) = 0$ for any $i > 0$ (i.e. $A$ is weakly equivalent to the constant simplicial $k$-algebra $\pi_0(A)$).

For a simplicial commutative $k$-algebra $A$, the associated *affine derived scheme* is the $\infty$-functor

$$ \mathbf{Spec} A := \mathbf{Map}_{\mathbf{sAlg}_k}(A, -) : \mathbf{sAlg}_k \to \mathbf{sSets} = \mathbf{Top} $$

We let $\mathbf{dAff}_k$, the $\infty$-category of *affine derived schemes* over $k$, be the full $\infty$-subcategory of $\mathbf{Fun}_\infty(\mathbf{sAlg}_k, \mathbf{Top})$ essential image of the $\infty$-functor $\mathbf{Spec}$. The $\infty$-category $\mathbf{dAff}_k$ has arbitrary pullbacks: for $B \leftarrow A \to C$ a diagram in $\mathbf{sAlg}_k$, the corresponding pullback in $\mathbf{dAff}_k$ is given by $\mathbf{Spec}(B \otimes_A^L C)$, where $B \otimes_A^L C$ is the derived tensor product of $B$ and $C$ in $A/\mathbf{sAlg}_k$, obtained by replacing either $B$ or $C$ by a cofibrant replacement in the undercategory $A/\mathbf{sAlg}_k$, before taking the usual (strict) tensor product.

The inclusion of usual discrete commutative $k$-algebras in $\mathbf{sAlg}_k$, as constant simplicial object, induces a fully faithful functor $j : \mathbf{Aff}_k \to \mathbf{dAff}_k$ having as right adjoint $t_0 : \mathbf{Spec} A \mapsto \mathbf{Spec}(\pi_0(A))$, called the *truncation functor*. An important property of $j$ is that it does not preserve general pullbacks: if $S \leftarrow R \to T$ is a diagram in $\mathbf{Alg}_k$, then $S \otimes_R^L T \neq S \otimes_R T$ unless $S$ or $T$ are flat over $R$.

If, as often in the main text, the fixed base ring $k$ is a commutative $\mathbb{Q}$-algebra, then we may replace $\mathbf{sAlg}_k$ with the model category $\mathbf{cdga}_k$ of commutative differential non-positively graded (aka *connective*) $k$-algebras, since we have an equivalence of the underlying $\infty$-categories $\mathbf{sAlg}_k \simeq \mathbf{cdga}_k$. If, under this equivalence, $A \in \mathbf{sAlg}_k$ corresponds to $A' \in \mathbf{cdga}_k$, we have $\pi_i(A) \simeq H^{-i}(A')$ for any $i \ge 0$. For this reason, we will sometimes abuse notations and write $\pi_i(B)$ for $H^{-i}(B)$ even when $B \in \mathbf{cdga}_k$.

For $A \in \mathbf{sAlg}_k$, its normalization $\mathbf{N}(A)$ is a commutative differential graded $k$-algebras, and we denote by $\mathbf{Mod}_A$ the model category of *not necessarily bounded* $\mathbf{N}(A)$-dg modules (see [TV08, Ch. 2.2] for more details). As usual, $\mathbf{Mod}_A$ will be the underlying $\infty$-category of $\mathbf{Mod}_A$. The same notation is used for $A \in \mathbf{cdga}_k$. For $M \in \mathbf{Mod}_A$ and $i \in \mathbb{Z}$, we will write

<footnote>$^3$This is, moreover, a cofibrantly generated simplicial model structure; see [Hir03] for the meaning of these words.</footnote>
$\pi_i(M)$ or $H^{-i}(M)$ interchangeably.

If $\mathbf{C}$ denotes either $\mathbf{sCAlg}_k$ or $\mathbf{cdga}_k$, a morphism $A \to B$ in $\mathbf{C}$ is called locally of finite presentation if the functor $\mathbf{Map}_{\mathbf{A}/\mathbf{C}}(B, -)$ commutes with filtered colimits (i.e. $B$ is a compact object inside $\mathbf{A}/\mathbf{C}$). Since the forgetful $\infty$-functor $\mathbf{A}/\mathbf{C} \to \mathbf{Mod}_A$ preserves and reflects filtered colimits, this condition is equivalent to $\mathbf{Map}_{\mathbf{Mod}_A}(B, -) : \mathbf{Mod}_A \to \mathbf{Top}$ preserving filtered colimits. A morphism $A \to B$ in $\mathbf{C}$ is called locally almost of finite presentation (sometimes abbreviated to lafp) $\mathbf{Map}_{\mathbf{Mod}_A}(B, -)$ preserves filtered colimits when restricted to each full $\infty$-subcategory $\mathbf{Mod}^{\le n}_{\mathbf{A}}$ of truncated $\mathbf{A}$-modules (i.e. $\mathbf{A}$-modules $M$ such that $\pi_i(M) = 0$ for $i > n$. For example, if $A$ is Noetherian, then $A \to B$ is lafp iff $B$ is Noetherian, and $\pi_0(B)$ is a finitely presented $\pi_0(A)$-algebra in the usual sense of commutative algebra.

One can consider different $\infty$-topologies ( [Lur09, 6.2.2], [TV05, 3.1]) on the $\infty$-category $\mathbf{dAff}_k$. For example, the étale $\infty$-topology is generated by the $\infty$-pre-topology defined as follows: for $X = \mathbf{Spec} A \in \mathbf{dAff}_k$ where $A \in \mathbf{sCAlg}_k$ or $A \in \mathbf{cdga}_k$, étale covering families of $X$ are given by arbitrary families $\{A \to A_i\}_{i \in I}$ such that the induced family $\pi_0(A) \to \pi_0(A_i)_{i \in I}$ is a usual étale covering family of $\mathbf{t}_0(\mathbf{Spec} A) = \mathrm{Spec}(\pi_0(A))$, and moreover, the canonical maps

$$ \pi_i(A) \otimes_{\pi_0(A)} \pi_0(A_i) \to \pi_i(A_i) $$

are isomorphisms<fnref n="4" /> for all $i \ge 0$ (and we say, then, that each $A \to A_i$ is strongly étale). Once an $\infty$-topology $\tau$ on $\mathbf{dAff}_k$ is fixed, we define the $\infty$-category of derived $\tau$-stacks

$$ \mathbf{dSt}_k^\tau := \mathrm{Sh}((\mathbf{dAff}_k, \tau), \mathbf{Top}) $$

the category of Top-valued $\infty$-sheaves on the $\infty$-site $(\mathbf{dAff}_k, \tau)$ ( [Lur09, Def. 6.2.2.6], [TV05, Def. 3.4.9]). Therefore, a derived $\tau$-stack is an $\infty$-functor $F : \mathbf{dAff}_k^{\mathrm{op}} \to \mathbf{Top}$ (i.e. a $\infty$-preseheaf on $\mathbf{dAff}_k$) satisfying $\tau$-hyperdescent, i.e. the canonical morphism $F(X) \to F(U_\bullet)$ is an equivalence in $\mathbf{Top}$, for any $\tau$-hypercover $U_\bullet \to X$.

In order to select geometrically meaningful and interesting derived stacks, one imposes recursively defined geometricity conditions on atlases and representable morphisms (see [TV08, Ch. 1.3], [Toe14, 3.3] and [PV21, §4]). In particular, a derived Deligne-Mumford stack is a derived étale stack admitting an étale atlas (by derived affine schemes), while a derived Artin (or algebraic) stack is a derived étale stack admitting a smooth atlas.

Any connective $k$-linear cdga $A$ possesses a cotangent complex $\mathbb{L}_{A/k}$ relative to $k$. By definition $\mathbb{L}_{A/k}$ is an $A$-dg-module such that derivations $A \to M$ are equivalent to $A$-dg-modules maps $\mathbb{L}_{A/k} \to M$ (see [TV08, §1.4.1]). More precisely, for $M \in \mathbf{dg}_A$ an $A$-dg-module we can form the trivial square zero extension $A \oplus M$, which is a new cdga augmented to $A$. We then

<footnote n="4">⁴Recall our convention $\pi_i(A) := H^{-i}(A)$ for $A \in \mathbf{cdga}_k$, and $i \ge 0$.</footnote>
have functorial equivalences of mapping spaces

$$ \mathbf{Map}_{\mathbf{dg}_A}(\mathbb{L}_{A/k}, M) \simeq \mathbf{Map}_{\mathbf{cdga}_k/A}(A, A \oplus M), $$

and the right hand side is, by definition, the space of derivations on A with coefficients in M. In particular, the identity of $\mathbb{L}_{A/k}$ defines a universal derivation $dR: A \to \mathbb{L}_{A/k}$. The cotangent complex of non-affine derived schemes and more generally of derived Artin stacks is discussed below in §A.5.

### A.3. Formal completion

Let A be a connective cdga over a field $k$ of characteristic zero endowed with au augmentation $x: A \to k$. We can define a pro-Artinian cdga $\hat{A}_x$ (or simply $\hat{A}_x$), called the formal completion of A at $x$, as follows. We consider $A/\mathbf{dgart}_k^*$ the full sub-$\infty$-category of the $\infty$-category of augmented connective cdga under A consisting of all morphisms $A \to A'$ where $A'$ is a local Artinian cdga. Recall that Artinian here means that $H^0(A')$ is a local Artinian $k$-algebra and moreover that $H^*(A')$ is finite dimensional (so all $H^i(A')$ are finite dimensional and vanish for $i << 0$). We note that $A/\mathbf{dgart}_k^*$ is a filtered $\infty$-category as it admits finite limits. The following definition therefore makes sense.

**DEFINITION A.3.0.1.** With the above notations, the formal completion of A at $x$ is the pro-object in $\mathbf{dgart}_k^*$ defined by

$$ \hat{A}_x := \text{``lim''}_{A' \in (A/\mathbf{dgart}_k^*)} A'. $$

The pro-object $\hat{A}$ has a realization $\hat{A} = \lim_{A' \in (A/\mathbf{dgart}_k^*)} A'$ which is a cdga. We will see below that $\hat{A}$ is connective when A is almost of finite presentation. This does not follow formally from the definition (as connective cdga's are not stable by limits inside the bigger $\infty$-category of all cdga's). This will be a consequence of a more explicit description of the pro-object $\hat{A}$.

Assume now that A is a connective cdga which is moreover almost of finite presentation. We consider the ideal $I \subset H^0(A)$ of the induced augmentation $H^0(A) \to k$, and pick $(f_1, \dots, f_p)$ a set of generators for $I$. For each integer $n > 0$ we define an object $A \to A_n$ of $A/\mathbf{dgart}_k^*$ as follows. Start with the Koszul cdga $K(A, f_n^*)$, which is the free A-linear cdga generated by elements $h_1, \dots, h_p$ in degree $-1$ with $dh_i = f_i^n$. We let $A_n := \tau_{\le n} K(A, f_n^*)$ the n-truncation of $K(A, f_n^*)$. For $m > n$, we have a canonical morphism $p_{m,n}: A_m \to A_n$ which sends $h_i$ to $f_i^{m-n} h_i$. making the $A_n$ into a projective system in $A/\mathbf{cdga}$. Moreover, $H^0(A_n) \simeq H^0(A)/(f_1^n, \dots, f_p^n)$ is a local Artinian algebra, and each $H^i(A_n)$ is of finite type over $H^0(A_n)$. Indeed, we can see this by induction on the number of elements $f_i$, using that $K(A, f_*) \simeq K(K(A, f_{*<k}), f_k)$, and the long exact sequence

$$ H^{i-1}(A) \to H^i(A) \xrightarrow{f} H^i(A) \to H^i(K(A, f)) \to H^{i+1}(A) \to \dots $$
for the Koszul algebra $K(A, f)$ of a single element $f \in H^0(A)$. Finally, $A_n$ are truncated by construction. The family of objects $A_n$ thus form a pro-object "$\lim_n$" $A_n$ in $A/\mathrm{dgart}_k^*$. There is an obvious morphism of pro-object $\hat{A} \to \lim_n$" $A_n$, as the diagram given by the canonical morphisms $A \to A_n$.

**PROPOSITON A.3.0.2.** With the notations above, the natural morphism $\hat{A} \to \lim_n$" $A_n$ is an equivalence of pro-objects in $A/\mathrm{dgart}_k^*$.

*Proof.* Let $\mathcal{N}$ be the category of natural numbers with its natural order. The family of objects $A_n$ defines an $\infty$-functor $\phi: \mathcal{N}^{op} \to A/\mathrm{dgart}_k^*$. The Proposition will follow from the fact that $\phi$ is cofinal. As a first observation, for any object $A \to A'$ in $A/\mathrm{dgart}_k^*$, the image of all $f_i$ in $H^0(A')$ are nilpotent. Therefore, we can chose $n$ such that for each $i$, the image of $f_i^n$ is homotopic to zero in $A'$. By the universal property of the A-cdga $A_n = K(A, f_*^n)$, there exists a factorization $A \to A_n \to A''$.

Let $A' \in A/\mathrm{dgart}_k^*$, and let $\phi/A'$ be the slice $\infty$-category of $\phi$. It is enough to show that $\phi/A'$ is cofiltered. For this, let $B_* : J \to \phi/A'$ be an $\infty$-functor $j \mapsto B_j$ with $J$ a finite $\infty$-category. We can for the limit along $J$ and get this way an dg-Artinian local algebra $A_J = \lim_{j \in J} B_j$ and a factorization $A \to A_J \to A'$. By our previous observation, there exists a factorization $A \to A_p \to A_J$.

For any $j \in J$, $B_j$ is of the form $A_{n_j}$ for some integer $n_j$. The morphism $A_p \to A_{n_j}$ is possibly not of the standard form, but it can be made so by increasing $p$ thanks to the following lemma.

**LEMMA A.3.0.3.** For any integer $p \ge n$, and any morphism of A-cdga's $u: A_p \to A_n$, there exists $q \ge p$ such that the composition

$$A_q \xrightarrow{p_{q,p}} A_p \xrightarrow{u} A_n$$

is homotopic to the standard morphism $p_{q,n}$.

*Proof of the lemma.* This follows easily from the universal property of Koszul algebras. Indeed, for any A-cdga $B$, the space of A-cdga morphisms $Map_{A-\mathrm{cdga}}(A_p, B)$ is $\Omega_{0,f_p}B$, the space of homotopies between 0 and the image of $f^p$ in $B$. It is therefore either empty or non-canonically equivalent to the loop space $\Omega_0 B$, and a choice of a homotopy $0 \sim f^p$ induces an equivalence $\Omega_{0,f_p}B \simeq \Omega_0 B$. As a result, for any $p \ge n$, the set of homotopy classes of A-cdga morphisms $\pi_0(Map_{A-\mathrm{cdga}}(A_p, A_n))$ is isomorphic to $H^{-1}(A_n)$, and moreover this isomorphism is canonical as the universal homotopy $0 \sim f^n$ in $A_n$ induces a homotopy $0 \sim f^p$ by multiplication by $f^{p-n}$.

Under the bijection $\pi_0(Map_{A-\mathrm{cdga}}(A_p, A_n)) \simeq H^{-1}(A_n)$, the canonical morphism $p_{p,n}$ corresponds to 0 in $H^{-1}(A_n)$. Therefore, the lemma simply follows by observing that for $q \gg p$ (and precisely $q \ge 2p$), the morphism $p_{q,p}$ induces the zero map $H^{-1}(A_q) \to H^{-1}(A_p)$. $\square$

Coming back to the proof of the Proposition. The lemma tells us that if $p$ is chosen big enough, then all the morphisms $A_p \to B_j$ are the standard morphisms. This means that the
$A_p$ defines indeed an object in $\phi/A'$ that admits a morphism to the diagram $B_*$.

An important consequence of Proposition A.3.0.2 is the description of the pro-$\hat{A}$-modules $\wedge^p \mathbb{L}_{\hat{A}} := \lim_{A \to A'} \wedge^p \mathbb{L}_{A'}$ of $p$-forms.

**COROLLARY A.3.0.4.** *The natural morphism*

$$ \lim_{A \to A'} \mathbb{L}_A \otimes_A A' \to \mathbb{L}_{\hat{A}} $$

is an equivalence of pro-$\hat{A}$-modules.

**Proof of Corollary A.3.0.4.** For all $A \to A'$ we have an exact triangle of pro-$\hat{A}$-modules

$$ \mathbb{L}_A \otimes_A A' \to \mathbb{L}_{A'} \to \mathbb{L}_{A'/A} $$

Therefore, in order to prove the statement we need to prove that the pro-object " $\lim_{A \to A'} \mathbb{L}_{A'/A}$ is zero. To see this, it is enough by Proposition A.3.0.2 to see that " $\lim_n \mathbb{L}_{A_n/A}$ is the zero pro-" $\lim_n A_n$"-module. For each $n$, $\mathbb{L}_{A_n/A} \simeq A_n[1]^p$, a basis being given by the de Rham differential $dR(h_i)$ (relative to $A$) of $h_i$, of the canonical elements $h_i$ of degree $-1$ in $A_n$. Moreover, for $p \ge 2n$, the canonical morphism $A_p \to A_n$ sends $dR(h_i)$ to $f^{p-n}dR(h_i)$, and thus is homotopic to the zero map because $f^{p-n}$ is homotopic to zero for $p \ge 2n$. This implies that the pro-" $\lim_n A_n$"-module " $\lim_{A \to A'} \mathbb{L}_{A'/A}$ is zero as wanted."

By taking $p$-th wedge powers we find the following consequence.

**COROLLARY A.3.0.5.** For any $p \ge 0$, the natural morphism

$$ \lim_{A \to A'} (\wedge_A^p \mathbb{L}_A) \otimes_A A' \to \wedge_{\hat{A}}^p \mathbb{L}_{\hat{A}} \simeq \lim_{A \to A'} \wedge_{A'}^p \mathbb{L}_{A'} $$

is an equivalence of pro-$\hat{A}$-modules.

Another important consequence is the following flatness result, stating the cohomology groups of the formal completion are the formal completion of the cohomology groups.

**COROLLARY A.3.0.6.** Let $x : A \to k$ be an augmented connective cdga which is almost of finite presentation and $\hat{A} = \lim_{A \to A'} A'$ be the realization of its formal completion at $x$. Then, we have

$$ H^0(\hat{A}) \simeq \widehat{H^0(A)} \qquad H^i(\hat{A}) \simeq \widehat{H^i(A)} \simeq H^i(A) \otimes_{H^0(A)} \widehat{H^0(A)}. $$

In particular, the morphism $A \to \hat{A}$ is a flat morphism of connected cdga's, which is moreover faithfully flat locally at $x$.

**Proof of Corollary A.3.0.6.** As $A$ is almost of finite presentation over $k$, we can find a strict quasi-free model as a cdga of the form

$$ \dots \to A^{-i} \to A^{-i+1} \to \dots \to A^0 $$
where $A^0 \simeq k[x_1, \dots, x_d]$ as a $k$-algebra, and each $A^i$ is a free $A^0$-module of finite rank. We can moreover assume that the augmentation $A \to k$ precomposed with $A^0 \to A$ is given by the augmentation ideal $I = (x_1, \dots, x_q)$. Therefore, the $f_i$ are here represented by the $x_i$.

By definition of the Koszul algebra $A_n = K(A, f_*^n)$ is thus given explicitly by

$$K(A, f_*^n) \simeq A \otimes_{A^0} K(A^0, x_*^n).$$

However, the $x_i^n$ form a regular sequence and thus $K(A^0, x_*^n)$ is naturally quasi-isomorphic to the $k$-algebra $k[x_1, \dots, x_q]/(x_1^n, \dots, x_q^n)$. As $A$ is cofibrant as an $A^0$-module, we thus have

$$K(A, f_*^n) := A \otimes_{A^0} K(A^0, x_*^n) \simeq A \otimes_{A^0} (A^0/(x_1^n, \dots, x_q^n)).$$

Therefore, by the Proposition A.3.0.2, the realization $\hat{A}$ is simply given by

$$\hat{A} \simeq A \otimes_{A^0} \hat{A}^0,$$

where $\hat{A}^0$ is the formal completion of $A^0$ at the ideal $(x_1, \dots, x_0)$. As formal completions of noetherian rings are flat, we deduce the Corollary. $\square$

## A.4. $\mathcal{O}_X$-modules, quasi-coherent and perfect modules

The $\infty$-topos $\mathbf{dSt}_k$, of derived stacks over $k$, admits a tautological sheaf of cdga's, namely the left Kan extension of the tautological $\infty$-functor sending $\mathbf{Spec} A$ to $A$. It is denoted by $\mathcal{O}_k$, and its restriction to $\mathbf{dSt}_k/X$ will be denoted by $\mathcal{O}_X$. Associated to this sheaf of cdga's we have $\infty$-categories of sheaves of modules, simply denoted by $\mathcal{O}_k - \mathbf{dg}$ and $\mathcal{O}_X - \mathbf{dg}$. In concrete terms, for $X \in \mathbf{dSt}_k$, and object $E$ in $\mathcal{O}_X - \mathbf{dg}$ consists of the following data:

* for all $u : \mathbf{Spec} A \to X$ an $A$-dg-module $E(u)$,

* for all $\mathbf{Spec} A \xrightarrow{f} \mathbf{Spec} B \xrightarrow{u} X$ a morphism of $B$-dg-modules $\phi_f : E(u \circ f) \to E(u)$,

together with the usual coherence conditions. In different terms, the $\infty$-category $\mathcal{O}_X - \mathbf{dg}$ can also be described as a lax limit

$$\mathcal{O}_X - \mathbf{dg} \simeq \lim_{\mathbf{Spec} A \to X}^{\text{lax}} A - \mathbf{dg}.$$

An object $E \in \mathcal{O}_X - \mathbf{dg}$ will be called *quasi-coherent* if all the transition morphisms $\phi_f : E(u \circ f) \to E(u)$ induces equivalences $A \otimes_B E(u \circ f) \simeq E(u)$. The full sub-$\infty$-category of quasi-coherent $\mathcal{O}_X$-dg-modules (also called quasi-coherent complexes) will be denoted by

$$j : \mathbf{QCoh}(X) \subset \mathcal{O}_X - \mathbf{dg}.$$

This is a full embedding of presentable $\infty$-categories, and obviously the inclusion $\infty$-functor commutes with arbitrary colimits. Therefore, the inclusion $j$ admits a right adjoint

$$(-)^{\text{coh}} : \mathcal{O}_X - \mathbf{dg} \to \mathbf{QCoh}(X),$$

called the *quasi-coherator* construction.
We remind also two more full sub-∞-categories **Perf**(X) ⊂ **APerf**(X) ⊂ **QCoh**(X). As a start let A be a connective cdga over k. We recall that an A-dg-module E is perfect if it is a compact object of A − dg. It is equivalent to as for E being dualizable with respect to the natural symmetric monoidal structure on A − dg, and also equivalent to the fact that E is a retract of a finite cell A-dg-module (see [TV07, Prop. 2.2]). Similarly, an A-dg-module E is almost perfect if it can be approximated by perfect objects: for all n, there exists a perfect A-dg-module E_n and a morphism E_n → E such that the induced morphism H^i(E_n) → H^i(E) is an isomorphism for all i > n. As A is connective, perfect A-dg-modules are bounded above, and thus so are almost perfect A-dg-modules. Moreover, if A is almost of finite presentation (i.e. H^0(A) is of finite type over k and H^i(A) is a finite type H^0(A)-module for all i) then E ∈ A − dg is almost perfect if and only if it satisfies the following two conditions:

• E is bounded above,

• H^i(E) is a H^0(A)-module of finite type for all i.

To finish this section, remind that $\mathcal{O}_X - \mathrm{dg}$ comes equipped with a canonical symmetric monoidal structure $\otimes_{\mathcal{O}_X}$, determined for any $u : \mathrm{Spec} A \to X$ by the rule $(E \otimes_{\mathcal{O}_X} F)(u) := E(u) \otimes_A F(u)$ plus sheafification. This symmetric monoidal structure preserves the full sub-∞-categories $\mathrm{Perf}(X) \subset \mathrm{APerf}(X) \subset \mathrm{QCoh}(X) \subset \mathcal{O}_X - \mathrm{dg}$, and thus induces symmetric monoidal structures on all of them. Note also that $\mathrm{Perf}(X)$ consists of all dualizable objects in $\mathrm{QCoh}(X)$, and thus any perfect complex is also dualizable as an $\mathcal{O}_X$-dg-module.

We finish with an important statements concerning quasi-coherent sheaves on formal derived completions of derived affine schemes. Let $X = \mathrm{Spec} A$ be a derived affine scheme almost of finite presentation over a field $k$ (so in particular $H^0(A)$ is noetherian and each $\mathrm{Hi}(A)$ is a $H^0(A)$-module of finite type). Let $x \in X(k)$ be a global point and $\hat{A}_x$ be the formal completion of $A$ at the point $x$. It is a pro-object in $\mathrm{cdga}_k$ whose limit is denoted by $\hat{A}_x$.

**PROPOSITON A.4.0.1.** *The natural $\infty$-functor*

$$
\phi : \mathbf{APerf}(\hat{A}_x) \to \mathbf{Perf}(\hat{A}_x) = \lim_{A \to A'} \mathbf{APerf}(A'),
$$

sending E to the pro-module " lim<sub>A→A'</sub> " (A' ⊗<sub>Â<sub>x</sub></sub> E), is an equivalence of ∞-categories.

*Proof.* From proposition A.3.0.2 $\hat{A}_x$ can be represented as pro-object of the form

$$
\hat{A}_x \longrightarrow \dots \longrightarrow A_n \longrightarrow A_{n-1} \longrightarrow \dots \longrightarrow A_0 \longrightarrow k
$$

where each $A_i$ is a connective local artinian cdga over $k$. Moreover each morphism $H^0(A_n) \to H^0(A_{n-1})$ is a surjective map of artinian local $k$-algebras. The proposition then becomes of statement about the $\infty$-functor

$$
\phi : \mathbf{APerf}(\hat{A}_x) \to \lim_n \mathbf{APerf}(A_n).
$$
The $\infty$-functor $\phi$ is the restriction of a left adjoint defined on the level of quasi-coherent complexes

$$ \phi : \mathbf{QCoh}(\hat{A}_x) \stackrel{\sim}{\longrightarrow} \lim_n \mathbf{QCoh}(A_n) : \psi. $$

The left adjoint has already been described, it sends $E$ to $\lim_n A_n \otimes_{\hat{A}_x} E$. The right adjoint sends a pro-object $\lim_n E_n$ to its realization $\lim_n E_n \in \mathbf{QCoh}(\lim_n A_n)$.

A first observation is that the unit of the adjunction $\hat{A}_x \to \psi\phi(\hat{A}_x)$ is obviously an equivalence, and thus this is also the case for any $E \in \mathbf{Perf}(\hat{A}_x)$ (as $\hat{A}_x$ generates $\mathbf{Perf}(\hat{A}_x)$ by finite colimits, shifts and retracts).

**LEMMA A.4.0.2.** Suppose that $E \in \mathbf{QCoh}(\hat{A}_x)$ is $k$-connective ($H^i(E) \simeq 0$ for $i > -k$) and $H^k(E)$ is of finite type as a $H^0(\hat{A}_x)$-module. Then, $\psi\phi(E)$ remains $k$-connective and the adjunction morphism $E \to \psi\phi(E)$ induces an isomorphism

$$ H^k(E) \simeq H^k(\psi\phi(E)). $$

*Proof of the lemma.* We use the Milnor short exact sequences

$$ 0 \longrightarrow \lim_n^1 H^{j-1}(E_n) \longrightarrow H^j(\lim_n E_n) \longrightarrow \lim_n^0 H^j(E_n) \longrightarrow 0 $$

for any $j \in \mathbb{Z}$ and any projective system of complexes

$$ \dots \longrightarrow E_n \longrightarrow E_{n-1} \longrightarrow \dots \longrightarrow E_1 \longrightarrow E_0. $$

Here the $\lim_n^i$ denote the $i$-th right derived functor associated to the $\lim_n$ functor, and $\lim_n E_n$ is understood in the $\infty$-category $\mathbf{dg}_k$ of complexes of $k$-modules. To see that $\psi\phi(E)$ remains $k$-connective we thus have to show that $\lim_n^1 H^k(E_n) = 0$. But this follows from the fact that the system $n \mapsto H^k(E_n)$ is of the form $n \mapsto H^0(A_n) \otimes_{H^0(\hat{A}_x)} H^k(E)$ and thus all maps $H^k(E_n) \to H^k(E_{n-1})$ are surjective. Once $\psi\phi(E)$ is known to be $k$-connective, the unit of the adjunction, on the $k$-th cohomology group can be written as

$$ H^k(E) \to \lim_n H^0(A_n) \otimes_{H^0(\hat{A}_x)} H^k(E). $$

This is an isomorphism because $H^0(\hat{A}_x) \simeq \lim_n H^0(A_n)$ and $H^k(E)$ is a module of finite type. $\square$

Let $E \in \mathbf{APerf}(\hat{A}_x)$. By definition, for any $j$, we can find $E_0 \in \mathbf{Perf}(\hat{A}_x)$ and a morphism $E_0 \to E$ whose cone $E'$ is $j$-connective. We have a commutative diagram with exact rows

$$ \begin{tikzcd} E_0 \arrow[r] \arrow[d] & E \arrow[r] \arrow[d] & E' \arrow[d] \\ \psi\phi(E_0) \arrow[r] & \psi\phi(E) \arrow[r] & \psi\phi(E') \arrow[d] \end{tikzcd} $$

The left vertical morphism is an equivalence because $E_0$ is perfect. Moreover, by the lemma A.4.0.2 both $E'$ and $\phi(E')$ are $j$-connective. In particular, considering the long exact sequences
on cohomology groups we see that the vertical morphism of the middle must induce isomorphisms $H^k$ for all $k > -j$. As $j$ is arbitrary this shows the unit of the adjunction $E\psi\phi(E)$ is an equivalence, and thus that the $\infty$-functor $\phi$ of the proposition is fully faithful.

To finish the proof we have to show that the right adjoint $\psi : \lim_n \mathbf{QCoh}(A_n) \stackrel{\sim}{\to} \mathbf{QCoh}(\hat{A}_x)$ sends objects in $\lim_n \mathbf{APerf}(A_n)$ to almost perfect complexes and that the induced $\infty$-functor

$$ \psi : \lim_n \mathbf{APerf}(A_n) \to \mathbf{APerf}(\hat{A}_x) $$

is furthermore conservative. As we have seen during the proof of the lemma A.4.0.2, the Milnor short exact sequences implies that $\psi$ preserves the $j$-connective objects. Any object $E = \{E_n\} \in \lim_n \mathbf{APerf}(A_n)$ is $j$-connective for some $j$ (uniform in $n$). Indeed, if $E_n \in \mathbf{APerf}(A_n)$, then it is $l$-connective for some integer $l$. Moreover, $A_n$ is an artinian local connective cdga, and thus the Yoneda lemma implies that if $E_n \otimes_{A_n} k$ is $j$-connective for some $j$ then $E_n$ was already $j$-connective as well. In particular, if $E_0$ is $j$-connective then all $E_n$ are $j$-connective as well (for the same $j$), as

$$ E_n \otimes_{A_n} k \simeq (E_n \otimes_{A_n} A_0) \otimes_{A_0} k \simeq E_0 \otimes_{A_0} k. $$

Therefore, $\psi(E)$ is also $j$-connective.

We now proceed as before by approximation by perfect complexes as before. For any $E = \{E_n\} \in \mathbf{APerf}(\hat{A}_x)$, which is $j$-connective for some $j$, we can find a free $A_0$-module $D_0 = A_0^r$ and a morphism

$$ u_0 : D_0[j] \to E_0 $$

whose cones $E'_0$ is $(j+1)$-connective. As $D_0$ is free we can lift $u_0$ to a morphism $u : D[j] \to E$ whose cone $E'$ is $(j+1)$-connective, and $D$ is here of the form $\phi(\hat{A}_x^r)$ (i.e. free of rank $r$). Continuing the resolution on $E'$, we construct inductively an object $F_r \in \mathbf{Perf}(\hat{A}_x)$ and a morphism $\phi(F_r) \to E$ whose cone $E'_r$ is now $(j+r)$-connective. For a fixed $r$, we have a commutative diagram with exact rows

<figure><img src="image_57.png" /></figure>

As $E'_r$ and $\phi\psi(E'_r)$ are both $(j+r)$-connective, we see that the vertical morphism in the middle induces bijection on $H^l$ for all $l \ge -r - j + 1$. As $r$ is arbitray, we deduce that $\phi\psi(E) \to E$ is an equivalence.

Finally, the same exact sequence $F_r \to E \to E'_r$ show that the cohomology groups of $\psi(E)$ are of finite type as $H^0(\hat{A}_x)$-modules, and thus that $\psi(E)$ is almost perfect as required. $\square$
## A.5. Cotangent complexes of derived Artin stacks

For a derived Artin stack $X$ over $k$, we have its big or fake cotangent complex $\mathbb{L}_{X/k}^{big}$. This is a non-necessarily quasi-coherent $\mathcal{O}_X$-dg-module. As an $\infty$-functor, it sends $u : \mathbf{Spec} A \to X$ to the $A$-dg-module $\mathbb{L}_{A/k}$. There is a canonical pull-back morphism $u^*\mathbb{L}_{X/k} \to \mathbb{L}_{A/k}$, which defines a morphism of $\mathcal{O}_X$-dg-modules

$$ \mathbb{L}_{X/k} \to \mathbb{L}_{X/k}^{big} $$

where $\mathbb{L}_{X/k}$ is the global cotangent complex of $X$ relative to $k$ as constructed in [TV08, §1.4]. As $\mathbb{L}_{X/k}$ is itself quasi-coherent, we get a canonical morphism of quasi-coherent complexes $\mathbb{L}_{X/k} \to (\mathbb{L}_{X/k}^{big})^{qcoh}$. Considering exterior product, we also get this way natural morphisms for all $i \ge 0$

$$ \wedge_{\mathcal{O}_X}^i \mathbb{L}_{X/k} \to (\wedge_{\mathcal{O}_X}^i \mathbb{L}_{X/k}^{big})^{qcoh}. $$

**THEOREM A.5.0.1.** Let $X$ be a derived Artin stack locally of finite presentation over $k$. For all $i \ge 0$ the natural morphism

$$ \wedge_{\mathcal{O}_X}^i \mathbb{L}_{X/k} \to (\wedge_{\mathcal{O}_X}^i \mathbb{L}_{X/k}^{big})^{qcoh} $$

is an equivalence of quasi-coherent complexes on $X$.

*Proof.* The proof is essentially the same as in [PTVV13, Prop. 1.14], together with the additional result from [Mon21]. Note that the statement of the theorem implies [PTVV13, Prop. 1.14] by taking global sections.

The proof goes by induction on the geometricity of $X$. When $X$ is affine, the statement follows formally from the shape of the adjunction

$$ i : \mathbf{QCoh}(X) \hookrightarrow \mathcal{O}_X - \mathbf{dg} : (-)^{qcoh}. $$

Indeed, the $\infty$-functor $E \mapsto E^{qcoh}$ is explicitely given by sending $E$ to the $A$-dg-module of global sections $\Gamma(X, E) \in A - \mathbf{dg} \simeq \mathbf{QCoh}(X)$. From this, it is obvious that the canonical morphisms $\wedge_{\mathcal{O}_X}^i \mathbb{L}_{X/k} \to (\wedge_{\mathcal{O}_X}^i \mathbb{L}_{X/k}^{big})^{qcoh}$ are indeed equivalence. In the same manner, the result remains true for disjoint union of derived affine schemes.

We assume now that the statement has been proved for $(n-1)$-geometric derived Artin stack. We let $f : U \to X$ be a smooth atlas with $U$ a disjoint union of affine derived schemes. Let $U_* \to X$ be the nerve of $f$, so each $U_i$ is a $(n-1)$-geometric derived Artin stack. For a derived stack $Y$ we denote by $TY[1]$ its shifted total tangent stack. It is formally defined by $\mathbf{Map}(\mathbf{Spec} k[\epsilon], Y)$, where $\epsilon$ sits in cohomological degree $-1$. The derived stack $TY[1]$ sits over $Y$, and as such is of the form $\mathcal{W}(\mathbb{L}_{Y/k}[-1])$, the linear stack associated to the perfect complex $\mathbb{L}_{Y/k}[-1]$. This is a $\mathbb{G}_m$-equivariant derived stack over $Y$, and according to [Mon21], the projection $\pi : [TY[1]/\mathbb{G}_m] \to Y \times B\mathbb{G}_m$ is such that the quasi-coherent push-forward $\pi_*(\mathcal{O})$ is given by

$$ \pi_*(\mathcal{O}) \simeq \bigoplus_p \mathrm{Sym}_{\mathcal{O}_Y}^p (\mathbb{L}_{Y/k}[-1]), $$
where the right hand side is a graded quasi-coherent complex on $Y$ defined by the natural grading of the $\mathrm{Sym}$. In other words, the weight $i$ piece of $\pi_*(\mathcal{O})$ is $\wedge^i \mathbb{L}_{Y/k}[-i]$.

It is very easy to check, for instance using the infinitesimal lifting properties, that $TU_0[1] \to TX[1]$ remains a smooth epimorphism. In particular, $TU_*[1]$, which is equivalent to the nerve of $TU_0[1] \to TX[1]$, induces an equivalence of derived $\mathbb{G}_m$-equivariant stacks over $X$

$$TX[1] \simeq \mathrm{Colim}_p TU_p[1].$$

If we denote by $p_i : TU_p[1] \to X$ the natural projection, and similarly $p : TX[1] \to X$, we thus have an equivalence of $\mathbb{G}_m$-equivariant quasi-coherent complexes $p_*(\mathcal{O}) \simeq \mathrm{Lim}_p(p_p)_*(\mathcal{O})$. Applying the main result of [Mon21] and considering the weight $p$ pieces, we get a canonical equivalence of quasi-coherent complexes

$$\wedge^i \mathbb{L}_{X/k} \simeq \mathrm{Lim}_p(f_p)_*(\wedge^i \mathbb{L}_{U_p/k})$$

where $f_i : U_i \to X$ is the natural projection.

Because the natural morphism $\mathrm{Colim}_p U_p \to X$ is an equivalence, we get, by descent, an equivalence of $\mathcal{O}_X$-dg-modules $\wedge^i \mathbb{L}_{X/k}^{\mathrm{big}} \simeq \mathrm{Lim}_p(f_p)_*(f_p^*)(\wedge^i \mathbb{L}_{X/k}^{\mathrm{big}})$. But we have a tautological identification $(f_p^*)(\wedge^i \mathbb{L}_{X/k}^{\mathrm{big}}) \simeq \wedge^i \mathbb{L}_{U_p/k}^{\mathrm{big}}$, and thus we get $\wedge^i \mathbb{L}_{X/k}^{\mathrm{big}} \simeq \mathrm{Lim}_p(f_p)_*(\wedge^i \mathbb{L}_{U_p/k}^{\mathrm{big}})$. As the $\infty$-functor $(-)^{qcoh}$ is right adjoint it commutes with limits. Also, as pull-backs of $\mathcal{O}$-dg-modules preserves quasi-coherent complexes, $(-)^{qcoh}$ also commutes with direct images. Finally, as each $U_p$ is $(n-1)$-geometric, we know that the theorem holds for the $U_p$'s. We thus get a natural equivalence

$$(\wedge^i \mathbb{L}_{X/k}^{\mathrm{big}})^{qcoh} \simeq \mathrm{Lim}_p(f_p)_*(\wedge^i \mathbb{L}_{U_p/k}).$$

But we have already seen that the right hand side is naturally equivalent to $\wedge^i \mathbb{L}_{X/k}$, and thus we have

$$(\wedge^i \mathbb{L}_{X/k}^{\mathrm{big}})^{qcoh} \simeq \wedge^i \mathbb{L}_{X/k}.$$

Unfolding the various equivalences we see that this equivalence is indeed induced by the canonical morphism $\wedge^i \mathbb{L}_{X/k} \to \wedge^i \mathbb{L}_{X/k}^{\mathrm{big}}$, which proves the theorem. $\square$

## A.6. Cotangent complexes of formal completion

As in the previous section let $X$ be a derived Artin stack of finite presentation over $k$. We assume here that $k$ is a field and that we have given ourselves a global point $x : \mathrm{Spec} k \to X$ (over more general rings $k$, we can always reduce to this special case by base change). As explained in §2.3.2 we have the formal completion $(X, x)^\wedge$. This formal completion can be described as a ringed site as follows. We let $X^{\mathrm{art}}$ be the $\infty$-category $\mathrm{dgart}_k^*/X$, of derived local Artinian affine schemes over $X$ (preserving the base point). The $\infty$-category is a ringed site, for the induced etale topology and for the canonical stack of simplicial rings sending $\mathrm{Spec} A \to X$
to A. We let $\mathcal{O}_{X^{art}} - \mathrm{dg}$ be the $\infty$-category of all $\mathcal{O}_{X^{art}}$-dg-modules. We have a full sub-$\infty$-category $\mathrm{QCoh}(X^{art}) \subset \mathcal{O}_{X^{art}} - \mathrm{dg}$ consisting of objects E such that for all morphism in $X^{art}$

<figure><img src="image_58.png" /></figure>

such that the induced morphism $E(B) \otimes_B A \to E(A)$ is a quasi-isomorphism. Note that by definition $\mathrm{QCoh}(X^{art})$ identifies canonically with $\mathrm{QCoh}((X, x)^\wedge)$, the $\infty$-category of quasi-coherent complexes on the formal completion $(X, x)^\wedge$ of X at $x$.

By Freyd representability the natural inclusion $j: \mathrm{QCoh}(X^{art}) \subset \mathcal{O}_{X^{art}} - \mathrm{dg}$ possesses a right adjoint denoted $E \mapsto E^{qcoh}$. We also have a canonical restriction $\infty$-functor

$$ \mathcal{O}_X - \mathrm{dg} \to \mathcal{O}_{X^{art}} - \mathrm{dg}, $$

and thus the cotangent complex of X (relative to k) provides a canonical object $\mathbb{L}_{X^{art}/k} \in \mathcal{O}_{X^{art}} - \mathrm{dg}$ called the *formal cotangent complex of X at x*. It is obviously a quasi-coherent complex over $X^{art}$. In the same manner, we have the big or fake cotangent complex $\mathbb{L}_{X^{art}/k}^{\mathrm{big}} \in \mathcal{O}_{X^{art}} - \mathrm{dg}$ given by $\mathbb{L}_{X^{art}/k}(\mathrm{Spec} A) := \mathbb{L}_{A/k}$.

The following theorem is the formal version of our previous result theorem A.5.0.1.

**THEOREM A.6.0.1.** With the notations and conditions as above, for all $i \ge 0$ the canonical morphism

$$ \wedge_{\mathcal{O}_{X^{art}}}^i \mathbb{L}_{X^{art}/k} \to (\wedge_{\mathcal{O}_{X^{art}}}^i \mathbb{L}_{X^{art}/k}^{\mathrm{big}})^{qcoh}. $$

*Proof.* We proceed as in theorem by induction on the geometricity of X. We let $f_*: U_* \to X$ be the nerve of a smooth atlas for X with $U_0$ a disjoint union of derived affine schemes. Moreover, by galois descent we can replace k by a algebraic separable extension $k'/k$, and thus assume that x lifts to a k-point u in $U_*$.

As we have already seen during the proof of A.5.0.1 we have equivalences in $\mathcal{O}_X - \mathrm{dg}$ and in $\mathrm{QCoh}(X)$

$$ \wedge^i \mathbb{L}_{X/k}^{\mathrm{big}} \simeq \mathrm{Lim}_p (f_p)_* (\wedge^i \mathbb{L}_{U_p/k}^{\mathrm{big}}) $$

$$ \wedge^i \mathbb{L}_{X/k} \simeq \mathrm{Lim}_p (f_p)_* \wedge^i \mathbb{L}_{U_p/k}. $$

As the restriction $\infty$-functor $\mathcal{O}_X - \mathrm{dg} \to \mathcal{O}_{X^{art}} - \mathrm{dg}$ commutes with limits, we see that we have a commutative square in $\mathcal{O}_{X^{art}} - \mathrm{dg}$

<figure><img src="image_59.png" /></figure>
where $\hat{f}_*: U_*^{art} \to X^{art}$ is the induced atlas on formal completions. Using induction on the geometricity of $X$ we see that we can reduce the theorem to the case $X$ is disjoint union of affines. Restricting to the component containing $x$ we can further assume that $X$ is an affine derived scheme $X = \mathbf{Spec} A$, where $A$ is of finite presentation over $k$.

We now fix $i \ge 0$, and consider the exact triangle of $\mathcal{O}_{X^{art}}$-dg-modules $\wedge^i \mathbb{L}_{X^{art}} \to \wedge^i \mathbb{L}_{X^{art}}^{big} \to E$. By construction, the $\mathcal{O}_{X^{art}}$-dg-module $E$ sends $\mathbf{Spec} A' \to X$ to the cone of $\wedge^i \mathbb{L}_A \otimes_A A' \to \wedge^i \mathbb{L}_{A'}$. For any quasi-coherent complex $E_0 \in \mathbf{QCoh}(X^{art})$ we have a natural equivalence

$$ \mathbf{Map}_{\mathcal{O}_{X^{art}}-\mathbf{dg}}(E_0, E) \simeq \lim_{A \to A'} \mathbf{Map}_{A'-\mathbf{dg}}(E_0(A'), E(A')). $$

In other words, this mapping space if also the mapping space of pro-$\hat{A}$-modules between $\lim_{A \to A'} E_0(A')$ and $\lim_{A \to A'} E(A')$. By Corollary A.3.0.5 we know that $E$ is equivalent to zero as a pro-object, and thus that $\mathbf{Map}_{\mathcal{O}_{X^{art}}-\mathbf{dg}}(E_0, E) \simeq 0$ for any quasi-coherent complex $E_0$, or equivalently that $E^{qcoh} \simeq 0$. This shows as required that $\wedge^i \mathbb{L}_{X^{art}} \simeq (\wedge^i \mathbb{L}_{X^{art}}^{big})^{qcoh}$. $\square$

### A.7. Deformation to the normal bundle

In this section we denote by $\mathcal{A}$ the quotient stack $[\mathbb{A}^1/\mathbb{G}_m]$. As explained in [Mou21], derived stacks over $\mathcal{A}$ should be considered as filtered derived stacks. We remind in particular (see §1.3.2), that there exists a natural symmetric monoidal equivalence of $\infty$-categories

$$ \mathbf{QCoh}(\mathcal{A}) \simeq \mathbf{dg}_k^{fil}, $$

between quasi-coherent complexes over $\hat{A}$ and filtered complexes of $k$-modules. This equivalence is induced by a natural (decreasing) filtered object $\mathcal{O}(-)$ in $\mathbf{QCoh}(\mathcal{A})$, namely the families of $\mathcal{O}(-)$ for $i \in \mathbb{Z}$. Here $\mathcal{O}(-)$ stands for the line bundle on $\mathcal{A}$ which is trivial on $\mathbb{A}^1$ together with its canonical $\mathbb{G}_m$-action of weight $-i$. The families of $\mathcal{O}(-)$ is endowed with canonical morphisms $\mathcal{O}(-) \to \mathcal{O}(-i+1)$ induced by multiplication by $X \in \Gamma(\hat{A}, \mathcal{O}(1))$. The $\infty$-functor $\mathbb{R}\mathrm{Hom}(\mathcal{O}(\cdot), -) : \mathbf{QCoh}(\hat{A}) \to \mathbf{dg}_k^{fil}$ is the equivalence mentioned above.

We consider the inclusion of the origin $\mathbf{Spec} k \to \mathbb{A}^1$, and the induced morphism on quotient stacks

$$ j : B\mathbb{G}_m \hookrightarrow [\mathbb{A}^1/\mathbb{G}_m] = \mathcal{A}. $$

As an object over $\mathcal{A}$, $B\mathbb{G}_m$ will be denoted by $\mathcal{Q} \in \mathbf{dSt}_k/\mathcal{A}$. This is a relative derived affine scheme, whose fiber at any non-zero point is empty, and whose fiber at 0 is $\mathbf{Spec} k[\epsilon_{-1}]$, where $\epsilon_{-1}$ is of degree $-1$ and $\epsilon_{-1}^2 = 0$. The induced $\mathbb{G}_m$-action acts by weight 1 on $\epsilon_{-1}$.

We now let $X \to Y$ be any morphism of derived stacks. we consider $Y \times \mathcal{Q}$ and $X \times \mathcal{A}$ as objects over $Y \times \mathcal{A}$, and we form the relative derived mapping stack

$$ Def(X/Y) := \underline{\mathbf{Map}}_{Y \times \mathcal{A}}(Y \times \mathcal{Q}, X \times \mathcal{A}) \in \mathbf{dSt}_k/Y \times \mathcal{A}. $$
DEFINITION A.7.0.1. The deformation to the normal bundle of a morphism $f: X \to Y$ is the derived stack defined above

$$ Def(X/Y) = Def(f) := \underline{\mathrm{Map}}_{/Y \times A}(Y \times \mathcal{Q}, X \times A) \in \mathrm{dSt}_k/Y \times A. $$

The deformation to the normal bundle $Def(X/Y)$ has the following two important properties. Its fiber over the open $Y = [\mathbb{G}_m/\mathbb{G}_m] \times Y \hookrightarrow A \times Y$ is $\mathrm{Map}_{/Y}(\emptyset, X) \simeq Y$, whereas its fiber at zero $Y \times B\mathbb{G}_m \hookrightarrow Y \times A$ is $[\mathrm{Map}_{/Y}(Y \times \mathrm{Spec} k[\epsilon_{-1}], X)/\mathbb{G}_m]$. Note that $[\mathrm{Map}_{/Y}(Y \times \mathrm{Spec} k[\epsilon_{-1}], X)]$ is identified with the shifted relative tangent bundle $T(X/Y)[1]$, where $\mathbb{G}_m$ acts by weight $-1$ dilatation along the fibers of $T(X/Y)[1] \to X$. When $X$ and $Y$ are derived Artin stacks locally of finite presentation, $T(X/Y)[1]$ is also the linear stack $\mathcal{V}(\mathbb{L}_{X/Y}[-1])$, associated to the perfect complex $\mathbb{L}_{X/Y}[-1]$ on $X$, endowed with its natural $\mathbb{G}_m$-action. Therefore, $Def(X/Y)$ provides indeed an interpolation, by means of filtration, between $X$ and $T(X/Y)[1]$, which is at the origin of the relative Hodge filtration.

Note that the projection $Y \times \mathcal{Q} \to Y \times A$ induces on derived mapping stacks a morphism $j: X \times A \to Def(f)$. Over the open point $[\mathbb{G}_m/\mathbb{G}_m]$ of $A$ the morphism $j$ recovers $f: X \to Y$, whereas over $B\mathbb{G}_m$ it is the zero section $X \to T(X/Y)[1]$.

We finish this section by the following comment. When $f: X = \mathcal{V}(E') \to Y = \mathcal{V}(E)$ is a morphism of derived linear stacks over $\mathrm{Spec} A$, induced by a morphism of bounded above $A$-dg-modules $u: E \to E'$, then $Def(f)$ is itself globally, as a derived stack over $\mathrm{Spec} A \times A$, a derived linear stack. Indeed, this can be seen directly using that for any derived stack $Z \to \mathrm{Spec} A$ we have functorial equivalences

$$ \mathrm{Map}_{\mathrm{dst}_A}(Z, \mathcal{V}(E)) \simeq \mathrm{Map}_{\mathrm{dg}_A}(E, \mathcal{O}_Z(Z)). $$

In this case, the derived linear stack $Def(f)$, relative to $\mathrm{Spec} A \times A$, can be described explicitly as $\mathcal{V}(u)$, where $u: E \to E'$ is now considered as a one step filtered quasi-coherent complex on $\mathrm{Spec} A$, and thus as a quasi-coherent complex over $\mathrm{Spec} A \times A$.

## A.8. Internal derived de Rham theory

We remind from [MRT22] the group stack (over $k$) $\mathcal{H}_0 := B\mathbb{G}_a \times \mathbb{G}_m$, semi-direct product of the multiplicative group acting, by its standard weight 1 action, on $B\mathbb{G}_a$. As we have seen, the $\infty$-category $\mathrm{QCoh}(B\mathcal{H}_0)$ is canonically equivalent to the $\infty$-category of graded mixed complexes

$$ \mathrm{QCoh}(B\mathcal{H}_0) \simeq \epsilon - \mathrm{dg}_k^{gr}. $$

In the same manner, there is an equivalence of $\infty$-categories of commutative algebras

$$ \mathcal{O}_{B\mathcal{H}_0} - \mathrm{cdga}_{qcoh} \simeq \epsilon - \mathrm{cdga}_k^{gr}, $$

where $\mathcal{O}_{B\mathcal{H}_0} - \mathrm{cdga}_{qcoh}$ is the $\infty$-category of quasi-coherent cdga's over the stack $B\mathcal{H}_0$, defined by the usual left Kan extension formula

$$ \mathcal{O}_{B\mathcal{H}_0} - \mathrm{cdga}_{qcoh} = \lim_{\mathrm{Spec} A \to B\mathcal{H}_0} \mathrm{cdga}_A. $$
For any derived stack $X$, we define the $\infty$-category of quasi-coherent graded mixed $\mathcal{O}_X$-cdga's by

$$ \mathcal{O}_X - \epsilon - \mathbf{cdga}_{qcoh}^{gr} := \mathcal{O}_{X \times B\mathcal{H}_0} - \mathbf{cdga}_{qcoh}. $$

Note that, by definition, an object in $\mathcal{O}_X - \epsilon - \mathbf{cdga}_{qcoh}^{gr}$ is a sheaf of graded mixed cdga's over $X$ which is moreover linear over $\mathcal{O}_X$ and quasi-coherent. That is, it is given by a family of graded mixed $A$-linear cdga $D_A$, for each $\mathbf{Spec} A \to X$, together with compatible by base change equivalences

$$ D_A \otimes_A B \simeq D_B $$

for each $\mathbf{Spec} B \to \mathbf{Spec} A \to X$.

**REMARK A.8.0.1.** One should not confuse $\mathcal{O}_X - \epsilon - \mathbf{cdga}_{qcoh}^{gr}$ with the notion of graded mixed cdga's over $X$ used in the definition of derived foliations over the derived stack $X$: the latter objects are not quasi-coherent by any means. We hope that the wording "quasi-coherent graded mixed $\mathcal{O}_X$-cdga" will help avoiding the confusion.

Taking the weight 0 part defines an $\infty$-functor $\mathcal{O}_X - \epsilon - \mathbf{cdga}_{qcoh}^{gr} \to \mathcal{O}_X - \mathbf{cdga}_{qcoh}$, where $\mathcal{O}_X - \mathbf{cdga}_{qcoh}$ is the $\infty$-category of sheaves of quasi-coherent $\mathcal{O}_X$-linear cdga's over $X$ (defined by $\mathcal{O}_X - \mathbf{cdga}_{qcoh} = \lim_{\mathbf{Spec} A \to F} \mathbf{cdga}_A$). This forgetful $\infty$-functor possesses a left adjoint called the relative or internal de Rham algebra construction

$$ \mathbf{DR}^{\text{int}}(-/\mathcal{O}_X) : \mathcal{O}_X - \mathbf{cdga}_{qcoh} \to \mathcal{O}_X - \epsilon - \mathbf{cdga}_{qcoh}^{gr}. $$

Concretely, if $B \in \mathcal{O}_X - \mathbf{cdga}_{qcoh}$, then for any $\mathbf{Spec} A \to X$ we consider $\mathbf{DR}(B_A/A) \in \epsilon - \mathbf{cdga}_A^{gr}$. The base change equivalences for $B$ induces base change equivalences for $\mathbf{DR}(B_A/A)$ making it into a quasi-coherent $\mathcal{O}_X$-linear graded mixed cdga.

We will now be interested in the specific case where $X = B\mathcal{H}_0$. We get this way an $\infty$-functor

$$ \mathcal{O}_{B\mathcal{H}_0} - \mathbf{cdga}_{qcoh} \to \mathcal{O}_{B\mathcal{H}_0} - \epsilon - \mathbf{cdga}_{qcoh}^{gr} \simeq \mathcal{O}_{B\mathcal{H}_0 \times B\mathcal{H}_0} \mathbf{cdga}_{qcoh}. $$

Note that, using the same argument as in the proof of the equivalence $\mathcal{O}_{B\mathcal{H}_0} - \mathbf{cdga}_{qcoh} \simeq \epsilon - \mathbf{cdga}^{gr}$, the $\infty$-category $\mathcal{O}_{B\mathcal{H}_0 \times B\mathcal{H}_0} - \mathbf{cdga}_{qcoh}$ is naturally equivalent to the $\infty$-category of double graded mixed cdga: that a cdga endowed with two multiplicative $\mathbb{Z}$-graduations and two compatible mixed structures $\epsilon_1$ and $\epsilon_2$ such that $\epsilon_1\epsilon_2 + \epsilon_2\epsilon_1 = 0$ ($\epsilon_1$ is of weight bigrading $(1,0)$ and $\epsilon_2$ of bigrading $(0,1)$, and both are of cohomological grading $-1$). The internal de Rham construction is then explicitly given by sending an graded mixed cdga $A$ to its graded de Rham algebra $\mathbf{DR}^{gr}(A/k)$, which is endowed with a double graded mixed structure: $\epsilon_1$ is induced by the original mixed structure on $A$, and $\epsilon_2$ is the de Rham differential of $\mathbf{DR}^{gr}(A/k)$.

Double graded mixed cdga's have two different realizations constructions, depending if we realize the first of the second mixed structure. These corresponds to considering direct images along the two projections

$$ B\mathcal{H}_0 \leftarrow B\mathcal{H}_0 \times B\mathcal{H}_0 \to B\mathcal{H}_0 $$
which induced the two (non-Tate) realization $\infty$-functors

$$|-|_1, |-|_2 : \mathcal{O}_{B\mathcal{H}_0 \times B\mathcal{H}_0} - \mathbf{cdga}_{qcoh} \to \mathcal{O}_{B\mathcal{H}_0} - \mathbf{cdga}_{qcoh}.$$

Applied to $\mathbf{DR}^{\text{int}}(A/k)$, for $A$ a graded mixed cdga, we get this way two possible de Rham complexes of $A$

$$\hat{C}_{DR}^{I}(A/k) := |\mathbf{DR}(A/k)|_{1} \qquad \hat{C}_{DR}^{II}(A/k) := |\mathbf{DR}(A/k)|_{2},$$

which both of them are graded mixed cdga's.

**DEFINITION A.8.0.2.** Let $A$ be a graded mixed cdga (over $k$).

(1) The (derived) de Rham complex of the first kind of $A$ relative to $k$ is the graded mixed cdga given by

$$\hat{C}_{DR}^{I}(A/k) := |\mathbf{DR}^{\text{int}}(A/k)|_{1}.$$

(2) The (derived) de Rham complex of the second kind of $A$ relative to $k$ is the graded mixed cdga given by

$$\hat{C}_{DR}^{II}(A/k) := |\mathbf{DR}^{\text{int}}(A/k)|_{2}.$$

(3) The absolute (derived) de Rham complex of $A$ relative to $k$ is the graded mixed cdga given by

$$\hat{C}_{DR}^{abs}(A/k) := ||\mathbf{DR}^{\text{int}}(A/k)|_1|_2 \simeq ||\mathbf{DR}^{\text{int}}(A/k)|_2|_1.$$

Visually, the two versions of the de Rham complexes of definition above can be summarized as follows.

$$\hat{C}_{DR}^{I}(A/k) = |A| \to |\mathbb{L}_{A/k}| \to |\wedge_{A}^{2} \mathbb{L}_{A/k}| \to \dots \to |\wedge_{A}^{i} \mathbb{L}_{A/k}| \to |\wedge_{A}^{i+1} \mathbb{L}_{A/k}| \to \dots$$

$$\hat{C}_{DR}^{II}(A/k) = |A| \to \mathbb{L}_{A/k} \to \wedge_{A}^{2} \mathbb{L}_{A/k} \to \dots \wedge_{A}^{i} \mathbb{L}_{A/k} \to \wedge_{A}^{i+1} \mathbb{L}_{A/k} \to \dots$$

Note also that the two graded mixed cdga's $\hat{C}_{DR}^{I}(A/k)$ and $\hat{C}_{DR}^{II}(A/k)$ are different, but their realizations are canonically equivalent as cdga's

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

In a more explicit form, $\hat{C}_{DR}^{I}(A/k)$ is a graded mixed cdgas whose piece of weight $i$ is $|\wedge_{A}^{i} \mathbb{L}_{A/k}|$, where $\mathbb{L}_{A/k}$ is the graded cotangent complexe of $A$, endowed with the induced mixed structure $\epsilon(dR(a)) = -dR(\epsilon(a))$, where $\epsilon$ is the original mixed structure on $A$. The mixed structure is then induced by the de Rham differential of $A$.

On the other hand, the graded piece of weight $i$ of $\hat{C}_{DR}^{II}(A/k)$ is given by the complex

$$\prod_{j \ge 0} (\wedge^j \mathbb{L}_{A/k})^{(i)} [-j],$$
endowed with the total differential $d + dR$. The mixed structure relating these graded pieces is then induced by the mixed structure $\epsilon$ of $A$. Note that for each $i$, we have a decomposition

$$ (\wedge^j \mathbb{L}_{A/k})^{(i)} \simeq \bigoplus_{n_1 k_1 + \cdots + n_j k_j = i} \bigotimes_l \wedge^{k_l} (\mathbb{L}_{A/k}^{(n_l)})[-j], $$

so the weight $i$ piece of $\hat{C}_{DR}^{*,II}(A/k)$ can also be written

$$ \prod_{j \ge 0} \left( \bigoplus_{n_1 k_1 + \cdots + n_j k_j = i} \bigotimes_l \wedge^{k_l} (\mathbb{L}_{A/k}^{(n_l)})[-j] \right). $$

As an example of applications of the notions of internal de Rham complex we prove the following well known lemma.

**PROPOSITON A.8.0.3.** Let $A \to k$ be a local augmented $k$-linear Artinian dg-algebra, almost of finite presentation over $k$. Then, the canonical morphism induces a quasi-isomorphism of cdga's

$$ k \simeq \hat{C}_{DR}(A/k). $$

*Proof.* Let $D := \mathbf{DR}(k/A) \in \epsilon - \text{cdga}_k^{gr}$, and we consider the canonical morphism $A \to \mathbf{DR}(k/A)$, where $A$ is considered as a graded mixed cdga of weight 0 with trivial mixed structure. This morphism induces a morphism on internal de Rham algebras

$$ \phi : \mathbf{DR}(A/k) \simeq \mathbf{DR}^{\text{int}}(A/k) \to \mathbf{DR}^{\text{int}}(D/k). $$

This is a morphism of double graded mixed cdga's, which can be realized in the absolute sense to get a morphism of cdga's

$$ \hat{C}_{DR}(A/k) \to \hat{C}_{DR}^{\text{abs}}(D/k). $$

This morphism can be described as first realizing along the first graded mixed structure and then realizing along the second one. The morphism $|\phi|_1$ is given by $\mathbf{DR}(A/k) \to \hat{C}_{DR}^I(D/k)$, as the first graded mixed structure on $\mathbf{DR}(A/k)$ is trivial of weight 0. On the piece of weight $i$ the morphism $|\phi|_1$ is the canonical morphism

$$ \wedge^i \mathbb{L}_{A/k}[i] \to |\wedge^i \mathbb{L}_{D/k}[i]|_1. $$

For $i = 0$, this morphism is the canonical morphism of cdga's $A \to \hat{C}_{DR}(k/A)$, which is a quasi-isomorphism as shown in [CPT$^{+}$17, Lem. 2.2.4]. The morphism $|\phi|_1$ in weights $i > 0$ is the canonical morphism

$$ \wedge^i \mathbb{L}_{A/k} \to \hat{C}_{DR}(k/A, \wedge^i \mathbb{L}_{A/k} \otimes_A k), $$

where the right hand side is the derived de Rham cohomology of $k$ relative to $A$, with coefficients in $\mathbb{L}_{A/k} \otimes_A k$. As $\mathbb{L}_{A/k}$ is almost perfect $A$-dg-module, we also know that this morphism is a quasi-isomorphism (see [CPT$^{+}$17, Cor. 2.2.5] for the case of perfect modules which can be extended to almost perfect modules in a straightforward manner by postnikov approximation).

This shows that $|\phi|_1 : \mathbf{DR}(A/k) \to \hat{C}_{DR}^I(D/k)$ is a quasi-isomorphism of graded mixed cdga's, and thus induces an equivalence after realizations $\hat{C}_{DR}(A/k) \simeq \hat{C}_{DR}^{\text{abs}}(D/k)$. Now,
$\hat{C}_{DR}^{abs}(D/k)$ can also be computed as $||\mathbf{DR}^{int}(D/k)|_2|_1$. As a graded cdga, we have $D \simeq \mathrm{Sym}_k(E[2])$, where $E := \mathbb{L}_{A/k} \otimes_A k$. Therefore, its de Rham cohomology is trivial, and we thus have $|\mathbf{DR}^{int}(D/k)|_2 \simeq k$. This shows that $\hat{C}_{DR}^{abs}(D/k) \simeq k$ and finishes the proof of the Proposition. $\square$

REMARK A.8.0.4. Proposition A.8.0.3 is valid in a more general context, with a similar proof. For any derived affine $k$-scheme $X = \mathbf{Spec} A$ almost of finite presentation over $k$, the natural morphism

$$ \hat{C}_{DR}(A/k) \to \hat{C}_{DR}(A_{red}/k) $$

is an equivalence.

Another useful statement is the following reconstruction result.

PROPOSITON A.8.0.5. Let $B$ be a graded mixed cdga with $B^{(0)} = A$ connective and $\mathrm{Sym}_A(B^{(1)}) \simeq B$ as graded cdgas. Then, there is a canonical equivalence of graded mixed cdga's

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

*Proof.* We first notice that the canonical morphism $B \to \mathbf{DR}^{int}(A/B)$ induces an equivalence of graded mixed cdga's $B \simeq \hat{C}_{DR}^{II}(A/B)$. Indeed, this statement is independent of the mixed structure on $B$, as it is enough to show that $B \to \hat{C}_{DR}^{II}(A/B)$ is an equivalence of graded cdga's. To see this it is enough to write down $B = \mathrm{Sym}_A(E)$ for a perfect $A$-dg-module $E$, and write $A$ as $\mathrm{Sym}_A(E')$ where $E'$ is the cone of the identity $E \to E$. as a graded cdga, the de Rham cohomology of the second kind $\hat{C}_{DR}^{II}(A/B)$ can be modeled by the (graded) de Rham complex of $\mathrm{Sym}_A(E')$ over $B$, which is explicitly given by $\mathrm{Sym}_A(K[-1])$ where $K$ is the cone of the natural projection $E' \to E[1]$. The $A$-dg-module $K$ is canonically quasi-isomorphic to $E[1]$ and the corresponding morphism $B \to \mathrm{Sym}_A(K[-1])$ sends $E$ to $K[-1]$ by its canonical identification and thus is an equivalence.

To finish the proof of the Proposition it is thus enough to identify $\hat{C}_{DR}^{I}(A/B)$ canonically with $\hat{C}_{DR}^{II}(A/B)$. For this we use the following statement about double graded mixed complexes.

LEMMA A.8.0.6. There exists a fully faithful symmetric monoidal $\infty$-functor

$$ \mathrm{QCoh}(B\mathcal{H}_0) \hookrightarrow \mathrm{QCoh}(B\mathcal{H}_0 \times B\mathcal{H}_0) $$

from graded mixed complexes to double graded mixed complexes, whose essential image consists of all object $E$ which, as a bigraded complex, is concentrated on the diagonal: $E^{(i,j)} \simeq 0$ for all $i \neq j$.

*Proof of the lemma.* The most easy proof consists of using the equivalence between graded mixed complexes and complete filtered complexes (see Proposition 1.3.3.3 and Corollary 1.3.4.4). Via this translation the lemma becomes obvious: the right adjoint $d_*$ consists of sending a filtered complex to the corresponding "diagonal" bilfiltered complex. More explicitly, if $\{F^j\}_{j \in \mathbb{Z}}$ is a filtered complex, the corresponding bifiltered complex $X$ is given by $X^{i,j} := F^{\mathrm{min}(i,j)}$ with
the natural transition morphisms. In terms of functors category, if we denote by $\mathbb{Z}$ the poset of integers, this full embedding is given by

$$d_1: \mathrm{Fun}(\mathbb{Z}, \mathbf{dg}_k) \to \mathrm{Fun}(\mathbb{Z} \times \mathbb{Z}, \mathbf{dg}_k)$$

given by left Kan extension along the diagonal $d: \mathbb{Z} \hookrightarrow \mathbb{Z} \times \mathbb{Z}$. This is clearly a fully faithful construction and its essential image correspond precisely to complete bifiltered complexes $E$ whose associated bigraded are concentrated on the diagonal. $\square$

The full embedding of the previous lemma being symmetric monoidal $\infty$-functor and thus extends to a similar statement concerning commutative algebras object. This means that in particular, that any double graded mixed cdga $D$, which is concentrated on the diagonal as a bi-graded complex, is naturally symmetric with respect to the action of $\mathbb{Z}/2$ which exchanges the two gradings and the two mixed structures. In particular, for any such object $D$, we have a canonical equivalence $|D|_1 \simeq |D|_2$, and both are equivalent to $|E|$, where $E$ is the graded mixed cdga obtained from the lemma A.8.0.6. This can therefore be applied to $D = \mathbf{DR}^{\mathrm{int}}(A/B)$, which as a bi-graded cdga is equivalent to $\mathrm{Sym}_A(E[2])$ and thus is indeed concentrated on the diagonal. $\square$

## A.9. Derived affine stacks

**A.9.1. Cosimplicial-simplicial algebras.** We let denote by $sCR$ be the category of simplicial commutative rings. It is endowed with its usual *model category structure* for which equivalences and fibrations are defined on the underlying simplicial sets.

Similarly, we denote by $csCR$ the category of cosimplicial objects in $sCR$, or equivalently of cosimplicial-simplicial commutative rings. Its objects will be denoted by $A_*^*$, where the superscript will refer to the cosimplicial direction and subscript to the simplicial direction. The category $csCR$ can be endowed with a *tensored* and *cotensored* structure over the category $sSet$ of simplicial sets. We warn the reader here that there are several natural such structures and that the choice we make here is important for the sequel. For a simplicial set $K$ and an object $A$ of $csCR$, we define the cotensored structure by $A^K$ by the formula

$$(A^K)_p^q := (A_p^q)^{K_q}$$

for $[p] \in \Delta^{op}$ and $[q] \in \Delta$. The tensored structure $K \otimes A$, for $A \in csCR$ and $K \in sSets$, is defined by the usual adjunction formula $\mathrm{Hom}(K \otimes A, B) \simeq \mathrm{Hom}(A, B^K)$, for arbitrary $B \in csCR$. Similarly, we have a canonical simplicial enrichment for which the simplicial Homs are denoted by $\underline{\mathrm{Hom}}$ and are defined by

$$\underline{\mathrm{Hom}}(A, B)_p := \mathrm{Hom}_{csCR}(A, B^{\Delta^p}).$$

By considering the natural embedding $\mathrm{Mod}_{\mathbb{Z}} \subset C(\mathbb{Z})$, from $\mathbb{Z}$-modules to cochain complexes, any object $A \in csCR$ can be considered, by forgetting the multiplicative structure, as a cosimplicial-simplicial object in $C(\mathbb{Z})$. We denote by $\mathrm{Tot}^\pi(A)$ the complex equals to $\prod_{q-p=n} A_p^q$
in cohomological degree $n$ and whose differential is given by the alternated sums of the faces and cofaces of the cosimplicial-simplicial diagram. Equivalently, $A$ can be first turned into a bi-complex, by individual totalisations along the cosimplicial and the simplicial directions, and $Tot^\pi(A)$ can then be realized as the product-total complex of this bicomplex. Note also that $Tot^\pi(A)$ is a model for the homotopy limit-colimit of $A$

$$Tot^\pi(A) \simeq \operatorname{holim}_{[p] \in \Delta^{op}} (\operatorname{hocolim}_{[q] \in \Delta} A_p^q) \in C(\mathbb{Z}).$$

**DEFINITION A.9.1.1.** A morphism $A \to B$ in $csCR$ is a completed quasi-isomorphism if the induced morphism $Tot^\pi(A) \to Tot^\pi(B)$ is a quasi-isomorphism of complexes.

We warn the reader that even though the categories $csCR$ and $scCR$, of cosimplicial-simplicial commutative rings and of simplicial-cosimplicial rings, are equivalent, the notion of completed quasi-isomorphisms is not the same as the notion of weak equivalences used for instance in [BCN21].

**THEOREM A.9.1.2.** *The category $csCR$ is endowed with a model category structure for which the weak equivalences are the completed quasi-isomorphism, and the fibrations are the epimorphisms. This model category structure is moreover cofibrantly generated and is a simplicial model category for the above mentioned simplicial enrichment.*

*Proof.* As fibrations<fnref n="5" /> and equivalences are defined via the functor $Tot^\pi$, the proof of the theorem consists of transferring the projective model structure on $C(\mathbb{Z})$ to $csCR$ via the functor $Tot^\pi$. For this we apply the path object argument (see for instance [BM03, §2.6]).

The functor $Tot^\pi$ is a right adjoint whose left adjoint sends a complex $E \in C(\mathbb{Z})$ to the free commutative ring over the cosimplicial-simplicial module $\phi(E)$ given by

$$(p, q) \mapsto (C^*(\Delta^p) \otimes C_*(\Delta^q) \otimes E)^0 / \operatorname{Im} d^{-1},$$

where $C^*(\Delta^p)$ and $C_*(\Delta^q)$ are the cohomology and homology complexes of the standard simplex $\Delta^p$ and $\Delta^q$. This left adjoint will be denoted by $L\phi$, as being the composition of the free commutative ring functor $L:csMod \to csCR$, with the functor $\phi: C(\mathbb{Z}) \to csMod$ defined by the above formula. Clearly, the functor $Tot^\pi$ preserves small objects (but does not commute with filtered colimits in general). In order to apply [BM03, §2.6] we thus simply have to prove that $csCR$ possesses a fibrant replacement functor and that all fibrant object possesses a path object. But all objects are fibrant by definition, and we are thus reduced to show the existence of path objects. For this we use the following lemma.

**LEMMA A.9.1.3.** For any simplicial set $K$ and any $A \in csCR$, there exists a canonical quasi-isomorphism

$$Tot^\pi(A^K) \simeq Tot^\pi(A)^K, \\ \text{where for } E \in C(\mathbb{Z}) \text{ and } K \in sSet \text{ we define}$$

$$E^K := \operatorname{holim}_{[p] \in \Delta} E^{K_p}.$$

<footnote n="5">⁵Note that $f: A \to B$ in $csCR$ is surjective iff $Tot^\pi(f)$ is.</footnote>
Proof of the lemma. We remind that for any functor $F : \Delta \times \Delta \to \mathbf{dg}$ (or more generally any bi-cosimplicial object in a complete $\infty$-category), the homotopy limit of $F$ can be computed, up to a quasi-isomorphism, either component-wise, or diagonally

$$ \operatorname{holim}_{[p] \in \Delta} F(p, p) \simeq \operatorname{holim}_{[p] \in \Delta} \operatorname{holim}_{[q] \in \Delta} F(p, q). $$

We apply this to the functor sending $([p], [q])$ to the complex $N(A_*^q)^{K_p}$, where $N : s\mathbf{Mod} \to \mathbf{dg}$ is the normalization construction. Computing the homotopy limit diagonally yields $\operatorname{Tot}^\pi(A^K)$, whereas the double homotopy limit construction yields $\operatorname{Tot}^\pi(A)^K$. $\square$

Lemma A.9.1.3 now implies the existence of path objects. Indeed, for $A \in \mathbf{csCR}$ an explicit path object is given by $A \to A^{\Delta^1} \to A^{\partial \Delta^1} = A \times A$. The fact that the constant map morphism $A \to A^{\Delta^1}$ is a completed quasi-isomorphism follows from the lemma A.9.1.3. The fact that $A^{\Delta^1} \to A^{\partial \Delta^1} = A \times A$ is a fibration simply follows from the fact that it is a levelwise epimorphism as this can be checked easily using the explicit formula for $A^K$.

In order to finish the proof of the theorem, we are left to showing that the simplicial enrichment is compatible with the model category structure. In other words, we must show that for any cofibration of simplicial sets $i : K \hookrightarrow L$ and any fibration $f : A \to B$ in $\mathbf{csCR}$, the induced morphism

$$ f^i : A^L \to B^L \times_{B^K} A^K $$

is a fibration, which is also an equivalence if $i$ is a trivial cofibration or $f$ is a trivial fibration. The fact that the morphism $f^i$ is a fibration is clear because fibrations are epimorphisms and because of the explicit formula for the exponentiation by a simplicial set in $\mathbf{csCR}$. Finally, the fact that $f^i$ is also a weak equivalence when $i$ or $f$ is so simply follows from Lemma A.9.1.3. $\square$

**DEFINITION A.9.1.4.** The $\infty$-category of cs-rings is the $\infty$-category obtained from $\mathbf{csCR}$ by inverting the completed quasi-isomorphisms. It is denoted by $\mathbf{csCR}$ (or by $\mathbf{csCR}_U$ if one wants to specify cosimplicial-simplicial rings belonging to a given universe $U$).

**REMARK A.9.1.5.** We note also that the model structure on $\mathbf{csCR}$ can be equivalently be obtained by first constructing a model structure on $\mathbf{csMod}_Z$, the category of cosimplicial-simplicial abelian groups, where again the equivalences and fibrations are defined via the $\operatorname{Tot}^\pi$ functor. The model structure on $\mathbf{csCR}$ can then be obtained by transferring along the forgetful functor $\mathbf{csCR} \to \mathbf{csMod}_Z$, with left adjoint given by the free commutative cosimplicial-simplicial ring.

**REMARK A.9.1.6.** It is possible to construct an $\infty$-functor $\operatorname{Tot}^\pi : \mathbf{csCR} \to \mathbf{LSym} - \mathbf{Alg}$, from our $\infty$-category of cosimplicial-simplicial commutative rings to the $\infty$-category of derived rings, namely modules over the $\mathbf{LSym}$-monad. Here we denote by $\mathbf{LSym}$ the monad associated to the derived commutative operad as in [BCN21, Ex. 3.71]. This $\infty$-functor simply sends $A$ to $\lim_q A_*^q$, where the limit is taken in $\mathbf{LSym} - \mathbf{Alg}$ and $A_*^q$ is considered as a connective $\mathbf{LSym}$-algebra and thus as an object in $\mathbf{LSym} - \mathbf{Alg}$. This $\infty$-functor is the right adjoint of
an adjunction, and preserves free objects over perfect complexes. It is likely that $Tot^\pi$ is an equivalence of $\infty$-categories when restricted to nice enough objects.

**A.9.2. Spectrum of cosimplicial-simplicial rings.** For a fixed commutative simplicial ring $k \in sCR$, we can work relatively over $k$ and define the model category $k - csCR$ of cosimplicial-simplicial commutative $k$-algebras, and its associated $\infty$-category $k - \mathbf{csCR}$. As usual, we have a natural equivalence of $\infty$-categories

$$k - \mathbf{csCR} \simeq k/\mathbf{csCR},$$

where $k$ is considered as an object in $\mathbf{csCR}$ which is constant is the cosimplicial direction.

We consider $dAff_k := (k - sCR)^{op}$, the model category of derived affine $k$-schemes, which is defined to be the opposite category of that of simplicial commutative $k$-algebras. Remind from [TV08] that it can be endowed with the fpqc model topology, and that we can consider the model category of (hyper-complete) stacks $dAff_k^{\sim,fpqc}$. There are here some set-theoretical issues, that can be solved, as usual, by fixing two universes $U \in \mathcal{V}$. By definition $k - sCR$ refers here to $U$-small simplicial $k$-algebras, and $dAff_k^{\sim,fpqc}$ is then the category of functors $k - sCR \to sSet_{\mathcal{V}}$ to $\mathcal{V}$-small simplicial sets.

We then consider a functor

$$\mathbf{Spec}^{\Delta} : k - \mathbf{csCR}_{\mathcal{V}}^{op} \to dAff_k^{\sim,fpqc},$$

from $\mathcal{V}$-small cosimplicial-simplicial commutative $k$-algebras to $dAff_k^{\sim,fpqc}$. It is defined by sending $A \in k - \mathbf{csCR}_{\mathcal{V}}$ to the functor $\mathbf{Spec}^{\Delta} A : sCR \to sSet_{\mathcal{V}}$ defined by

$$\mathbf{Spec}^{\Delta} A : B \mapsto \underline{Hom}(A, B) = \underline{Hom}_{\mathbf{csCR}}(A, B^{\Delta}),$$

where $\underline{Hom}$ are the simplicial $\underline{Hom}$'s of the simplicial enrichement described in the previous section, and $B$ is considered as an object in $k - \mathbf{csCR}_{\mathcal{V}}$ by viewing it as constant in the cosimplicial direction.

**PROPOSITON A.9.2.1.** The functor $\mathbf{Spec}^{\Delta} : k - \mathbf{csCR}_{\mathcal{V}}^{op} \to dAff_k^{\sim,fpqc}$ is right Quillen, and the induced functor on $\infty$-categories

$$\mathbb{R}\mathbf{Spec}^{\Delta} : k - \mathbf{csCR}_{\mathbb{U}} \to \mathbf{dSt}_k$$

is fully faithful. The essential image of $\mathbf{Spec}^{\Delta}$ is the smallest full sub-$\infty$-category of $\mathbf{dSt}_k$ containing the objects $K(\mathbb{G}_a, n)$ for various $n$, and which is stable by $U$-small limits.

*Proof.* This is proven in a very similar manner than [Toe06, Cor. 2.2.3]. The left adjoint to $\mathbf{Spec}^{\Delta}$, denoted by $\mathcal{O}$, sends a representable presheaf $h^B = \underline{Hom}(B, -) : k - sCR \to sSet$ to $B \in k - \mathbf{csCR}$, which is considered as constant in the cosimplicial direction. It also sends an object of the form $K \times h^B$, for a simplicial set $K \in sSet$, to $B^K \in k - \mathbf{csCR}$. Finally, it is uniquely defined by these properties together with the requirement that it sends colimits in $dAff_k^{\sim,fpqc}$ to limits in $k - \mathbf{csCR}$.

To prove that $\mathbf{Spec}^{\Delta}$ is right Quillen we use that $dAff_k^{\sim,fpqc}$ is a left Bousfield localization of the levelwise projective model structure on the category of simplicial presheaves
Fun($k - sCR_U, sSet$) obtained by inverting fpqc-local equivalences and equivalences in $sCR$ (see [TV08]). Therefore, by general facts about Bousfield localizations, it is enough to show that $\mathbf{Spec}^{\Delta}$ is a right Quillen functor for the levelwise projective model structure, and moreover that for any cofibrant $A \in k - csCR$, $\mathbf{Spec}^{\Delta}(A)$ is a fibrant object in $dAff_{k}^{\sim,fpqc}$. The first of these statements easily follows from the fact that $k - csCR$ is a simplicial model category in which every object is fibrant, and thus that if $A \to A'$ is a (trivial) cofibration in $k - csCR$, for any $C \in k - sCR$ the morphism $\mathbf{Hom}(A', C) \to \mathbf{Hom}(A, C)$ is a (trivial) fibration of simplicial sets. For the second of these statements, let $A \in k - csCR$ be a cofibrant object. Any equivalence $B \to B'$ in $k - sCR$ obviously induces an equivalence in $k - csCR$ when considered both $B$ and $B'$ as constant in the cosimplicial direction, and thus $\mathbf{Hom}(A, B) \to \mathbf{Hom}(A, B')$ is an equivalence of simplicial sets. This shows that $\mathbf{Spec}^{\Delta}(A)$ is fibrant when the topology is the trivial topology. Finally, if $B = \text{holim}_i B_i$ is a homotopy limit in $k - sCR$, it is also a homotopy limit in $k - csCR$, and thus the natural morphism $\mathbf{Hom}(A, B) \to \text{holim}_i \mathbf{Hom}(A, B_i)$ is an equivalence. As the fpqc model topology is subcanonical, this implies that $\mathbf{Spec}^{\Delta}(A)$ is indeed a stack for the fpqc topology, and thus fibrant as an object in $dAff_{k}^{\sim,fpqc}$ (see [TV08]).

To finish the proof of the Proposition, let $A \in k - csCR_U$ and $X = \mathbf{Spec}^{\Delta}(A)$. By definition of the simplicial structure on $k - csCR$, the functor $X$ is the geometric realization of the simplicial object $q \mapsto h_{*}^{A_q}$, where $h_{*}^{A_q} : k - sCR \to sSet$ is corepresented by $A_q \in k - sCR$. In other words, we have $\mathbf{Spec}^{\Delta}(A) \simeq \text{hocolim}_q h_{*}^{A_q}$. Therefore, the adjunction morphism

$$A \to \mathcal{O}(\mathbf{Spec}^{\Delta}(A))$$

is the canonical morphism $A \to \text{holim}_q A_q$ in $\text{Ho}(k - csCR_U)$. This canonical morphism is obviously an equivalence, showing the fully faithfulness property in the Proposition. It is then formal that the essential image of $\mathbb{R}\mathbf{Spec}^{\Delta}$ is stable by limits. It also contains the objects $K(\mathbb{G}_a, n)$, as these are the images of $L\phi(\mathbb{Z}[-n])$, where $L\phi : C(\mathbb{Z}) \to k - csCR$ is the left adjoint to $\text{Tot}^{\pi}$. Finally, as $k - csCR$ is cofibrantly generated, any object is equivalent to a $I$-cell object, where $I$ is the image by $L\phi$ of the generating cofibrations in $C(\mathbb{Z})$. The images of $I$ by $\mathbf{Spec}^{\Delta}$ are equivalent to morphisms of the form $* \to K(\mathbb{G}_a, n)$, so that any object is the essential image of $\mathbb{R}\mathbf{Spec}^{\Delta}$ lies in the smallest sub-$\infty$-category containing the $K(\mathbb{G}_a, n)$ and stable by limits. $\square$

**DEFINITION A.9.2.2.** The $\infty$-category of (k-linear) derived affine stacks is the essential image of $\mathbb{R}\mathbf{Spec}^{\Delta} : k - \mathbf{csCR}_U^{\text{op}} \to \mathbf{dSt}_k$. It is denoted by $\mathbf{dChAff}_k$.

The above notions and results also have *graded versions*, for which the details are left to the reader. We denote by $k - csCR^{gr}$ the category of graded cosimplicial-simplicial commutative $k$-algebras. We endow this category with the model category structure for which equivalences and fibrations are defined by the graded Tot functor to graded complexes $\text{Tot}^{\pi} : k - csCR^{gr} \to C(\mathbb{Z})^{gr}$, defined by taking the $\text{Tot}^{\pi}$ of each graded component. The proof of the existence of this model category structure follows the same lines as in the non-graded case. The corresponding $\infty$-category will be denoted by $k - \mathbf{csCR}^{gr}$.
We consider the strict multiplicative group object $\mathbb{G}_m^{st}$ in the model category $dAff_k^{\sim,fpqc}$ of fpqc (hyper-complete) stacks (see Section A.9.2). $\mathbb{G}_m^{st}$ is defined as the functor sending $B \in k - sCR$ to the group $B_0^*$ of invertible elements in the ring $B_0$ of 0-simplices in $B$. This is a group object in the model category $dAff_k^{\sim,fpqc}$, which is not a fibrant object (it does not preservers equivalences in $B$), but its image in $dSt_k$ is the usual multiplicative group scheme $\mathbb{G}_m$. Indeed, $\mathbb{G}_m^{st} = h^{k[t,t^{-1}]}$ is corepresented by $k[t, t^{-1}]$, and, as shown in [TV08], a fibrant model for $h^A$ is the derived scheme $\mathbf{Spec} A : B \mapsto \mathbf{Map}(A, B)$. In particular, we have that the model category $\mathbb{G}_m^{st} - dAff_k^{\sim,fpqc}$, of $\mathbb{G}_m^{st}$-equivariant objects in $dAff_k^{\sim,fpqc}$, is a model for the $\infty$-category $dSt_k^{\mathbb{G}_m}$, of $\mathbb{G}_m$-equivariant derived stacks, called graded derived k-stacks.

The graded version of the spec functor is the functor

$$ \mathbf{Spec}^{\Delta,gr} : (k - csCR^{gr})^{op} \to \mathbb{G}_m^{st} - dAff_k^{\sim,fpqc}, $$

sending $A \in k - csCR^{gr}$ to $\underline{Hom}(A, -)$, endowed with the natural $\mathbb{G}_m^{st}$-action coming from the grading on $A$. More explicitly, for $B \in k - sCR$, the set of $q$-simplices of $\mathbf{Spec}^{\Delta,gr}(A)(B)$ is the set of morphisms $\underline{Hom}(A^q, B)$, which is endowed with a $B_0^*$-action as follows. For $b \in B_0^*$ and $f : A^q \to B$, we define $bf : A^q \to B$ by the formula $(bf)(x) = b^n.f(x)$ for a homogeneous element $x$ of degree $n$. As in the non-graded case, $\mathbf{Spec}^{\Delta,gr}$ is a right Quillen functor, and the induced $\infty$-functor

$$ \mathbb{R}\mathbf{Spec}^{\Delta,gr} : (k - \mathbf{csCR}_U^{gr})^{op} \to \mathbf{dSt}_k^{\mathbb{G}_m} $$

is fully faithful. Its essential image is the smallest sub-$\infty$-category containing all the objects $K(\mathbb{G}_a^\chi, n)$, for $n \ge 0$ and $\chi \in \mathbb{Z}$, where $\mathbb{G}_m$ acts on $\mathbb{G}_a$ with weight $\chi \in \mathbb{Z}$, and which is stable by $U$-small limits. This is the graded analog of Proposition A.9.2.1.

**DEFINITION A.9.2.3.** The $\infty$-category of graded derived affine $k$-stacks is the essential image of $\mathbb{R}\mathbf{Spec}^{\Delta,gr} : (k - \mathbf{csCR}_U^{gr})^{op} \hookrightarrow \mathbf{dSt}_k^{\mathbb{G}_m}$. It is denoted by $\mathbf{dChAff}_k^{gr}$.

We finish this section by recalling the notion of linear derived stack, already introduced and studied in [Mon21]. We first notice that the Tot functor $Tot^\pi : \mathbf{csMod}_k \to C(\mathbb{Z})$ admits a natural lax monoidal structure. Therefore, it gives rise to an $\infty$-functor

$$ Tot^\pi : \mathbf{csMod}_k \to \mathbf{dg}_k, $$

where $\mathbf{csMod}_k$ is the $\infty$-category of cosimplicial-simplicial $k$-modules. This $\infty$-functor turns out to be an equivalence of $\infty$-categories, and we denote by $\phi : \mathbf{dg}_k \to \mathbf{csMod}_k$ its inverse.

For any object $E \in \mathbf{dg}_k$, we consider $\phi(E) \in \mathbf{csMod}_k$, and define

$$ \mathrm{Sym}^{\Delta}(E) := L\phi(E) \in \mathbf{csCR}^{gr} $$

the free cosimplicial-simplicial commutative ring generated by $\phi(E)$ (here, $L\phi$ is the left adjoint to the $Tot^\pi$ functor, same notation as in the proof of Theorem A.9.1.2). Being a free commutative ring, $\mathrm{Sym}^{\Delta}(E)$ comes naturally equipped with a graduation, and thus is considered as an object in the $\infty$-category $\mathbf{csCR}^{gr}$.
**DEFINITION A.9.2.4.** The $k$-linear derived stack associated to a complex $E \in \mathbf{dg}_k$, is the graded derived affine $k$-stack defined by

$$ \mathcal{V}(E) := \mathbb{R}\mathbf{Spec}^{\Delta,gr}(\mathrm{Sym}^{\Delta}(E)) \in \mathbf{dSt}_k^{\mathbb{G}_m}. $$

Note that by construction the functor of points of $\mathcal{V}(E)$ sends $B \in k-sCR$ to $\mathrm{Map}_{\mathbf{dg}_k}(E, N(B))$, where $N(B)$ is the normalisation of $B$ as a simplicial $k$-module. The $\mathbb{G}_m$-action is then the natural action of invertible elements of $B$ on $B$ itself.

It is proven in [Mon21] that the $\infty$-functor sending $E$ to $\mathcal{V}(E)$ is fully faithful when restricted to bounded above $k$-dg-modules $E$, and in particular for perfect $k$-dg-modules $E$.
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