# Derived Foliations

Bertrand Toen, Gabriele Vezzosi

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Bertrand Toen, Gabriele Vezzosi. Derived Foliations. 2025. hal-05193556

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# Derived Foliations

Bertrand Toën and Gabriele Vezzosi
<figure><img src="image_2.png" /></figure>
# Contents

<table><tr><td>Introduction</td><td>5</td></tr><tr><td>Contents of individual chapters</td><td>12</td></tr><tr><td>Guide to our notations</td><td>15</td></tr><tr><td>Chapter 1. Algebraic preliminaries</td><td>17</td></tr><tr><td>1.1. Homological conventions</td><td>17</td></tr><tr><td>1.2. Graded mixed complexes</td><td>20</td></tr><tr><td>1.3. Filtered complexes</td><td>25</td></tr><tr><td>1.4. Graded mixed cdga's and derived de Rham theory</td><td>36</td></tr><tr><td>1.5. The geometry of graded mixed objects</td><td>43</td></tr><tr><td>1.6. The classifying spaces of graded mixed structures</td><td>44</td></tr><tr><td>Chapter 2. Derived foliations</td><td>49</td></tr><tr><td>2.1. First definitions and examples</td><td>49</td></tr><tr><td>2.2. Comparison with derived Lie algebroids</td><td>73</td></tr><tr><td>2.3. Formal structures</td><td>76</td></tr><tr><td>2.4. Existence of derived enhancements</td><td>92</td></tr><tr><td>2.5. Direct images</td><td>98</td></tr><tr><td>Chapter 3. De Rham cohomology of derived foliations</td><td>103</td></tr><tr><td>3.1. Crystals along derived foliations</td><td>103</td></tr><tr><td>3.2. Foliated de Rham cohomology</td><td>106</td></tr><tr><td>3.3. The Hodge filtration</td><td>108</td></tr><tr><td>3.4. Transversal geometry</td><td>111</td></tr><tr><td>Chapter 4. Operations on crystals</td><td>117</td></tr><tr><td>4.1. Sheaves of differential operators</td><td>117</td></tr><tr><td>4.2. Direct and inverse images of filtered crystals</td><td>128</td></tr><tr><td>4.3. Characteristic cycles</td><td>134</td></tr><tr><td>4.4. Grothendieck-Riemann-Roch for derived foliations</td><td>140</td></tr><tr><td>4.5. Examples and applications</td><td>144</td></tr><tr><td>Chapter 5. Analytic aspects</td><td>149</td></tr><tr><td>5.1. Quick reminders on derived analytic geometry</td><td>149</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>5.2. De Rham algebras in the holomorphic context</td>
      <td>158</td>
    </tr>
    <tr>
      <td>5.3. Analytic derived foliations</td>
      <td>159</td>
    </tr>
    <tr>
      <td>5.4. Analytic integrability and existence of derived enhancements</td>
      <td>162</td>
    </tr>
    <tr>
      <td>5.5. Analytification of derived foliations: the GAGA theorem</td>
      <td>165</td>
    </tr>
    <tr>
      <td>Chapter 6. Leaf spaces and holonomy</td>
      <td>173</td>
    </tr>
    <tr>
      <td>6.1. Formal, algebraic and analytic leaves</td>
      <td>173</td>
    </tr>
    <tr>
      <td>6.2. Existence of leaf space and holonomy</td>
      <td>175</td>
    </tr>
    <tr>
      <td>6.3. Relative homotopy theory</td>
      <td>184</td>
    </tr>
    <tr>
      <td>6.4. The Riemann-Hilbert correspondence</td>
      <td>187</td>
    </tr>
    <tr>
      <td>6.5. Reeb stability and algebraic integrability</td>
      <td>191</td>
    </tr>
    <tr>
      <td>Chapter 7. Derived foliations on schemes and stacks in non-zero characteristic</td>
      <td>197</td>
    </tr>
    <tr>
      <td>7.1. The graded circle and graded derived loop spaces</td>
      <td>197</td>
    </tr>
    <tr>
      <td>7.2. Derived foliations via graded derived loop spaces</td>
      <td>201</td>
    </tr>
    <tr>
      <td>7.3. Foliated de Rham theory</td>
      <td>204</td>
    </tr>
    <tr>
      <td>7.4. Characteristic classes</td>
      <td>206</td>
    </tr>
    <tr>
      <td>7.5. Residues of singular foliations in positive characteristics</td>
      <td>210</td>
    </tr>
    <tr>
      <td>7.6. Infinitesimal derived foliations</td>
      <td>214</td>
    </tr>
    <tr>
      <td>Appendix A.</td>
      <td>227</td>
    </tr>
    <tr>
      <td>A.1. Model categories and &infin;-categories</td>
      <td>227</td>
    </tr>
    <tr>
      <td>A.2. Derived schemes and stacks: basic facts and notations</td>
      <td>229</td>
    </tr>
    <tr>
      <td>A.3. Formal completion</td>
      <td>232</td>
    </tr>
    <tr>
      <td>A.4. O<sub>X</sub>-modules, quasi-coherent and perfect modules</td>
      <td>235</td>
    </tr>
    <tr>
      <td>A.5. Cotangent complexes of derived Artin stacks</td>
      <td>239</td>
    </tr>
    <tr>
      <td>A.6. Cotangent complexes of formal completion</td>
      <td>240</td>
    </tr>
    <tr>
      <td>A.7. Deformation to the normal bundle</td>
      <td>242</td>
    </tr>
    <tr>
      <td>A.8. Internal derived de Rham theory</td>
      <td>243</td>
    </tr>
    <tr>
      <td>A.9. Derived affine stacks</td>
      <td>248</td>
    </tr>
    <tr>
      <td>Bibliography</td>
      <td>255</td>
    </tr>
  </tbody>
</table>
## Introduction

One important feature of derived algebraic geometry (in the sense of [TV08, Toe14, PV21]) is the observation that singularities of varieties and schemes encountered in real life are most of the time tamed by natural derived structures. This is particularly visible in moduli theory. In many instances, classical/underived moduli spaces are very singular, but their derived counterparts typically have finite dimensional cotangent complexes, and in the best scenarios, local complete intersection singularities. As a result, derived moduli spaces carry important information of geometric origin which can not be seen directly on their classical truncations. Typical examples are: virtual fundamental classes (see [STV15, Kha19]), and shifted symmetric structures (see [PTVV13]).

The main subject of the present book is guided by a similar principle: foliations encountered in real life often come equipped with natural derived structures which tame their singularities. More precisely, there exists a natural notion of derived foliation, with a natural truncation functor to classical, possibly singular, foliations, and this notion helps when dealing with foliation singularities in general. The purpose of this book is to make precise the notion of derived foliation, and to prove results that support this principle.

**From classical foliations to graded mixed algebras.** Before presenting the main definition of this book, we would like to contemplate the classical situation. For a smooth algebraic complex variety $X$, a (algebraic) foliation $\mathcal{F}$ on $X$ is usually defined as a sub-bundle$^1$ $\mathbb{T}_{\mathcal{F}} \subset \mathbb{T}_X$ of the tangent bundle $\mathbb{T}_X$, which is furthermore stable by the Lie bracket. To avoid possible confusions, we will say that such a $\mathbb{T}_{\mathcal{F}} \subset \mathbb{T}_X$ is a classical smooth foliation on $X$. By duality, the sub-bundle $\mathbb{T}_{\mathcal{F}}$ defines a quotient bundle

$$p : \Omega_X^1 \to \Omega_{\mathcal{F}}^1 = \mathbb{T}_{\mathcal{F}}^{\vee}$$

where $\Omega_X^1$ is the sheaf of Kähler differential forms on $X$. Using the following standard formula relating the Lie bracket $[-, -]$ of vector fields and the de Rham differential $dR$ (for $\omega$ a 1-form, and $u$ and $v$ vector fields on $X$)

$$dR(\omega)(u, v) = u(\omega(v)) - v(\omega(u)) + \omega([u, v])$$

we see that the kernel $K_{\mathcal{F}} \subset \Omega_X^1$ of the above quotient map $p$ is a differential ideal: the (sheaf) image of $dR: K_{\mathcal{F}} \to \Omega_X^2$ is contained in $K_{\mathcal{F}} \wedge \Omega_X^1$, where $K_{\mathcal{F}} \wedge \Omega_X^1$ denotes the (sheaf) image of the composition

$$K_{\mathcal{F}} \otimes \Omega_X^1 \to \Omega^1 \otimes \Omega_X^1 \xrightarrow{\wedge} \Omega_X^2.$$

Conversely, any differential ideal $K_{\mathcal{F}}$, i.e. a sub-bundle with the property that $dR(K_{\mathcal{F}}) \subset K_{\mathcal{F}} \wedge \Omega_X^1 \subset \Omega_X^2$, is of the form $\mathbb{T}_{\mathcal{F}}^{\vee}$ for a sub-bundle $T_{\mathcal{F}} \subset \mathbb{T}_X$ stable by the Lie bracket. More importantly for our purposes, the differential ideal property of $K_{\mathcal{F}} \subset \Omega_X^1$ is also equivalent to

<footnote>$^1$We warn the reader here that all along this book the expression *sub-bundle* of a vector bundle $V$ means a sub-coherent sheaf $W \subset V$ such that both coherent sheaves $W$ and $V/W$ are in fact vector bundles.</footnote>
the property that the de Rham differential $dR$ on the graded ring of forms $\Omega_X^*$, descends to a differential $dR_F$ on the quotient graded ring

$$ \Omega_F^* = (\Omega_X^*/K_F \cdot \Omega_X^*). $$

It is easy to check the association $\mathcal{F} \mapsto (K_F, dR_F)$ provides a one-to-one correspondence between classical smooth foliations $\mathcal{F}$ on $X$ and sub-bundles $K_F \subset \Omega_X^1$ together with a multiplicative differential $dR_F$ on $(\Omega_X^*/K_F \cdot \Omega_X^*)$ compatible with the de Rham differential via the quotient map $\Omega_X^* \to \Omega_F^*.$

We now want to consider the sheaf $\Omega_X^1$ as sitting in cohomological degree $-1$, and we thus consider $\Omega_X^1[*] = \oplus_p \Omega_X^p[p]$. This unusual convention, as opposed to $\Omega_X^1$ being considered in cohomological degrees 1, is explained by the geometric interpretation of forms in terms of derived loop spaces and for which we refer to our sections §2.5.2 and §7.2. The sheaf $\Omega_F^*[*]$ is now a sheaf of bi-graded commutative rings, were the bi-degree is given on the one side by the degree of differential forms, and on the other side by the cohomological degree. The differential $dR_F$ on $\Omega_F^*[*]$ is now a multiplicative differential of bi-degree $(1, -1)$: it increases the degree of forms by 1 but decreases the cohomological degree by 1. Moreover, with these conventions we have

$$ \Omega_F^*[*] \simeq \mathrm{Sym}_{\mathcal{O}_X}(\Omega_X^1[1]), $$

making the $\Omega_F^*[*]$ into the free bi-graded commutative ring over $\Omega_F^1[1]$. Again, the association $\mathcal{F} \mapsto (\Omega_F^*[*], dR_F)$ can be seen to define a one-to-one correspondence between classical smooth foliations $\mathcal{F}$ on $X$ and quotient bi-graded commutative rings $\Omega_X^*[*] \to \Omega_F^*[*]$ of the form $\mathrm{Sym}_{\mathcal{O}_X}(\Omega_F^1[1])$ such that the de Rham differential descends to a compatible differential on $\Omega_F^*[*]$.

This somehow complicated alternative point of view on classical smooth foliations on $X$ has the advantage of introducing (and motivating) one of the major actor of this book: graded mixed commutative differential graded algebras, or in short graded mixed cdga's. These are bi-graded commutative $\mathbb{C}$-algebras $A^*(*)$, together with two differentials

$$ d : A^*(*) \to A^{*+1}(*) \quad \epsilon : A^*(*) \to A^{*-1}(*+1) $$

satisfying natural compatibilities (see Definition 1.4.0.1). In the example above arising from a smooth classical foliation on $X$, $\Omega_F^*[*]$ is a graded mixed cdga with $d = 0$ and $\epsilon = dR_F$. Clearly, smooth classical foliations on $X$ can be understood as quotients of $\Omega_X^*[*]$ in the category of graded mixed cdga's. This observation, which is not very helpful when dealing only with classical smooth foliations, is central in the definition of derived foliations in general.

**The general notion of derived foliations.** We turn to a more general situation where $X$ is an affine derived scheme, say over some base field of characteristic zero $k$. Such a derived scheme $X = \mathrm{Spec} A$ is the spectrum of a connective $k$-linear cdga $A$. The cdga $A$ possesses a cotangent complex $\mathbb{L}_{A/k}$ (relative to $k$) which comes equipped with a universal derivation $dR : A \to \mathbb{L}_{A/k}$ (see Appendix §A.2). This derivation can be extended uniquely to an extra differential $\epsilon$ on the graded cdga $\mathrm{Sym}_A(\mathbb{L}_{A/k}[1])$, making it into a graded mixed cdga. This
graded mixed cdga is denoted by $\mathbf{DR}(X/k)$ or $\mathbf{DR}(A/k)$ and is called the de Rham graded mixed cdga associated to $X$. The underlying bi-graded algebra is $\mathrm{Sym}_A(\mathbb{L}_{A/k}[1])$ where the bi-degrees are the cohomological degrees on one hand, and the natural power degree in the symmetric algebra on the other. The two differentials $d$ and $\epsilon$ are respectively induced by the natural cohomological differential of $A$ and by the universal de Rham derivation $dR: A \to \mathbb{L}_{A/k}$.

We can now state the major definition of this book (see Definition 2.1.1.1).

**Definition.** A derived foliation $\mathcal{F}$ on $X$ (relative to $k$) consists of a graded mixed cdga $\mathbf{DR}(\mathcal{F})$ together with a morphism of graded mixed cdga's $\mathbf{DR}(X/k) \to \mathbf{DR}(\mathcal{F})$, such that the underlying graded cdga of $\mathbf{DR}(\mathcal{F})$ is quasi-isomorphic to $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$, for some $A$-dg-module $\mathbb{L}_{\mathcal{F}}$.

In the definition above the $A$-dg-module $\mathbb{L}_{\mathcal{F}}$ is called the cotangent complex of the derived foliation $\mathcal{F}$; it comes together with an anchor map $\mathbb{L}_{X/k} \to \mathbb{L}_{\mathcal{F}}$ and plays the role of the quotient $\Omega_X^1 \to \Omega_{\mathcal{F}}^1$ previously discussed in the setting of classical smooth foliations. The mixed structure on $\mathbf{DR}(\mathcal{F})$ together with the quasi-isomorphism $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}}[1]) \simeq \mathbf{DR}(\mathcal{F})$ of graded cdga's, define on $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$ a de Rham-like differential

$$ \epsilon: \Lambda_A^i \mathbb{L}_{\mathcal{F}} \to \Lambda_A^{i+1} \mathbb{L}_{\mathcal{F}} $$

which is well defined as a morphism in the $\infty$-category of $k$-dg-modules. This de Rham differential carries (an infinite number of) homotopy coherent structures expressing that it squares to 0, that is multiplicative and compatible with the de Rham differential on $\mathbf{DR}(X/k)$. These homotopy coherent structures are encapsulated and hidden in the full graded mixed cdga structure $\mathbf{DR}(\mathcal{F})$ and are implicit in our approach, as the de Rham differential $\epsilon$ exists by definition on $\mathbf{DR}(\mathcal{F})$ (which is only quasi-isomorphic to $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$), and does not exist strictly speaking on $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$ itself. We refer to Corollary 1.6.0.4 for more on the shape of homotopy coherences involved, but the point of our definition of derived foliations, and more generally the point of view of this book is to avoid, as much as possible, to work with explicit homotopy coherences and try to keep them implicit in the picture.

This being said, the derived foliations on $X$ as defined above form an $\infty$-category $\mathcal{F}ol(X/k)$. The assignment $X \mapsto \mathcal{F}ol(X/k)$ is functorial in $X$ in the sense that a derived foliation $\mathcal{F}$ on $X$ can be pulled back along any morphism $f: Y \to X$ of derived affine schemes. This pull-back operation is of fundamental importance, as even when $Y$ and $X$ are smooth schemes, and $\mathcal{F}$ is a classical smooth foliation on $X$, its pull-back $f^*(\mathcal{F})$ is a derived foliation on $Y$ which is, in general, not a classical foliation anymore. As a result, pull-backs offer a way to create interesting examples of derived foliations starting from a classical situation. This is, of course, parallel to the fact that the (derived) fiber product of smooth schemes usually is a non-trivial derived scheme.

The functoriality $X \mapsto \mathcal{F}ol(X/k)$, on derived affine schemes $X$, can be used in order to provide a definition of derived foliation on any derived stack $X$, simply by imposing the usual
descent condition

$$ \mathcal{F}ol(X/k) := \lim_{\mathbf{Spec} A \to X} \mathcal{F}ol(\mathbf{Spec} A/k). $$

However, the behaviour of derived foliations on general derived stacks brings some difficulties, as for instance cotangent complexes of derived foliations are not stable by pull-backs. As a result the existence of well defined global cotangent complex of a derived foliation over a general derived stack is not obvious. In that direction we prove several descent results showing that, in spite of that, cotangent complexes still have a reasonable meaning when working with derived Artin stacks.

Finally, a given derived foliation $\mathcal{F}$ has an attached *de Rham cohomology*, that is the generalisation of the well known de Rham cohomology along the leaves, or *foliated de Rham cohomology*. Within our approach, this foliated de Rham cohomology is very easy and natural to define by means of the *Tate-realization functor*, relating graded mixed cdga's and complete filtered cdga's (see Corollary 1.3.4.4). The image of $\mathbf{DR}(\mathcal{F})$ by this realization, $\hat{C}_{\mathbf{DR}}^*(\mathcal{F})$, is by definition the commutative dg-algebra of *de Rham cohomology along* $\mathcal{F}$, and the filtration is here of course the natural Hodge filtration. This de Rham cohomology is however only a piece of a refined cohomological invariant called the *Hodge filtration* associated to a derived foliation $\mathcal{F}$. The terminology is here unfortunate, as it should not be confused with the filtration on $\hat{C}_{\mathbf{DR}}^*(\mathcal{F})$. The Hodge filtration associated with $\mathcal{F}$ is indeed a filtration on the de Rham cohomology of $X$ (suitably derived and completed) $\hat{C}_{\mathbf{DR}}^*(X)$, whose first graded piece recovers $\hat{C}_{\mathbf{DR}}^*(\mathcal{F})$. More generally, the higher graded pieces can also be described as foliated de Rham cohomology with coefficients in the powers of the normal complex (see § 3.3). The existence of this Hodge filtration seems to us more important than the coarser notion of foliated de Rham cohomology, and it will turn out to be very helpful in the study of characteristic classes associated to derived foliations (see Corollary 7.4.0.3).

**Formal aspects of derived foliations.** One specific and important aspect of derived foliations is their *formal integrability* property. This is a major difference with the classical situation, where certain singular foliations might not be formally integrable (see Remark 2.4.1.3). It can be shown, under very mild assumptions, that any derived foliation $\mathcal{F}$ on a derived Artin stack $X$ is formally integrable around each point in $X$ (see Corollary 2.3.3.3). In particular, this implies that for a field valued point $x \in X(K)$, there is a well defined notion of *formal leaf passing through* $x$, as well as a *formal leaf space at* $x$. As a result, it may happen that a given classical singular foliation cannot be enhanced to a derived foliation (i.e. it is not the truncation of a derived foliation). This can be seen as either a positive result, or a negative result, depending of the situation. Nevertheless, this shows that the situation diverges from what is happening in derived algebraic geometry, where any scheme can be endowed with the trivial derived structure.

The precise connection between classical singular foliations and derived foliations can be made clearer. For a smooth scheme $X$ we have a notion of *quasi-smooth and rigid* derived foliation on $X$. This is an important class of derived foliations as they can be considered as
the derived foliations which are the closest to be classical. These are given by $\mathcal{F} \in \mathcal{F}ol(X/k)$ for which the cotangent complex sits in an exact triangle of perfect complexes on $X$

$$ \mathcal{N}_{\mathcal{F}}^* \longrightarrow \Omega_X^1 \longrightarrow \mathbb{L}_{\mathcal{F}} $$

where $\mathcal{N}_{\mathcal{F}}^*$ is a vector bundle on $X$. In many situation $\mathcal{N}_{\mathcal{F}}^* \to \Omega_X^1$ is a monomorphism, and thus $\mathcal{F}$ is a classical smooth foliation on the open $U \subset X$ where $\mathbb{L}_{\mathcal{F}}$ is a vector bundle. The closed complement of $U$ is by definition the *singular set of* $\mathcal{F}$. The derived foliation $\mathcal{F}$ admits a classical truncation which is the image $\mathcal{N}_{\mathcal{F}}^*$ in $\Omega_X^1$ and which defines a differential ideal $K_{\mathcal{F}}$ on $X$. As we have already mentioned, not all differential ideals arise by this construction. In fact we can show that the differential ideals obtained this way are the one for which an infinite *Godbillon-Vey* sequence exists (see §2.4). That of a Godbillon-Vey sequence is a classical notion in foliation theory and therefore this result sheds some light of how derived foliations are related to classical singular foliations.

**Derived foliations in the holomorphic setting.** Most of this book is written in the algebraic setting. However, except in some extremely specific situations, classical foliations cannot be integrated algebraically, and analytic integration is in most situations the best one can hope. It is therefore important to introduce derived foliations also in the holomorphic context. Our definition of derived foliation as certain graded mixed cdga's can be easily transported in the holomorphic setting, but with an extra technical component in order to keep track of the holomorphic structure. Nevertheless derived foliations on a derived analytic Artin stack makes sense, and formally behave as in the algebraic setting. We develop the strict minimum of the holomorphic theory with two objectives in sight: the *GAGA theorem*, and the existence of an *analytic leaf space* under appropriate assumptions.

The GAGA theorem for derived foliations states that for a proper derived Deligne-Mumford stack $X$, the analytification functor induces an equivalence between the $\infty$-categories of algebraic (and almost perfect) derived foliations on $X$ and the $\infty$-category of holomorphic derived foliations on the derived analytic stack $X^h$ associated to $X$ (see Theorem 5.5.0.4). This is not very surprising as derived foliations are defined in terms of coherent data via the derived de Rham algebra of forms, and comparison results relating algebraic and holomorphic cotangent complexes are well known.

As in the algebraic situation, holomorphic derived foliations are always formally integrable and it is thus natural to ask if they can be integrated locally in the analytic topology. For this, we can use classical results from [Mal76, Mal77] to provide interesting conditions ensuring that quasi-smooth and rigid holomorphic derived foliations are indeed locally analytically integrable. This is the starting point for a construction of a global holomorphic leaf space and for the global analytic integrability. Indeed, under certain natural assumption, it is possible to show that a quasi-smooth holomorphic derived foliation $\mathcal{F}$ on a smooth complex variety $X$ admits a holomorphic leaf space $\pi: X \to X \parallel \mathcal{F}$ (see Theorem 6.2.2.1). Here $X \parallel \mathcal{F}$ is a complex (highly non-separated) orbifold, $\pi$ is a flat surjective holomorphic map, and $\mathcal{F}$ is realized as the derived foliation induced by $\pi$. The situation is thus very similar to the case of smooth
classical foliation for which $X \parallel \mathcal{F}$ is represented by the well known holonomy groupoid of $\mathcal{F}$. The novelty here is that $\pi$ is only flat and thus can have singular fibers, corresponding to the possibly singular leaves of $\mathcal{F}$.

It is interesting to combine together the GAGA theorem and the existence of the analytic leaf space when we start with a derived foliation $\mathcal{F}$ on a smooth and proper complex variety $X$. As a first application, we prove a *Riemann-Hilbert correspondence* which provides an equivalence between the category of algebraic vector bundles with integrable connections along $\mathcal{F}$, and the category of relative local systems along the fibers of $\pi$ (see Theorem 6.4.0.4). This is a generalization to the singular setting of the Riemann-Hilbert correspondence appearing in [Del70]. In another direction, we can exploit the notion of the *holonomy groups* deduced from the existence of the analytic leaf space, which are, by definition, the stabilizers of the orbifold $X \parallel \mathcal{F}$. Classically holonomy groups have been useful in stability results, such as the famous *Reeb stability theorem* (see [Ree52]). In our setting, it is in fact possible to prove that $\mathcal{F}$ is globally algebraic integrable on $X$, if and only if $\mathcal{F}$ possesses at least one compact leaf with finite holonomy groups (see Theorem 6.5.0.1). This is again an extension to the singular setting of previously known results for classical smooth foliations (see [Per01]).

**Crystals along derived foliations.** We have already mentioned that a given derived foliation has an associated foliated de Rham cohomology. This can be extended by the introduction of a category of coefficients for foliated de Rham cohomology called *crystals along a derived foliation F*. Morally speaking these are quasi-coherent complexes endowed with partial flat connections along $\mathcal{F}$. Concretely, they can be described in terms of graded mixed DR($\mathcal{F}$)-dg-modules (see Definition 3.1.0.1). They also have another description in terms of modules over a sheaf of associative dg-algebras $D_\mathcal{F}$, called the *sheaf of differential operators along F* (see Theorem 4.1.2.1). The advantage of the latter description is the existence of a natural filtration on $D_\mathcal{F}$ induced by the order of differential operators. This is useful to define the notion of *good filtration* on a given crystal, analogue to the well known notion of a good filtration on a classical $D$-module. This notion can then be used in order to define the *singular support*, as well as the *characteristic cycle* of a crystal endowed with a good filtration, which is a $K$-theory class inside the total space of the foliated cotangent complex of $\mathcal{F}$ (see Definition 4.3.2.3). Even thought-the notion of good filtration has some drawbacks for derived foliations with singularities (see § 4.3.2.2), it is enough to establish a *Grothendieck-Riemann-Roch theorem* and *index theorem* for crystals along derived foliations (see Theorem 4.4.1.1 and Corollary 4.5.4.2).

**Derived foliations over bases of arbitrary characteristics.** Classical foliations in positive characteristics is a rich subject (see for instance [Eke87, Bos01]), and we have therefore decided to include a last chapter on derived foliations over arbitrary base ring. It turns out that there are two, non-equivalent, possible extensions of our notion of derived foliations outside of characteristic zero, which roughly speaking correspond to the two existing notion of differential operators: *crystalline differential operators* and *Grothendieck differential operators*. These are
also related to two different extension of de Rham cohomology, namely *crystalline cohomology* and *infinitesimal cohomology*.

The best way to understand the difference between these two possible definitions of derived foliations in arbitrary characteristics is to adopt a geometric point of view on derived foliations based on group actions on derived linear stacks (see Definition 2.5.2.1). We introduce two group stacks $\mathcal{H}$ and $\mathcal{H}_\pi$. The first one, $\mathcal{H}$ is the semi-direct product of $G_m$ with $BG_a^\sharp$, where $G_a^\sharp$ is the additive group with divided powers. The second one, $\mathcal{H}_\pi$ is the semi-direct product of $G_m$ with $G_a[-1] = \Omega_0 G_a$ the additive group in degree $-1$. Both of these groups control graded mixed complexes, in the sense that the symmetric monoidal $\infty$-categories $\mathbf{QCoh}(\mathcal{B}\mathcal{H})$ and $\mathbf{QCoh}(\mathcal{B}\mathcal{H}_\pi)$ are both equivalent to the symmetric monoidal $\infty$-category of graded mixed complexes. However, the two groups $\mathcal{H}$ and $\mathcal{H}_\pi$ carry different informations. A complex $E$ together with an action of $\mathcal{H}$ on the corresponding derived linear stack $\mathcal{V}(E)$ is called a *derived foliation* (see Definition 7.2.0.1). On the other hand, a complex $E$ together with an action of $\mathcal{H}_\pi$ on the corresponding derived linear stack $\mathcal{V}(E)$ is called an *infinitesimal derived foliation* (see Definition 7.6.1.1). Derived foliations are directly related to (derived and Hodge completed) de Rham cohomology, and thus to crystalline cohomology as well (see Proposition 7.5.0.1). On the other hand, infinitesimal derived foliations are closely related to infinitesimal cohomology, suitably derived and Hodge completed, if necessary (see Corollary 7.6.2.2).

Both, derived foliations and infinitesimal derived foliations have a generic behaviour similar to derived foliations in characteristic zero: they have pull-backs, can be defined over general derived stacks, have an associated foliated cohomology theory, etc. However, their difference can be seen in applications. Derived foliations, due to their relation with crystalline cohomology, are useful to study characteristic classes. For instance they can be used to prove a characteristic $p > 0$ version of the vanishing theorem of *Baum-Bott* (and its generalization on the existence of *residues*, see [BB72a]). On the other hand infinitesimal derived foliations are always formally integrable, as opposed to derived foliations in general (see Corollary 7.6.3.2). This is a generalization of the fact that not all classical smooth foliations in characteristic $p > 0$ can be formally integrated, because of the well known obstruction associated to $p$-curvature (see e.g. [Kat70, Eke87]). In a way, infinitesimal derived foliations in characteristic $p > 0$ should be thought as *p-restricted derived foliation*, even though we did not try to make this notion precise.

**Relations to other works.** We finish this introduction by mentioning several relations with other works.

To our knowledge the notion of derived foliation first appears in a talk by Tony Pantev in 2014 (see [Pan14]). He explained in particular how derived foliations appear naturally in the setting of shifted symplectic and Poisson structures of [PTVV13, CPT+17], and in particular how they are useful to reconstruct shifted potentials in order to write a $n$-shifted symplectic derived stack globally as the derived critical locus of a function of degree $n + 1$. The present book has pursued this direction, and owns a lot to the conversations among the
authors of [PTVV13, CPT$^+$17] on the subject. The reconstruction of potential is not part of
this book as we have decided to focus on slightly different aspects of derived foliation theory,
but has been carried out in details recently in [HHR25, §2.4].

In a similar direction, an idea conveyed by Damien Calaque during the last decade states
that shifted Poisson structures in the sense of [CPT$^+$17] can be considered as derived foliations
with shifted symplectic structures along the leaves (in short a derived symplectic foliation).
This is the derived analogue of the fact that a classical Poisson structure provides a symplectic
foliation. This has been realized recently in [Tom25], where it is shown that a shifted Poisson
structure is always given by a Lagrangian thickening, which can be identified with the formal
leaf space of the wanted symplectic derived foliation (see also [CS24]).

The formal integrability properties of derived foliations studied in this book has been gen-
eralized recently in [BMN25], including the non-zero characteristic situations. Our results of
§2.3.3.3 and §7.6.3.2 are therefore already subsumed by the results of [BMN25]. Moreover, the
results of the above work also provide a very general construction of a *global formal leaf space*
of a derived foliation in the form of a formal thickening. These type of quotients were already
mentioned in [Pan14] in the characteristic zero case, but are not included in this book.

Many of the filtrations appearing in this book are obtained by means of the equivalence
given by the Tate-realization functor, which identifies graded mixed complexes and complete
filtered complexes (see §1.3.4.4). Therefore, all the notions of this book could also have been
written in terms of complete filtered objects, such as complete filtered cdga or complete filtered
LSym-algebras. We are aware that Ben Antieau has also developed a theory of derived foliation
in the topological context based on this notion of filtered algebra, and that probably some of
the definitions of this book also appear in his work.

Mauro Porta has been keeping telling us that graded mixed cdga which are not necessarily
graded free (and thus do not correspond, strictly speaking, to derived foliations) are nonetheless
interesting. More recently, Philippe Eyssidieux has convinced the authors that there could be
indeed an interesting derived version of the notion of *exterior differential system* (EDS for
short). By definition, these derived EDS on a variety *X* are precisely the general graded mixed
cdga under the de Rham algebra *DR*(*X*). It is pretty clear that some of the notions and results
of this book extend to this more general setting of derived EDS. However, we have not included
any content in that direction.

**Contents of individual chapters**

This book is divided in seven chapters and one appendix. Chapter 1 is devoted to the
algebraic preliminaries necessary to approach the study of derived foliations. We have included
sections on graded mixed objects, filtered objects and the Tate-realization relating these two
notions. These notions will be used throughout the book. The two last sections of this chapter
include one incarnation of graded mixed complexes as representations of the group stack $BH_0$.
This geometric interpretation will turn out to be useful, typically to construct natural graded
mixed objects as push-forwards along a morphism to $BH_0$. The last section is devoted to a
(partial) description of the homotopy coherences encoded in a graded mixed structure. This
is a technical result that will be used occasionally, and does not quite follow the general spirit
of the general approach of this book which is to make these coherences implicit as much as
possible.

Chapter 2 contains the main definitions and notions on the subject. We define the $\infty$-category $\mathcal{Fol}(X/k)$ of derived foliations, first on an affine derived $k$-schemes $X$, then by descent on general derived $k$-stacks $X$. We provide examples, and compare derived foliations in particular with the theory of dg-Lie algebroids already existing in the literature. We present also the two important universal foliations: the initial foliation, also called the zero foliation, and the final foliation, also called the de Rham or tautological foliation. The third section is devoted to the formal properties of derived foliations. We have restricted ourselves to the behaviour formally at a field valued point only, which is a non-trivial restriction. However, we have treated the general case of almost perfect derived foliations on general derived Artin stacks locally of finite presentation. We explain in particular, how derived foliations on the formal completion at a point are governed purely in terms of dg-Lie algebras. This section is central in the sense that it allows to make sense of the notion of *formal leaf* and *formal leaf space* at a given point, whereas those notions are not totally obvious when working over general derived stacks. The subsequent section concerns formal integrability in the more restrictive context of quasi-smooth and rigid derived foliations on smooth varieties. It contains in particular a comparison with classical singular foliations, and provide interpretations of classical notions (such as Godbillon-Vey sequences and transversal jet spaces) in terms of derived foliations. The last section is taken from the work of [Alf25], and presents a useful construction tool of derived foliations by direct images (or Weil restriction). The geometric interpretation of derived foliations which is behind this direct image construction will be useful to understand and motivate Chapter 7 about derived foliations in arbitrary characteristics.

In Chapter 3 we introduce and study foliated de Rham cohomology. We start by the
introduction of the categories of crystals along a derived foliations, and their basic functorialities
by pull-backs. They give rise to the notion of foliated de Rham cohomology with coefficients in
a crystal that will be used quite often in the rest of the book. The third section of this chapter
introduces the Hodge filtration. This is a filtration on the de Rham cohomology of a derived
stack X induced by the datum of a derived foliation $\mathcal{F} \in \mathcal{Fol}(X/k)$. It is obtained by the
geometric point of view on derived foliations explained in § 2.5 together with a very general
deformation to the normal bundle construction. The final section of this chapter provides some
elements of transversal geometry by constructing the formal jets transversal to a given derived
foliation. We explain how these jets are endowed with a crystal structure, and how they give
rise to an algebraic notion of holonomy and monodromy by means of a Tannakian approach.
These include higher homotopical invariants captured in the form of a schematic homotopy
type in the sense of [Toe06], and closely related to Galois-differential homotopy theory.

Chapter 4 is devoted to the study of operations on crystals. We have not included a full
6 operations formalism, which seems to us out of reach at the moment, but have studied
direct images and direct images with proper support when they are defined. The first section introduces the sheaf of differential operators along a derived foliation $\mathcal{F}$, which is a sheaf of filtered dg-algebras whose associated graded object is of the form $\text{Sym}_{\mathcal{O}_X}(\mathbb{T}_{\mathcal{F}})$, the symmetric algebra over the tangent complex of the derived foliation $\mathcal{F}$. Modules over $\mathcal{D}_{\mathcal{F}}$ are equivalent to crystals along $\mathcal{F}$, but the existence of the filtration on $\mathcal{D}_{\mathcal{F}}$ can be used in order to define good filtrations on crystals and the corresponding notion of characteristic cycles. We prove in particular that characteristic cycles are stable by push-forward, i.e. a foliated version of the Grothendieck-Riemann-Roch formula.

The analytic theory of derived foliation is covered in chapter 5. In its first section we remind the basics of derived analytic geometry based on the notion of holomorphic cdga. In the second section we introduce the *holomorphic graded mixed cdga*, which can be used to defined holomorphic derived foliations along the same lines as in the algebraic setting. The two last sections are devoted to the proof of two main results. First of all we study the analytic integrability, locally in the analytic topology, of quasi-smooth and rigid derived holomorphic foliations. This allow us to make comparisons with previously known integrability results for holomorphic classical singular foliations, notably the results from [Mal76, Mal77]. In the last section of the chapter we prove a GAGA theorem for derived foliations on proper derived Deligne-Mumford stacks.

Chapter 6 contains the notion of analytic leaves for holomorphic derived foliations. By restricting to quasi-smooth and rigid derived foliations, we provide simple assumptions ensuring the existence of a nice analytic leaf space whose construction uses techniques arising in the study of *étendues* from SGA4, and is a generalization of the construction of the holonomy groupoid of a classical smooth holomorphic foliation. The possible singularities bring new features concerning the leaf space: one instance is the non-triviality of the *relative homotopy type* that encodes the variations of the homotopy types of the leaves. This relative homotopy type is not locally constant (even at the level of germs because of the existence of vanishing cycles). In the last two sections we give two main applications of the existence of an analytic leaf space and of our previously established GAGA theorem. The first application is a Riemann-Hibert correspondence, and the second application is an algebraic integrability result based on the classical Reeb stability theorem.

The last chapter of the book is chapter 7. It contains some results on derived foliations in non-zero characteristic situations. Its first two sections introduce an important group stack $\mathcal{H}$, that enables us to define derived foliations as linear derived stacks endowed with an action of $\mathcal{H}$. In the next section we explain how this notion is related to de Rham cohomology, and introduce extensions of the notion of crystals and of the Hodge filtration already encountered in characteristic 0. The fourth section is devoted to the construction of characteristic classes of derived foliations over any base ring, and explain how these interact nicely with the Hodge filtration. We then establish classical vanishing results for the characteristic classes. In the subsequent section of this chapter, we use these results to prove a characteristic $p > 0$ version of the existence of residues of foliations introduced by Baum-Bott in the complex setting. These
are proven in crystalline cohomology thanks to the comparison between derived de Rham and crystalline cohomology. Finally, the last section introduces the notion of infinitesimal derived foliations, explains its relation with infinitesimal cohomology, and how infinitesimal derived foliations can be always integrated by formal groupoids.

The Appendix contains some quick and basic reminders on topics in $\infty$-categories and derived algebraic geometry (global and formal), together with some important descent results that are used in the main text.

## Guide to our notations

Most of our notations will be defined in the main text. Here we only fix a few of the most recurring notations.

Throughout the book, $k$ will be a commutative ring, sometimes supposed to be a field: see the beginning of each chapter or section to understand what are the standing hypotheses on $k$.

We will be freely using the language of model categories and of $\infty$-categories (see §A.1 for a brief recapitulation about these, and for the corresponding notations we adopted). We will denote by **Top** the $\infty$-category of spaces (in [Lur09, §1.2.16]), and by **Cat$_\infty$** the $\infty$-category of $\infty$-categories (see [Lur09, Chapter 3]).

For a model category (or a category with weak equivalences) $C$, we will denote by $C$ its associated $\infty$-category obtained by inverting in the $\infty$-categorical sense the weak equivalences (see §A.1 for more details). For a functor $F : C \to D$ that induces a functor between the associated $\infty$-categories (e.g. a functor preserving weak equivalences or a left/right Quillen functor), we will denote with boldface fonts the induced functor $F : C \to D$.

The $\infty$-category of derived stacks over $k$ (mostly for the étale topology) will be denoted by $\mathbf{dSt}_k$ (see §A.2), and, for $G$ a group scheme or group stack over $k$, $\mathbf{dSt}_k^G$ will be the $\infty$-category of derived stacks endowed with a $G$-action.
<figure><img src="image_3.png" /></figure>
CHAPTER 1

Algebraic preliminaries

In this first chapter we gather preliminaries results that we will use all along the book. We fix the cohomological conventions, and present the central notion of graded mixed complexes. We show that it is related to complete filtered dg-modules by means of the Tate-realization functor. In the last two section we present a stacky interpretation of graded mixed complexes as quasi-coherent sheaves on the classifying stack of the group stack $\mathbb{G}_m \times B\mathbb{G}_a$. Finally, we study the moduli spaces of graded mixed structures.

All along this chapter we denoted by $k$ a ground commutative ring. It will be assumed to be a $\mathbb{Q}$-algebra starting from Section 1.4 on, but most of the results of the chapter are independent of the characteristics of the ground ring.

1.1. Homological conventions

In this section $k$ is of arbitrary characteristics. The sole purpose of this section is to fix notations and the conventions used in this book.

**1.1.1. Basics.** Complexes of $k$-modules are by definition cohomologically indexed and their differential $d$ will raise the cohomological degree by 1. More explicitly, a complex of $k$-modules consists of a $\mathbb{Z}$-indexed family of $k$-modules $\{E^n\}_{n\in\mathbb{Z}}$, together with $k$-linear morphisms

$$d^n : E^n \to E^{n+1}$$

such that $d^{n+1}d^n = 0$. We will often write symbolically such a complex as a direct sum $E = \oplus_n E^n$, and $d$ as a $k$-linear endomorphism $d : E \to E$ satisfying $d^2 = 0$.

For two complexes $E$ and $F$, a morphism $f : E \to F$ consists of a family of $k$-linear morphisms $f^n : E^n \to F^n$ which commute with the differentials of $E$ and $F$

$$d_F^n f^n = f^{n+1} d_E^n,$$

where we have denoted by $d_E$ and $d_F$ the differentials of $E$ and $F$, respectively. The complexes of $k$-modules and morphisms between them form a category denoted by $C(k)$.

For an object $E \in C(k)$, we define its $i$-th cohomology $k$-module as

$$H^i(E) := \frac{(Ker d^i : E^i \to E^{i+1})}{(Imd^{i-1} : E^{i-1} \to E^i)}.$$
This defines functors $H^i(-) : C(k) \to k - Mod$, for $i \in \mathbb{Z}$.

**1.1.2. Symmetric monoidal structure.** The category $C(k)$ is endowed with a symmetric monoidal structure $\otimes_k$ defined as follows. For two objects $E$ and $F$, we define $E \otimes_k F \in C(k)$ by letting for $n \in \mathbb{Z}$

$$ (E \otimes_k F)^n := \bigoplus_{i+j=n} E^i \otimes_k F^j, $$

the differential on $E \otimes_k F$ being defined as the $k$-linear map sending a tensor $x \otimes y$ to

$$ d(x \otimes y) := d(x) \otimes y + (-1)^{|x|} x \otimes d(y) $$

where $|x|$ is the cohomological degree of $x$, i.e. $x \in E^{|x|}$. In more concrete terms, the differential $(E \otimes_k F)^n \to (E \otimes_k F)^{n+1}$ is the sum of all maps

$$ (d_E^i \otimes \mathrm{id}, (-1)^{ij} \mathrm{id} \otimes d_F^j) : E^i \otimes_k F^j \to (E^{i+1} \otimes_k F^j) \oplus (E^i \oplus_k F^{j+1}) \subset \bigoplus_{a+b=n+1} E^a \otimes_k F^b. $$

This defines the functor $- \otimes_k - : C(k) \times C(k) \to C(k)$. It is furthermore endowed with the natural unity and associativity constraints isomorphisms induced from the usual unity and associativity constraints of tensor products of $k$-modules. The symmetry constraint is itself the usual isomorphism

$$ (1) \qquad V_{E,F} : E \otimes_k F \simeq F \otimes_k E $$

defined by sending $x \otimes y$ to $(-1)^{ij}y \otimes x$ for $x \in E^i$ and $y \in F^j$. With these definitions, $(C(k), \otimes_k)$ is a closed symmetric monoidal category.

**1.1.3. Shifts.** We denote by $k[1]$ the object of $C(k)$ which is defined by

$$ (k[1])^n = \begin{cases} k & \text{if } n = -1 \\ 0 & \text{if } n \neq -1 \end{cases} $$

obviously endowed with the zero differential. The shift endofunctor on $C(k)$ is defined to be

$$ E \mapsto E[1] := k[1] \otimes_k E. $$

Unfolding the definitions of the tensor product we see that $E[1]$ is explicitly given, up to a natural isomorphism, by $(E[1])^n = E^{n+1}$ with differential

$$ -d^{n+1} : (E[1])^n = E^{n+1} \to (E[1])^{n+2} = E^{n+2}. $$

The endofunctor $E \mapsto E[1]$ is an auto-equivalence of the category $C(k)$ and its inverse will be denoted by $E \mapsto E[-1]$. Finally, we set, for all $i \in \mathbb{Z}$ and $E \in C(k)$

$$ E[i] := (k[1])^{\otimes i} \otimes_k E $$

the $n$-th composite of either $E \mapsto E[1]$ if $i \ge 0$, or $E \mapsto E[-1]$ if $i < 0$. Explicitly, again up to a natural isomorphism, $E[i]$ is given by $(E[i])^n = E^{n+i}$ with differential $(-1)^i d^{n+i} : (E[i])^n \to$
$(E[i])^{n+1}.$

**1.1.4. Model category structure.** We remind that $C(k)$ possesses a model category structure whose fibrations are the morphisms $f: E \to F$ with $f^n: E^n \to F^n$ surjective for all $n \in \mathbb{Z}$, and weak equivalences are the quasi-isomorphisms, i.e. the morphisms $f: E \to F$ such that for all $i$ the induced map $H^i(f): H^i(E) \to H^i(F)$ is bijective. The cofibrations are defined by the left lifting property with respect to the class of fibrations that are also quasi-isomorphisms. This model category structure is suitably compatible with the symmetric monoidal structure and thus makes $C(k)$ into a *symmetric monoidal model category* in the sense of [Hov99, §4]. It satisfies moreover the monoid axiom of [SS00].

As explained in the Appendix A.1, the model category $C(k)$ gives rise to an $\infty$-category by localization along the weak equivalences. This $\infty$-category is called the *$\infty$-category of complexes*, and will be denoted by

$$ \mathbf{dg}_k := L(C(k)). $$

A concrete model for $\mathbf{dg}_k$ is given by considering the category of cofibrant<fnref n="1" /> objects $C(k)^c \subset C(k)$. Between two such cofibrant objects $E$ and $F$, we can define a simplicial set of maps $\mathrm{Hom}^\Delta(E, F) \in \mathrm{sSets}$, whose set of $n$-dimensional simplices is given by the formula

$$ \mathrm{Hom}^\Delta(E, F)_n := \mathrm{Hom}_{C(k)}(\mathcal{C}(\Delta^n) \otimes_k E, F) $$

where $\mathcal{C}(\Delta^n) \in C(k)$ is the normalized chain complex of homology of the simplicial set $\Delta^n$. More precisely, if $\mathbf{N}: \mathrm{sMod}_k \to \mathcal{C}^-(k)$ is the Dold-Kan normalization functor (see ...), $\mathcal{C}(\Delta^\bullet) := \mathbf{N}(\Delta^\bullet)$ is a cosimplicial object in $\mathcal{C}^-(k) \subset C(k)$, hence

$$ \mathrm{Hom}^\Delta(E, F) := \mathrm{Hom}_{C(k)}(\mathcal{C}(\Delta^\bullet) \otimes_k E, F) \in \mathrm{sSets}. $$

It is easy to see that these data organize themselves into an $\mathbb{S}$-category (i.e. a category enriched in $\mathrm{sSets}$) which is a concrete model for $\mathbf{dg}_k$ (see §A.1 for more details).

Finally, the symmetric monoidal model structure on $C(k)$ induces a natural symmetric monoidal structure on the $\infty$-category $\mathbf{dg}_k$ (see §A.1). It will be again denoted by $\otimes_k$. Note however, that the localization $\infty$-functor

$$ C(k) \to \mathbf{dg}_k $$

is not a monoidal $\infty$-functor, even though its restriction to cofibrant objects is so. Therefore, $E \otimes_k F$ is an ambiguous notation, whose meaning depends on whether $E$ and $F$ are considered as objects in $C(k)$ or as objects in $\mathbf{dg}_k$. We hope however that the context will provide the necessary extra information, each time we will use such a notation in the text.

For an associative and unital dg-algebra $A$, we will also denote by $\mathrm{Mod}(A)$ the category of (left) $A$-dg-modules. It is endowed with an induced model category structure whose weak equivalences (resp. fibrations) are morphisms which are quasi-isomorphisms (resp. epimorphisms) on the underlying complexes (see e.g. [SS00]). The corresponding $\infty$-category will be

<footnote n="1">¹Note that all objects in $C(k)$ are, obviously, fibrant.</footnote>
denoted by $dg_A$. For a morphism of dg-algebras $\varphi: A \to B$, we have the usual base change functor $B \otimes_A - : Mod(A) \to Mod(B)$, which is a left Quillen functor. It induces an adjunction on the corresponding $\infty$-categories

$$B \otimes_A - : dg_A \stackrel{\sim}{\to} dg_B : (-)_{[\varphi]}$$

When $\varphi$ will be clear from the context, the functor $(-)_{[\varphi]}$ (restriction of scalars) will be simply omitted from our notations.

## 1.2. Graded mixed complexes

In this section $k$ is of arbitrary characteristics.

**1.2.1. Graded complexes.** We consider $C(k)^{gr}$ the category of $\mathbb{Z}$-graded objects in $C(k)$. An object in $C(k)^{gr}$ is by definition a collection $E = \{E^{(n)}\}_{n \in \mathbb{Z}}$ of complexes $E^{(n)}$. Morphisms $f: E \to F$ in $C(k)^{gr}$ are defined levelwise as families of morphisms of complexes

$$f^{(n)} : E^{(n)} \to F^{(n)}.$$

In order to avoid confusions with the notion of cohomological degree we refer to $E^{(n)}$ as the part of weight $n$ of $E$, and in general refer to weights for the degree corresponding to the extra $\mathbb{Z}$-grading on objects in $C(k)^{gr}$.

The category $C(k)^{gr}$ is endowed with the usual monoidal structure of graded objects. For two objects $E$ and $F$ of $C(k)^{gr}$ we set

$$(E \otimes_k F)^{(n)} := \bigoplus_{i+j=n} E^{(i)} \otimes_k F^{(j)}.$$

This defines a monoidal structure on the category $C(k)^{gr}$. We define a symmetry constraints on $C(k)^{gr}$ induced from the symmetry constraint of the monoidal structure on $C(k)$. In formula, this is the family of isomorphisms $(E \otimes_k F)^{(n)} \simeq (F \otimes_k F)^{(n)}$ which is the sum over $i + j = n$ of the symmetry maps of complexes reminded in (1) on page 18

$$V_{E^{(i)}, F^{(j)}} : E^{(i)} \otimes_k F^{(j)} \to F^{(j)} \otimes_k E^{(i)}$$

We warn the reader here that the signs rule only involves the cohomological degrees and not the weights themselves. With these definitions the category $C(k)^{gr}$ is endowed with a symmetric monoidal structure $\otimes_k$.

The model category structure on $C(k)$ defines a natural model category structure on $C(k)^{gr}$ by defining fibrations, equivalences and cofibrations of graded complexes levelwise on their weight pieces. A morphism $f: E \to F$ in $C(k)^{gr}$ is thus a fibration (resp. cofibration, resp. weak equivalence) if for each $n \in \mathbb{Z}$ the morphism $f^{(n)}: E^{(n)} \to F^{(n)}$ is so in $C(k)$. This model category structure is compatible with the symmetric monoidal structure and makes $C(k)^{gr}$ into a symmetric monoidal model category in the sense of [Hov99, §4]
**1.2.2. Weight twists.** As for the shift of complexes, we have weight shifts, which will be rather referred to as *weight twists*. We denote the weight twists of an object *E* by *E(i)*, and it is formally defined by the formula

$$ (E(i))^{(n)} := E^{(n-i)}. $$

We warn the reader about the discrepancy between cohomological shifts and weight shifts: for $F \in C(k)$ and $E \in C(k)^{gr}$ we have

$$ (F[i])^n = F^{n+i} \qquad (E(i))^n = E^{(n-i)}. $$

Therefore, with these conventions, if $E$ is pure of weight 0, that is $E^{(i)} = 0$ if $i \neq 0$, then $E(1)$ is pure of weight 1.

**1.2.3. Mixed structures on graded complexes.** Because of its importance, we display the notion of mixed structure as a definition by its own.

**DEFINITION 1.2.3.1.** A graded mixed structure on a graded complex $E \in C(k)^{gr}$ consists of a family of morphisms of complexes

$$ \epsilon^{(n)} : E^{(n)} \to E^{(n+1)}[-1], $$

for $n \in \mathbb{Z}$, such that

$$ (2) \qquad (\epsilon^{(n+1)}[-1]) \circ \epsilon^{(n)} : E^{(n)} \to E^{(n+2)}[-2] $$

equals the zero map for all $n$.

A graded complex $E$ endowed with a graded mixed structure is called a graded mixed complex.

In order to keep notations simple we will most often write $\epsilon : E^{(n)} \to E^{(n+1)}[-1]$ instead of $\epsilon^{(n)}$, and the structural equation (2) by $\epsilon^2 = 0$. Note that graded mixed complexes can also be considered as bi-graded $k$-modules (for the cohomological degree and the weight) endowed with two differentials $d$ and $\epsilon$, satisfying the following conditions.

(1) The differential $d$ is of bidegree $(1, 0)$ and $\epsilon$ is of bidegree $(-1, 1)$.

(2) We have $d^2 = \epsilon^2 = 0$.

(3) We have $d\epsilon + \epsilon d = 0$.

Therefore, graded mixed complexes can also be considered as bicomplexes, by reindexing the weights in order to get a second differential of bidgree $(0, 1)$. However, we recommend the reader not doing so, as for instance the symmetry constraints we defined on graded complexes do not match the usual sign rules of tensor products of bicomplexes. It is also important, for many purposes in the sequel, to consider $\epsilon$ as an *external* differential which should not be treated on the same footing as the *internal* cohomological differential.

Graded mixed complexes form a category in an obvious manner. The morphisms between two graded mixed complexes $E$ and $F$ simply being the morphisms of graded complexes which commute with the mixed structures $\epsilon$ on both sides. More explicitly, a morphism from $E$ to $F$
consists of a family of morphisms $f(n) : E^{(n)} \to F^{(n)}$ in $C(k)$ such that the following diagram commutes

$$ \begin{tikzcd} E^{(n)} \arrow[r, "f^{(n)}"'] \arrow[d, "\epsilon^{(n)}"] & F^{(n)} \arrow[d, "\epsilon^{(n)}"'] \\ E^{(n+1)}[-1] \arrow[r, "f^{(n+1)}[-1]"'] & F^{(n+1)}[-1]. \end{tikzcd} $$

This category is denoted by $\epsilon - C(k)^{gr}$.

It is useful to present $\epsilon - C(k)^{gr}$ also as a category of modules over a certain algebra in the monoidal category $C(k)^{gr}$ of graded complexes. For this, we introduce $k[\epsilon] \in C(k)^{gr}$, the object defined as

$$ k[\epsilon] := k(0) \oplus k(1)[1]. $$

According to our conventions with shifts and twists (see section 1.2.2) this means that the weight pieces of $k[\epsilon]$ are given by

$$ k[\epsilon]^{(n)} = \begin{cases} k & \text{if } n = 0 \\ k[1] & \text{if } n = 1 \\ 0 & \text{if } n \neq 0, 1. \end{cases} $$

The graded complex $k[\epsilon]$ possesses a natural monoid structure, with respect to the tensor product of graded complexes mentioned in §1.2.1. It is given by a morphism of graded complexes

$$ k[\epsilon] \otimes_k k[\epsilon] \to k[\epsilon] $$

defined by

$$ k[\epsilon] \otimes_k k[\epsilon] \simeq k(0) \oplus k[1](1) \oplus k[1](1) \oplus k[2](2) \to k \oplus k[1](1) $$

sending $k[2](2)$ to zero, $k$ isomorphically to $k$ by the identity, and each factor $k[1](1)$ isomorphically to $k[1](1)$ by the identity as well. In other words, as a monoid in $C(k)^{gr}$, $k[\epsilon]$ is the trivial square zero extension of $k$ by $k[1](1)$ (see [TV08, §1.2.1]). It is, in particular, an associative, unital and commutative monoid.

We can now rephrase the definition of graded mixed complexes 1.2.3.1 as follows. A graded mixed structure on a graded complex $E \in C(k)^{gr}$ consists of the data of a (unital) $k[\epsilon]$-module structure

$$ k[\epsilon] \otimes_k E \to E. $$

Indeed, the structure map of such a module structure provides

$$ k[\epsilon] \otimes_k E \simeq E \oplus E(1)[1] \to E, $$

sum of the identity and a morphism $E \to E(-1)[-1]$ of graded complexes. In degree $n$ this defines a morphism of complexes

$$ \epsilon^{(n)} : E^{(n)} \to E^{(n+1)}[-1]. $$
The fact that $\epsilon$ defined as above satisfies $\epsilon^2 = 0$ is the consequence of the associativity of the module structure. This construction provides a functor

$$ \epsilon - C(k)^{gr} \to k[\epsilon] - Mod, $$

which is an isomorphism of categories. Note that the right hand side of this equivalence of categories consists of modules inside the monoidal category $C(k)^{gr}$, which by unfolding the definitions amount to graded dg-modules over the graded dg-algebra $k[\epsilon]$ (endowed with the zero cohomological differential).

The fact that the category $\epsilon - C(k)^{gr}$ can be written as a category of modules $k[\epsilon] - Mod$ possesses an important consequence, namely the existence of model category structures. Indeed, according to [Kel06, Thm. 3.2] there exists two model category structures on $\epsilon - C(k)^{gr}$, the projective model structure and the injective model structure, defined using the forgetful functor $\epsilon - C(k)^{gr} \to C(k)^{gr}$. The following Proposition summarizes the situation.

**PROPOSITON 1.2.3.2.** There exists a projective (resp. an injective) model category structure on $\epsilon - C(k)^{gr}$ whose fibrations (resp. cofibrations) and weak equivalences are defined on the underlying graded complexes obtained by forgetting the mixed structures. The forgetful functor $\epsilon - C(k)^{gr} \to C(k)^{gr}$ is a left (resp. right) Quillen functor for the projective (resp. injective) model category structure.

The $\infty$-category associated to the model category $\epsilon - C(k)^{gr}$ will be denoted by $\epsilon - \mathrm{dg}_k^{gr}$. By definition it does not depend on which of the two model category structure we choose as they both share the same notion of weak equivalences.

**REMARK 1.2.3.3.** In the proposition above, the expression *injective model structure* can be confusing. Indeed, the injective model structure on $\epsilon - C(k)^{gr}$ is here relative to the projective model structure on $C(k)^{gr}$. In particular, it is not the same as the usual (absolute) injective model structure for which cofibrations are defined to be monomorphisms. One important advantage of the injective model structure of proposition 1.2.3.2 is the fact that it remains a monoidal model category (which is not the case for the absolute injective model structure).

The category $\epsilon - C(k)^{gr}$, of graded mixed complexes, carries a natural symmetric monoidal structure denoted by $\otimes_k$ and defined as follows. For two objects $E$ and $F$ of $\epsilon - C(k)^{gr}$, we first form the tensor product $E \otimes_k F$ as a graded complex. We then define a mixed structure on $E \otimes_k F$ whose component of weight $n$ is given as a morphism of complexes

$$ \epsilon^{(n)} : \bigoplus_{i+j=n} E^{(i)} \otimes_k F^{(j)} \to \bigoplus_{a+b=n+1} E^{(a)} \otimes_k F^{(b)}[-1] $$

sending $x \otimes y \in E^{(i)} \otimes_k F^{(j)}$ to

$$ \epsilon(x) \otimes y + (-1)^{|x|} x \otimes \epsilon(y) \in E^{(i+1)} \otimes_k F^{(j)}[-1] \bigoplus E^{(i)} \otimes_k F^{(j+1)}[-1], $$
where as usual $|x|$ denotes the cohomological degree of $x$. It is straightforward to check that this defines a graded mixed structure on the graded complex $E \otimes_k F$, and thus an object, still denoted by $E \otimes_k F$, in the category $\epsilon - C(k)^{gr}$. From the formula, it is easy to see that the associativity, unital and symmetry constraints form the monoidal structure of graded complexes, are all compatible with this definition, and thus provide an associative, unital and symmetric monoidal structure on $\epsilon - C(k)^{gr}$. By construction, it is such that the faithful functor

$$F : \epsilon - C(k)^{gr} \to C(k)^{gr},$$

obtained by forgetting the mixed structure, is a symmetric monoidal functor when endowed with the canonical monoidal structure identifying $F(E \otimes_k F)$ with $F(E) \otimes_k F(F)$ by means of the identity morphism.

The symmetric monoidal structure on $\epsilon - C(k)^{gr}$ can also be described in terms of a commutative and cocommutative Hopf structure on the object $k[\epsilon] \in C(k)^{gr}$. The comultiplication for this Hopf structure is a morphism of monoids in $C(k)^{gr}$

$$\Delta : k[\epsilon] \to k[\epsilon] \otimes_k k[\epsilon]$$

obtained by sending $\epsilon$ to $\epsilon \otimes 1 + 1 \otimes \epsilon$. It should be noted here that the multiplication on $k[\epsilon] \otimes_k k[\epsilon]$ is defined as the composition

$$k[\epsilon] \otimes_k k[\epsilon] \otimes_k k[\epsilon] \otimes_k k[\epsilon] \xrightarrow{id \otimes V \otimes id} k[\epsilon] \otimes_k k[\epsilon] \otimes_k k[\epsilon] \otimes_k k[\epsilon] \xrightarrow{\mu \otimes \mu} k[\epsilon] \otimes_k k[\epsilon],$$

where $V = V_{k[\epsilon],k[\epsilon]}$ is the volte (see (1) on page 18), and $\mu$ the multiplication map. In particular, $(1 \otimes \epsilon).(\epsilon \otimes 1) = -(\epsilon \otimes 1).(1 \otimes \epsilon)$, and thus $(\epsilon \otimes 1+1 \otimes \epsilon)^2 = 0$ as required. This comultiplication makes $k[\epsilon]$ into a commutative and cocommutative bialgebra object in $C(k)^{gr}$ and thus induces a symmetric monoidal structure on its category of modules $k[\epsilon] - Mod \simeq \epsilon - C(k)^{gr}$. We leave to the reader that this monoidal structure is naturally isomorphic to the one already discussed above.

Finally, the two model category structures on $\epsilon - C(k)^{gr}$ are compatible with the monoidal structure, and makes $\epsilon - C(k)^{gr}$ into a symmetric monoidal model category, for both the injective and the projective model structures. As a result, the $\infty$-category $\epsilon - dg_k^{gr}$ of graded mixed complexes over $k$ carries a canonical symmetric monoidal structure, again denoted by $\otimes_k$ (see A.1.

We would like to finish this section by mentioning that the stable $\infty$-category $\epsilon - dg_k^{gr}$ carries a natural t-structure for which an object $E$ is connective if for any $n \in \mathbb{Z}$ the part of weight $n$ $E^{(n)}$ is $n$-connective: $H^i(E^{(n)}) = 0$ if $i < -n$. This t-structure is directly related to the Beilinson's t-structure on filtered complexes by the equivalence 1.3.4.4 proved in the next section, and therefore will be referred as the Beilinson's t-structure on graded mixed complexes as well.
**DEFINITION 1.2.3.4.** The Beilinson's t-structure on the $\infty$-category $\epsilon - \mathrm{dg}_k^{\mathrm{gr}}$ of graded mixed complexes is the t-structure for which an object $E$ is connective if $E^{(n)}$ is $n$-connective for all $n \in \mathbb{Z}$.

## 1.3. Filtered complexes

In this section $k$ is of arbitrary characteristics.

**1.3.1. The homotopy theory of filtered complexes.** We remind that a filtered complex $E$ (of $k$-modules) consists of a sequence of complexes $F^n E$, indexed by $n \in \mathbb{Z}$, together with morphisms of complexes $F^n E \to F^{n+1} E$ for all $n$. In another language, $E$ consists of a functor

$$ \mathbb{Z} \to C(k), $$

whose values at $n$ is $F^n E$, and where $\mathbb{Z}$ is the ordered group of integers considered as a category in the usual manner. Filtered complexes form a category in a natural way, for which the morphisms are defined to be the natural transformations of functors $\mathbb{Z} \to C(k)$. We denote by $C(k)^{\mathrm{fil}}$ this category, which is therefore the category of all functors from the ordered group $\mathbb{Z}$ to $C(k)$. Note that a morphism $E \to E'$ consists of a family of morphisms of complexes $F^n E \to F^n E'$ for $n \in \mathbb{Z}$, such that all the diagrams

<figure><img src="image_4.png" /></figure>

are commutative.

**REMARK 1.3.1.1.** Our conventions for filtered complexes differ from the one of [Lur15] and [Mou21]. In our setting the filtrations are increasing, whereas the filtrations of these references are decreasing by definition, and therefore correspond to functors $\mathbb{Z}^{\mathrm{op}} \to C(k)$ instead. These two notions are equivalent by means of the isomorphism of ordered groups $-1 : \mathbb{Z} \to \mathbb{Z}^{\mathrm{op}}$, sending $n$ to $-n$. In more concrete terms, we can turn an increasing filtered complex $E$ into a decreasing one $E'$ by setting $F^n E' := F^{-n} E$ for all $n \in \mathbb{Z}$.

Being a category of diagrams in $C(k)$, the category $C(k)^{\mathrm{fil}}$ comes equipped with a natural model category structure, for which the fibrations and the weak equivalence equivalences are defined levelwise (the projective model structure on the category of diagrams in a combinatorial model category). In a more explicit manner, a morphism $f : E \to E'$ is defined to be a weak equivalence (resp. a fibration) if and only if for all $n \in \mathbb{Z}$ the morphism of complexes $F^n E \to F^n E'$ is a quasi-isomorphism (resp. an epimorphism). The corresponding $\infty$-category (obtained by localization, see A.1) will be denoted by $\mathrm{dg}_k^{\mathrm{fil}}$.
The category $C(k)^{fil}$ is also endowed with a symmetric monoidal structure. It can be defined explicitly, for two filtered complexes $E$ and $E'$, by the following formula

$$F^n(E \otimes_k E') := \mathrm{Colim}_{i+j \le n} (F^i E) \otimes_k (F^j E).$$

The colimit is here taken over the category whose objects are pairs $(i, j)$ of integers with $i + j \le n$, and for which there is a unique morphism $(i, j) \to (i', j')$ if and only if $i \le i'$ and $j \le j'$. In a more functorial manner, this monoidal structure is obtained as the convolution product on the category of functors $\mathrm{Fun}(\mathbb{Z}, C(k))$, using that addition in $\mathbb{Z}$ to define the structure of a symmetric monoidal category. For two such functors $E, E' : \mathbb{Z} \to C(k)$, we can form their external product

$$E \boxtimes_k E' : \mathbb{Z} \times \mathbb{Z} \to C(k)$$

sending $(i, k)$ to $F^i E \otimes_k F^j E'$. The addition defines a functor $+ : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$, and we can thus form the left Kan extension along $+$ to get the tensor product

$$E \otimes_k E' := (+)_!(E \boxtimes_k E') \in \mathrm{Fun}(\mathbb{Z}, C(k)).$$

We refer the reader to [Lur15, Mou21] for more details on this construction.

The monoidal structure on $C(k)^{fil}$ comes equipped with natural associative, unital and symmetry constraints. Moreover, it is compatible with the model category structure on $C(k)^{fil}$, and makes it into a symmetric monoidal model category. As a consequence the $\infty$-category $\mathrm{dg}_k^{fil}$ comes equipped with a canonical symmetric monoidal structure (see A.1.

We have two important functors out of the category $C(k)^{fil}$, namely the underlying object and associated graded functors. The associated graded is the functor

$$\mathrm{Gr} : C(k)^{fil} \to C(k)^{gr}$$

sending a filtered complex $E$ to the graded complex $\mathrm{Gr}(E)$ whose component of weight $n$ is given by

$$\mathrm{Gr}(E)^{(n)} := \mathrm{Coker}(F^{n-1}E \to F^n E) \in C(k).$$

We will often use the more standard notation

$$\mathrm{Gr}^n(E) := \mathrm{Gr}(E)^{(n)}.$$

This functor is a left adjoint and part of a Quillen adjunction

$$\mathrm{Gr} : C(k)^{fil} \rightleftharpoons C(k)^{gr} : e.$$

The right adjoint $e$ sends a graded object $E$ to the filtered object $e(E)$ with $F^n(e(E)) := E^{(n)}$, and for which the morphism $E^{(n)} \to E^{(n+1)}$ are all defined to be zero. This adjunction is a Quillen adjunction as $e$ obviously preserves fibrations and equivalences. The left adjoint $\mathrm{Gr}$ is moreover endowed with a canonical symmetric monoidal structure

$$\mathrm{Gr}(E) \otimes_k \mathrm{Gr}(E') \simeq \mathrm{Gr}(E \otimes_k E').$$
These functorial isomorphisms are defined by the observation that for all $n \in \mathbb{Z}$ and $a+b < n$, the composite morphism

$$F^a(E) \otimes_k F^b(E') \to F^n(E \otimes_k E') \to Gr^n(E \otimes_k E')$$

is the zero map, which gives rise to natural morphisms

$$Gr^i(E) \otimes_k Gr^j(E') \to Gr^n(E \otimes_k E')$$

for $i + j = n$. We leave the reader to check that the sum of all these morphisms over $i + j = n$ is an isomorphism. The left derived functor of $Gr$ therefore induces a well defined symmetric monoidal $\infty$-functor (see A.1)

$$\mathbf{Gr} : \mathbf{dg}_k^{\mathbf{fil}} \to \mathbf{dg}_k^{\mathbf{gr}}.$$

**REMARK 1.3.1.2.** Note that the functor $Gr : C(k)^{\mathbf{fil}} \to C(k)^{\mathbf{gr}}$ and its left derived $\infty$-functor $\mathbf{Gr} : \mathbf{dg}_k^{\mathbf{fil}} \to \mathbf{dg}_k^{\mathbf{gr}}$ are related by the following diagram of $\infty$-functors

<figure><img src="image_5.png" /></figure>

where $h$ is a non-invertible natural transformation between $\infty$-functors $C(k)^{\mathbf{fil}} \to \mathbf{dg}_k^{\mathbf{gr}}$ (measuring the non-commutativity of $Gr$ with the vertical localization functors). We refer to A.1 for details on the general situation of deriving symmetric monoidal $\infty$-functors.

The second important functor out of the category $C(k)^{\mathbf{fil}}$ is the *underlying object functor*. It is the functor

$$(-)^u : C(k)^{\mathbf{fil}} \to C(k)$$

sending $E$ to the complex

$$E^u := \mathrm{Colim}_{n \in \mathbb{Z}} F^n E.$$

This functor is again the left adjoint of a Quillen adjunction. The right adjoint to this adjunction sends a complex $E$ to the constant diagram $F^n E = E$, where all morphisms $F^n E \to F^{n+1} E$ are identities. We will denote this filtered object by $E_\infty$. Note that in particular we have $Gr(E_\infty) \simeq 0$, whereas $(E_\infty)^u \simeq E$ by the counit of the adjunction. In particular $E \to E_\infty$ is fully faithful and $(-)^u$ is a localization functor.

The functor $(-)^u$ is again endowed with a natural symmetric monoidal structure

$$(E)^u \otimes_k (E')^u \simeq (E \otimes_k E')^u.$$

Both of these functors preserve quasi-isomorphisms, and after localization this adjunction induces an adjunction on the corresponding $\infty$-categories

$$(-)^u : \mathbf{dg}_k^{\mathbf{fil}} \to \mathbf{dg}_k : (-)_\infty.$$
REMARK 1.3.1.3. The $\infty$-category $\mathrm{dg}_k^{\mathrm{fil}}$ can be canonically identified with the $\infty$-category $\mathrm{QCoh}([\mathbb{A}^1/\mathbb{G}_m])$ of quasi-coherent complexes on the (geometric) quotient stack $[\mathbb{A}^1/\mathbb{G}_m]$. This idea goes back to C. Simpson (see [Mou21] for a recent account). Analogously (and more classically), the $\infty$-category $\mathrm{dg}_k^{\mathrm{gr}}$ can be canonically identified with the $\infty$-category $\mathrm{QCoh}(B\mathbb{G}_m)$ of quasi-coherent complexes on the (geometric) classifying stack $B\mathbb{G}_m$ of $\mathbb{G}_m$. Through these identifications, the underlying object functor $(-)^u : \mathrm{dg}_k^{\mathrm{fil}} \to \mathrm{dg}_k$ becomes the pullback on quasi-coherent complexes along the inclusion $\mathrm{Spec} k \simeq [\mathbb{G}_m/\mathbb{G}_m] \to [\mathbb{A}^1/\mathbb{G}_m]$, while the associated graded object functor $\mathrm{Gr} : \mathrm{dg}_k^{\mathrm{fil}} \to \mathrm{dg}_k^{\mathrm{gr}}$ is viewed as the pullback along the map $0 : B\mathbb{G}_m \to [\mathbb{A}^1/\mathbb{G}_m]$ corresponding to the $\mathbb{G}_m$-equivariant inclusion of the origin in $\mathbb{A}^1$. The interested reader is referred to [Mou21] for details.

### 1.3.2. Graded mixed complexes and filtered complexes.
We are now ready to compare filtered complexes with graded mixed complexes. This comparison states the existence of a Quillen adjunction between $C(k)^{\mathrm{fil}}$ and $\epsilon - C(k)^{\mathrm{gr}}$, which fully faithfully embeds the $\infty$-category of graded mixed complexes into the $\infty$-category of filtered complexes. The image of this embedding consists of complete filtered complexes, and thus we can state that the $\infty$-category of graded mixed complexes is equivalent to the $\infty$-category of complete filtered complexes. This equivalence can be enhanced into an equivalence of symmetric monoidal $\infty$-categories once the monoidal structure on filtered complexes is suitably completed.

We start by describing the *Tate realization functor*

$$|-|^t : \epsilon - C(k)^{gr} \to C(k)^{\mathrm{fil}}.$$

For a given graded mixed complex $E$ the $i$-th layer of the filtered complex $|E|^i$ is given by

$$(3) \qquad F^i|E|^i := \prod_{p \ge -i} E^{(p)}[-2p],$$

where the infinite product on the right is endowed with the total differential $D := d + \epsilon$. In formulas, for a family of elements $x := \{x_p\}$ in $\prod_{p \ge -i} E^{(p)}[-2p]$, the element $D(x) := \{D(x)_p\}$ is explicitly given by

$$D(x)_p := d(x_p) + \epsilon(x_{p-1}),$$

where by convention $x_p = 0$ if $p < -i$. The morphism $F^i|E|^i \hookrightarrow F^{i+1}|E|^i$ is itself defined to be the natural inclusion of infinite products

$$\prod_{p \ge -i} E^{(p)}[-2p] \hookrightarrow \prod_{p \ge -i-1} E^{(p)}[-2p],$$

that sets the first component to be zero.

This defines in a functor

$$|-|^t : \epsilon - C(k)^{gr} \to C(k)^{\mathrm{fil}}.$$
This functor commutes with arbitrary limits, and therefore by Freyd adjoint theorem (see [TV15, Prop. 2.9]), it admits a left adjoint denoted by $\psi$. The functor $\psi$ can be made explicit using complicated formula, but we will not need this and for our purpose its existence will be enough. Moreover, by definition of the various model structures, it is clear that $|-|^t$ preserves fibrations. The following lemma shows that it furthermore preserves equivalences.

**LEMMA 1.3.2.1.** *The functor $|-|^t : \epsilon - C(k)^{gr} \to C(k)^{fil}$ sends quasi-isomorphisms of graded mixed complexes to quasi-isomorphisms of filtered complexes.*

*Proof.* Let $E \to E'$ be a quasi-isomorphism of graded mixed complexes. We must show that for all $i \in \mathbb{Z}$ the induced morphism of complexes

$$F^i|E|^t \to F^i|E'|^t$$

is a quasi-isomorphism. For this, we use that sequence of subcomplexes $F^j|E|^t \subset F^i|E|^t$, for $j \le i$. Each successive quotient morphism

$$(F^i|E|^t)/(F^{j+1}|E|^t) \to (F^i|E|^t)/(F^j|E|^t)$$

is a fibration of complexes, and by the explicit formula for $|E|^t$ (3) on page 28) the natural map

$$F^i|E|^t \to \lim_{j \le i} (F^i|E|^t)/(F^j|E|^t)$$

is an isomorphism of complexes. In particular, the canonical map

$$F^i|E|^t \to \text{Holim}_{j \le i} (F^i|E|^t)/(F^j|E|^t)$$

is itself a quasi-isomorphism of complexes. As a consequence, the lemma will follow if we can prove that for all $j \le i$ fixed, the induced morphisme

$$(F^i|E|^t)/(F^j|E|^t) \to (F^i|E'|^t)/(F^j|E'|^t)$$

is a quasi-isomorphism. But this can be proved by induction on $j$ using the short exact sequences of complexes

$$0 \to (F^j|E|^t)/(F^{j-1}|E|^t) \to (F^i|E|^t)/(F^{j-1}|E|^t) \to (F^i|E|^t)/(F^j|E|^t) \to 0,$$

together with the identification of complexes $(F^j|E|^t)/(F^{j-1}|E|^t) \simeq E^{(-j)}[-2j]$.

An important consequence of the lemma 1.3.2.1 is that $|-|^t$ is a right Quillen functor. In particular, we obtain this way an induced adjunction of $\infty$-categories

$$\psi : \mathbf{dg}_k^{fil} \hookrightarrow \epsilon - \mathbf{dg}_k^{gr} : |-|^t.$$

Note that the associated graded of the realization can be described explicitly as (beware of the change of signs in the gradings !)

$$(4) \qquad \mathrm{Gr}^p(|E|) \simeq E^{-p}[-2p].$$

The importance of the realization $\infty$-functor is the following result, that will be used many times during this book.
PROPOSITON 1.3.2.2. The $\infty$-functor

$$| - |^t : \epsilon - \mathbf{dg}_k^{gr} \to \mathbf{dg}_k^{fil}$$

is fully faithful.

*Proof.* We must show that for any object $E \in \epsilon - \mathbf{dg}_k^{gr}$, the counit morphism

$$\alpha : \psi(|E|^t) \to E$$

is an equivalence in $\epsilon - \mathbf{dg}_k^{gr}$. For this, we combine the adjunction $(\psi, | - |^t)$ with the adjunction

$$\mathbf{oub} : \epsilon - \mathbf{dg}_k^{gr} \stackrel{\sim}{\to} \mathbf{dg}_k^{fil} : F,$$

where the left adjoint **oub** is the forgetful $\infty$-functor sending a graded mixed complex to its underlying graded complex. The fact that **oub** is a left adjoint follows formally from the description of $\epsilon - \mathbf{dg}_k^{gr}$ as the $\infty$-category of graded dg-modules over $k[\epsilon]$ (see §1.2.3). The $\infty$-functor **oub** is conservative, and thus in order to prove that $\alpha$ is an equivalence, it is enough to show the following fact: for any graded complex $M \in \mathbf{dg}_k^{gr}$, the induced morphism

$$\mathbf{Map}(E, F(M)) \to \mathbf{Map}(\psi(|E|^t), F(M))$$

is an equivalence of spaces. Using the adjunction property this is equivalent to the fact that for all $M \in \mathbf{dg}_k^{gr}$, the natural morphism

$$\mathbf{Map}(\mathbf{oub}(E), M) \to \mathbf{Map}(|E|^t, |F(M)|^t)$$

is an equivalence of spaces.

To prove this last statement on mapping spaces, it will be enough to have an explicit description of the filtered complex $|F(M)|^t$. As a start, $F(M)$ is the graded mixed complex whose piece of weight $p$ is $M^{(p)} \oplus M^{(p+1)}[-1]$. The mixed structure is defined by the obvious map

$$\epsilon : M^{(p)} \oplus M^{(p+1)}[-1] \to M^{(p+1)}[-1] \oplus M^{(p+1)}[-2]$$

which is the identity on the factor $M^{(p+1)}[-1]$ and zero on other factors. Using that $M$ can be written as an infinite product $M \simeq \prod_p M^{(p)}(p)$ inside the $\infty$-category $\mathbf{dg}_k^{gr}$, we see that $|F(M)|^t$ is itself given, up to an equivalence, as an infinite product of filtered complexes

$$|F(M)|^t \simeq \prod_{p \in \mathbb{Z}} W_p.$$

Here, each $W_p$ is the Tate realization of the graded mixed complex $F(M^{(p)}(p))$. Note that $F(M^{(p)}(p))$ only possesses non-zero weight pieces in weight $p$ and $p-1$, respectively $M^{(p)}$ and $M^{(p)}[-1]$. The mixed structure is itself non-zero only in weight $p-1$, for which it is the identity

$$F(M^{(p)}(p))(p-1) = M^{(p)}[-1] \to F(M^{(p)}(p))(p)[-1] = M^{(p)}[-1].$$
Using the explicit formula for the Tate realization $| - |^t$ (see (3) on page 28), we deduce that $W_p$ is, up to a natural equivalence, the filtered complex given by

$$F^i W_p = \begin{cases} 0 & \text{if } i \neq p \\ M^{(p)}[-2p] & \text{if } i = -p. \end{cases}$$

As a result, for any $F' \in \mathbf{dg}_k^{fil}$, we have functorial equivalences in $M \in \mathbf{dg}_k^{gr}$

$$\mathbf{Map}_{\mathbf{dg}_k^{fil}}(F', |F(M)|^t) \simeq \prod_{p \in \mathbb{Z}} \mathbf{Map}_{\mathbf{dg}_k}(Gr^{-p}(F')[2p], M^{(p)}).$$

On the other hand, for $E \in \epsilon - \mathbf{dg}_k^{gr}$ we have canonical equivalences of complexes (see (4) above)

$$E^{(-p)}[-2p] \simeq Gr^p(|E|^t).$$

Therefore, we have canonical equivalences of spaces

$$\mathbf{Map}_{\mathbf{dg}_k^{gr}}(\mathbf{oub}(E), M) \simeq \prod_{p \in \mathbb{Z}} \mathbf{Map}_{\mathbf{dg}_k}(E^{(p)}, M^{(p)}) \simeq \mathbf{Map}_{\mathbf{dg}_k^{fil}}(|E|^t, |F(M)|^t).$$

By construction, this equivalence is the natural morphism $Map(\mathbf{oub}(E), M) \to Map(|E|^t, |F(M)|^t)$ induced by adjunction. This concludes the proof that the counit $\alpha$ is itself an equivalence, and thus the proof of the Proposition. □

**1.3.3. Graded mixed complexes as complete filtered complexes.** The full embedding of Proposition 1.3.2.2 can be refined into an equivalence of $\infty$-categories, by characterizing the essential image of the $\infty$-functor $\psi$. For this, we introduce the notion of completeness for objects in $\mathbf{dg}_k^{fil}$.

**DEFINITION 1.3.3.1.** An object $E \in \mathbf{dg}_k^{fil}$ is complete if we have an equivalence in $\mathbf{dg}_k$

$$\lim_{i \le 0} F^i E \simeq 0.$$

Note that the limit and the equivalence of the above definition happen inside the $\infty$-category $\mathbf{dg}_k$ of complexes. In particular, $\lim_{i \le 0} F^i E$ is here the homotopy limit of the projective system of complexes

$$\dots \to F^i E \to F^{i-1} E \to \dots \to F^{-1} E \to F^0 E.$$

On the level of cohomology, this homotopy limit also involves the derived functor $\lim^1$. In particular, if $V$ is a $k$-module endowed with a filtration by submodules $F^i V \subset V$, being complete in the sense of Definition 1.3.3.1 is stronger than the naive separateness condition stating that $\cap_i F^i V = 0$. Indeed, if we denote by $F^0 \hat{V} := \lim_{i \le 0} F^0 V / F^i V$ the naive completion of $k$-modules, we have natural isomorphisms

$$H^0(\mathrm{Holim}_i F^i V) \simeq \cap_i F^i V \qquad H^1(\mathrm{Holim}_i F^i V) \simeq (F^0 \hat{V}) / (F^0 V),$$
and $H^i(\mathrm{Holim}_i F^i V) \simeq 0$ for all $i \neq 0, 1$. This implies that a filtered $k$-module $V$ is complete in the sense of Definition 1.3.3.1 if and only if it satisfies the following two conditions

• $\cap_{i \leq 0} F^i V = 0$

• the inclusion map $F^0 V \to F^0 \hat{V}$ is bijective.

**REMARK 1.3.3.2.** The completeness condition of Definition 1.3.3.1 can also be stated as follows. For a filtered complex $E \in \mathbf{dg}_k^{\mathrm{fil}}$, and any $j \in \mathbb{Z}$ we can form the natural morphism of complexes

$$F^j E \to \lim_{i \le j} (F^i E)/(F^j E).$$

Then, $E$ is complete if and only if for all $j$ the above morphism is a quasi-isomorphism. Indeed, the fiber of the above morphism, taken in $\mathbf{dg}_k$, is equivalent to $\lim_{i \le j} F^i E$. Moreover, if $j \ge 0$, the natural projection $\lim_{i \le j} F^i E \to \lim_{i \le 0} F^i E$ is a quasi-isomorphism, as the inclusion of posets $\{i \le 0\} \subset \{i \le j\}$ is final. In the same manner, if $j \le 0$, the projection $\lim_{i \le 0} F^i E \to \lim_{i \le j} F^i E$ is an equivalence.

**PROPOSITON 1.3.3.3.** The essential image of the full embedding

$$| - |^t : \epsilon - \mathbf{dg}_k^{gr} \hookrightarrow \mathbf{dg}_k^{\mathrm{fil}}$$

consists of all complete filtered complexes in the sense of Definition 1.3.3.1.

*Proof.* By the Proposition 1.3.2.2, the $\infty$-functor $| - |^t$ identifies $\epsilon - \mathbf{dg}_k^{gr}$ with a full reflexive sub-$\infty$-category $\mathcal{C}$ of $\mathbf{dg}_k^{\mathrm{fil}}$. We have to show that an object $E \in \mathbf{dg}_k^{\mathrm{fil}}$ lies in this sub-$\infty$-category if and only if it is complete.

As a start, from the explicit form of the Tate realization functor (see (3) on page 28) we see that $|E|^t$ is a complete filtered complex for any graded mixed complex $E$. Therefore, all objects of $\mathcal{C}$ are complete filtered complexes. Conversely, let $E$ be a complete filtered complex. By definition, $E$ lies in $\mathcal{C}$ if and only if the adjunction unit morphism $E \to |\psi(E)|^t$ is a quasi-isomorphism of filtered complexes. During the proof of Proposition 1.3.2.2, we have seen that the composition of $\psi$ with the forgetful $\infty$-functor $\mathrm{oub} : \epsilon - \mathbf{dg}_k^{gr} \to \mathbf{dg}_k^{gr}$, is naturally equivalent to $E \mapsto \mathrm{Gr}^{*-}(E)[2*]$. Here, $\mathrm{Gr}^{*-}(E)[2*]$ denotes the graded complexe whose weight $p$ piece is $\mathrm{Gr}^{-p}(E)[2p]$. We deduce from this the following lemma.

**LEMMA 1.3.3.4.** The $\infty$-functor

$$\psi : \mathbf{dg}_k^{\mathrm{fil}} \to \epsilon - \mathbf{dg}_k^{gr},$$

restricted to the full sub-$\infty$-category of complete filtered complexes, is conservative.

*Proof of the lemma.* As the forgetful $\infty$-functor $\mathrm{oub}$ is itself conservative, it is enough to prove the conservativity of the composition $\mathrm{oub} \circ \psi : \mathbf{dg}_k^{\mathrm{fil}} \to \mathbf{dg}_k^{gr}$, when restricted to complete filtered complexes. By the formula $\circ \mathrm{oub} \psi \simeq \mathrm{Gr}^{*-}[2*]$ we see that it is even enough to show that the family of $\infty$-functors $\mathrm{Gr}^i$ for $i \in \mathbb{Z}$ is a conservative family of $\infty$-functors when restricted to complete filtered complexes.
Let $E \to E'$ be a morphism of complete filtered complexes, and assume that for all $i \in \mathbb{Z}$ the induced morphism $Gr^i(E) \to Gr^i(E')$ is a quasi-isomorphism. For all $i \in \mathbb{Z}$, we have a canonical morphism of complexes

$$ \alpha_i : F^i E \to \lim_{j \le i} (F^i E / F^j E). $$

The fiber of this morphism is naturally equivalent to $\lim_{j \le i} F^j E$, which is the zero object by completeness of $E$. Therefore, for all $i$, the morphism $\alpha_i$ is an equivalence of complexes. Moreover, for all $j \le i$ we have a commutative diagram of complexes, whose horizontal rows are fibration sequences of complexes

$$ \begin{array}{ccc} \mathrm{Gr}^{-j}(E)[2j] & \to & (F^i|E|^t)/(F^{j-1}|E|^t) \to (F^i|E|^t)/(F^j|E|^t) \\ \downarrow & & \downarrow \\ \mathrm{Gr}^{-j}(E')[2j] & \to & (F^i|E'|^t)/(F^{j-1}|E'|^t) \to (F^i|E'|^t)/(F^j|E'|^t). \end{array} $$

We therefore see by induction on $j$, than for all $j \le i$ the induced morphism

$$ (F^i|E|^t)/(F^j|E|^t) \to (F^i|E'|^t)/(F^j|E'|^t) $$

is a quasi-isomorphism. Passing to the limit along $j \le i$ and using completeness as above we have that the induced morphism

$$ F^i E \simeq \lim_{j \le i} (F^i E / F^j E) \to F^i E' \simeq \lim_{j \le i} (F^i E' / F^j E') $$

is a quasi-isomorphism for all $i \in \mathbb{Z}$.

The lemma implies that complete filtered complexes are objects in $\mathcal{C}$, and thus finishes the proof of the Proposition. $\square$

**1.3.4. The completed monoidal structure.** By Proposition 1.3.3.3 we know that the Tate-realization $\infty$-functor $| - |^t : \epsilon - \mathrm{dg}_k^{gr} \to \mathrm{dg}_k^{fil}$ is fully faithful, and that its essential image consists of all complete filtered complexes in the sense of Definition 1.3.3.1. We will denote by $\widehat{\mathrm{dg}}_k^{fil} \subset \mathrm{dg}_k^{fil}$ the full sub-$\infty$-category of complete filtered complexes, so that $| - |^t$ induces an equivalence of $\infty$-categories

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

We will now study the monoidal behaviour of the above equivalence. For this, we start by considering the right Quillen functor $| - |^t : \epsilon - C(k) \to C(k)^{fil}$, and show that it carries a natural lax monoidal structure. For two graded mixed complexes $E$ and $F$, we construct a natural morphism

(5) $$ \alpha_{E,F} : |E|^t \otimes |F|^t \to |E \otimes F|^t $$
as follows. Such a morphism is determined by a compatible family of maps

$$ \alpha_{i,j} : F^i |E|^t \otimes F^j |F|^t \to F^{i+j} |E \otimes F|^t $$

for all integers $i$ and $j$. By the explicit formula for $| - |^t$ the morphism $\alpha_{i,j}$ is a morphism of complexes

$$ ( \prod_{p \ge -i} E^{(p)}[-2p] ) \otimes ( \prod_{q \ge -j} F^{(q)}[-2q] ) \to \prod_{n \ge -i-j} ( \bigoplus_{a+b=n} E^{(a)} \otimes F^{(b)}[-2n] ). $$

On the level of graded modules, the above morphism is given by a family of maps

$$ \alpha_n : ( \prod_{p \ge -i} E^{(p)}[-2p] ) \otimes ( \prod_{q \ge -j} F^{(q)}[-2q] ) \to \bigoplus_{a+b=n} E^{(a)} \otimes F^{(b)}[-2n]. $$

By definition, the morphism $\alpha_n$ symbolically sends a tensor of the form $\{x_p\} \otimes \{y_q\}$ to the element $\sum_{p+q=n} x_p \otimes y_q$ (note that this sum is finite as $p$ and $q$ are bounded below by $-i$ and $-j$). The family of $\alpha_n$ for $n \in \mathbb{Z}$ provides the morphism of complexes

$$ \alpha_{i,j} : F^i |E|^t \otimes F^j |F|^t \to F^{i+j} |E \otimes F|^t. $$

These morphisms are associative and symmetric with respect to the associativity and unit constraints. Moreover, the image by $| - |^t$ of the unit object $k(0) \in \epsilon - C(k)^{gr}$ is canonically isomorphic to $k$, considered as a filtered complex by

$$ F^i k = \begin{cases} 0 & \text{if } i < 0 \\ k & \text{if } i \ge 0 \end{cases} $$

In other words, there is a canonical isomorphism $u : |k(0)|^t \simeq k$, where $k$ is the unit object in $C(k)^{fil}$. All together, the $\alpha_{i,j}$ and the isomorphism $u$ defines a lax monoidal structure on the functor $| - |^t$, which is unital, associative and symmetric.

By adjunction, the lax monoidal structure on $| - |^t$ defines a unital, associative and symmetric colax monoidal structure on the left adjoint functor $\psi : C(k)^{fil} \to \epsilon - C(k)^{gr}$ as follows. We start by $E$ and $F$ two filtered complexes, and we consider the adjunction morphisms

$$ E \to |\psi(E)|^t \quad F \to |\psi(F)|^t. $$

Using the lax monoidal structure on $| - |^t$ constructed above (see (5) on page 33), we define a morphism

$$ E \otimes F \to |\psi(E)|^t \otimes |\psi(F)|^t \to |\psi(E) \otimes \psi(F)|^t. $$

Its image by $\psi$, composed with the counit $| - |^t \Rightarrow id$, gives a morphism

$$ \psi(E \otimes F) \to \psi(E) \otimes \psi(F), $$

which is a unital, associative and symmetric colax monoidal structure on the functor $\psi$.
LEMMA 1.3.4.1. The induced colax monoidal structure on the $\infty$-functor

$$ \psi : \mathbf{dg}_k^{\text{fil}} \to \epsilon - \mathbf{dg}_k^{\text{gr}} $$

is a monoidal structure. In other words, for all E and F objects in $\mathbf{dg}_k^{\text{fil}}$, the natural morphism

$$ \beta_{E,F} : \psi(E \otimes F) \to \psi(E) \otimes \psi(F) $$

is an equivalence of graded mixed complexes.

Proof. To check that $\beta_{E,F}$ is an equivalence in $\epsilon - \mathbf{dg}_k^{\text{gr}}$ we can use the conservative $\infty$-functor $\text{oub} : \epsilon - \mathbf{dg}_k^{\text{gr}} \to \mathbf{dg}_k^{\text{gr}}$ to graded complexes. Moreover, we know that $\text{oub} \circ \psi$ is naturally equivalent to $Gr^{-*}[-2*]$ (see the proof of Proposition 1.3.2.2). By observation, we the image of $\beta_{E,F}$ by $\text{oub}$ induces a morphism

$$ \mathrm{Gr}^{-*}(E \otimes F)[-2*] \to \mathrm{Gr}^{-*}(E)[-2*] \otimes \mathrm{Gr}^{-*}(F)[-2*] $$

which is nothing else than the standard monoidal structure on the $\infty$-functor $\mathrm{Gr}^{-*}[-2*]$. It is therefore an equivalence, proving that $\beta_{E,F}$ is indeed an equivalence of graded mixed complexes.

□

The consequence of the above lemma is that the colax monoidal structure on the $\infty$-functor $\psi$ is a symmetric monoidal structure. Therefore, we can use the equivalence

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

in order to transport the monoidal structure of $\epsilon - \mathbf{dg}_k^{\text{gr}}$ to a symmetric monoidal structure on $\widehat{\mathbf{dg}}_k^{\text{fil}}$. It will be denoted by $\hat{\otimes}$, and called the *completed tensor product*. The previous lemma, implies that the $\infty$-functor $\psi$ provides a symmetric monoidal $\infty$-functor

$$ \psi : \mathbf{dg}_k^{\text{fil}} \to \widehat{\mathbf{dg}}_k^{\text{fil}} \simeq \epsilon - \mathbf{dg}_k^{\text{gr}}. $$

This exhibits $\widehat{\mathbf{dg}}_k^{\text{fil}}$ has a symmetric monoidal localization of $\mathbf{dg}_k^{\text{fil}}$, obtained by inverting the *graded quasi-isomorphisms* (i.e. morphisms of filtered complexes inducing quasi-isomorphisms on the associated graded). The localization $\infty$-functor $\psi$ will also be called the *completion* $\infty$-functor, and denoted by

$$ \widehat{(-)} : \mathbf{dg}_k^{\text{fil}} \to \widehat{\mathbf{dg}}_k^{\text{fil}}. $$

Note that by construction we have canonical equivalences of complete filtered complexes

$$ (\widehat{E \otimes F}) \simeq \widehat{E} \hat{\otimes} \widehat{F}. $$

Note also, that the completion $\infty$-functor naturally commutes with forming the associated graded

$$ \mathrm{Gr}^*(\widehat{E}) \simeq \mathrm{Gr}^*(E), $$

as can be seen directly from the identification between $\text{oub} \circ \psi$ and $\mathrm{Gr}^{-*}[-2*]$.
On the level on notations, $|E|^t$ will always refer to the filtered complex associated to the graded mixed complex $E$. We will also use the following notation

**DEFINITION 1.3.4.2.** We define the (non Tate) realization $\infty$-functor

$$|-| : \epsilon - \mathbf{dg}_k^{gr} \to \mathbf{dg}_k, E \mapsto |E| := F^0|E|^t$$

sending a mixed graded module $E$ to the 0-th layer filtration of its Tate realization.

**REMARK 1.3.4.3.** It is easy to verify that the realization functor $|-|$ is equivalent to $\mathbb{R}\mathrm{Hom}(k(0), -)$, the derived enriched Hom complex from the unit graded mixed object $k(0)$. Note also that, when $E$ has only non-negative weight pieces, we have

$$|E| \simeq F^0|E|^t \simeq (|E|^t)^u$$

so $|E|$ is in fact the underlying complex of the filtered complexe $|E|^t$.

We can subsume our comparison results as follows.

**COROLLARY 1.3.4.4.** The Tate realization $\infty$-functor

$$|-|^t : \epsilon - \mathbf{dg}_k^{gr} \to \widehat{\mathbf{dg}}_k^{fil}$$

is a symmetric monoidal equivalence when $\widehat{\mathbf{dg}}_k^{fil}$ is endowed with the complete tensor product $\hat{\otimes}_k$. It induces an equivalence on the $\infty$-categories of commutative algebras

$$\mathrm{CAlg}(\epsilon - \mathbf{dg}_k^{gr}) \simeq \mathrm{CAlg}(\widehat{\mathbf{dg}}_k^{fil}).$$

We note here that the $\infty$-categories of commutative algebras of the previous corollary can be described in simple terms as localizations of model categories of strict commutative algebra objects, thanks to the rectification result of [Lur22a, Prop. 4.5.4.6]. In particular, $\mathrm{CAlg}(\epsilon - \mathbf{dg}_k^{gr})$ is canonically equivalent to $\epsilon - \mathrm{cdga}_k^{gr}$, the $\infty$-category of graded mixed cdga's described in the next section. In the same manner, $\mathrm{CAlg}(\widehat{\mathbf{dg}}_k^{fil})$ can be identified with the localization of the model category of complete filtered cdgas (i.e. commutative algebra objects in the model category $C(k)^{fil}$ of filtered complexes, whose underlying filtered complex is complete).

**REMARK 1.3.4.5.** It is easy to check that the (non Tate) realization functor $|-|$ of Definition 1.3.4.2 induces a functor

$$|-| : \mathrm{CAlg}(\epsilon - \mathbf{dg}_k^{gr}) \to \mathrm{cdga}_k.$$

## 1.4. Graded mixed cdga's and derived de Rham theory

In this section we show how graded mixed complexes provide a setting for de Rham theory of algebraic varieties and schemes, and more generally derived stacks. For this we assume further more that our base ring $k$ is a $\mathbb{Q}$-algebra. The extension to non-zero characteristics require substantial modification and will be treated separately in last chapter of this book (see chapter 7).
We remind the symmetric monoidal model category $\epsilon - C(k)^{gr}$ of graded mixed complexes, and its corresponding symmetric monoidal $\infty$-category $\epsilon - \mathbf{dg}_k^{gr}$. We will use in the sequel the category of commutative (unital and associative) monoids in $\epsilon - C(k)^{gr}$, for which we introduce the following notation

$$ \epsilon - CAlg(k)^{gr} := Comm(\epsilon - C(k)^{gr}). $$

Objects in $\epsilon - CAlg(k)^{gr}$ will be referred to as graded mixed cdga's. In a more explicit form, a graded mixed cdga $A$ consists of the data of a graded mixed complex $A$, together with

* (multiplication) morphisms of complexes

$$ (.) : A^{(i)} \otimes_k A^{(j)} \to A^{(i+j)}, $$

* (unit) a cocycle $1 \in A^{(0)}$ of cohomological degree 0.

The multiplication and the unit are required to satisfy the obvious associativity, unity and commutativity axioms. Moreover, the multiplication and the unit must be compatible with the mixed structure $\epsilon$ in the following sense:

$$ \epsilon(1) = 0 \qquad \epsilon(x.y) = \epsilon(x).y + (-1)^{|x|} x.\epsilon(y), $$

for all elements $x$ and $y$ in $A$ with $x$ of cohomological degree $|x| \in \mathbb{Z}$. We note in particular that the sign appearing the above formula does not depend on the weights of $x$ and $y$, but only on their cohomological degrees.

We can use the forgetful functor $\epsilon - CAlg(k)^{gr} \to \epsilon - C(k)^{gr}$, which forgets the multiplicative structure, in order to create a model category structure on $\epsilon - CAlg(k)^{gr}$ for which the fibrations and weak equivalences are defined in $\epsilon - C(k)^{gr}$. The existence of this model category structure uses in an essential manner that $k$ is assumed to be a $\mathbb{Q}$-algebra, and can be deduced for instance from the general statement [Lur22a, Prop. 4.5.4.6]. The homotopy theory of graded mixed cdga's is central in the definition and study of derived foliations, and we thus set the following important definition and notations.

**DEFINITION 1.4.0.1.** The model category of graded mixed cdga's is $\epsilon - CAlg(k)^{gr}$ defined above. The corresponding $\infty$-category will be denoted by $\epsilon - \mathbf{cdga}_k^{gr}$ and will be called the $\infty$-category of graded mixed cdga's over $k$.

The model category $\epsilon - CAlg(k)^{gr}$ comes equipped with two important Quillen adjunctions, which will be used often all along the text. As a start, we have the adjunction induced by the forgetful functor

$$ \mathrm{Sym}^{\epsilon} : \epsilon - C(k)^{gr} \rightleftharpoons \epsilon - CAlg(k)^{gr} : \mathrm{oub}. $$

Here the left adjoint sends a graded mixed complex $E$ to the usual symmetric algebra object

$$ \mathrm{Sym}^{\epsilon}(E) := \bigoplus_{n \ge 0} E^{\otimes n} / \Sigma_n, $$
where the sum, tensor products and quotients by symmetric groups are all understood in the category $\epsilon - C(k)^{gr}$ of graded mixed complexes. We have denoted this functor by $Sym^\epsilon$ to avoid confusions with other $Sym$ constructions that we will use later.

The second important Quillen adjunction is denoted by

$$DR : CAlg(k) \rightleftarrows \epsilon - CAlg(k)^{gr} : (-)^{(0)},$$

where $CAlg(k)$ denotes the model category of (non-graded and non-mixed) commutative dg-algebras. The right adjoint of this adjunction sends a graded mixed cdga $A$ to its weight 0 part $A^{(0)}$, which comes equipped with an induced commutative dg-algebra structure. The left adjoint is called the *de Rham functor* and will be studied in detail in the next two paragraphs.

The two mentioned above Quillen adjunctions of course induce natural adjunctions of $\infty$-categories (as usual by A.1)

$$Sym^\epsilon : \epsilon - \mathrm{dg}_k^{gr} \rightleftarrows \epsilon - \mathrm{cdga}_k^{gr} : \mathrm{oub} \qquad \mathrm{DR} : \mathrm{cdga}_k \rightleftarrows \epsilon - \mathrm{cdga}_k^{gr} : (-)^{(0)}.$$

**1.4.1. Derivations and the cotangent complex.** For a commutative dg-algebra $A \in CAlg(k)$, and a (left) $A$-dg-module $M$, we can form the trivial square zero extension of $A$ by $M$ in the usual manner. It is a new commutative dg-algebra, denoted by $A \oplus M$ or $A[M]$, whose underlying complex is the direct sum $A \oplus M$. The multiplication on $A \oplus M$ is the morphism

$$\mu : (A \oplus M) \otimes (A \oplus M) \simeq (A \otimes A) \oplus (A \otimes M) \oplus (M \otimes A) \oplus (M \otimes M) \longrightarrow A \oplus M,$$

defined as follows. On the factor $A \otimes A$ the morphism $\mu$ is the multiplication in $A$, followed by the natural inclusion $A \hookrightarrow A \oplus M$. On the factor $M \otimes M$ the morphism $\mu$ is the zero map. On the factor $A \otimes M$, $\mu$ is the given by the left $A$-dg-module structure on $M$, followed by the natural inclusion $M \hookrightarrow A \oplus M$. Finally, on the factor $M \otimes A$, $\mu$ is given by the right $A$-dg-module structure on $A$ followed by the inclusion $M \hookrightarrow A \oplus M$. We note here that the right $A$-dg-module structure on $M$ is defined using the symmetry constraints on complexes (see (1) on page 18), and is given explicitly for $a \in A^i$ and $m \in M^j$ by $m.a = (-1)^{i+j}a.m$. Unfolding the above description of the multiplication on $A \oplus M$, we can also write the following formula, for homogeneous elements $(a, m) \in A \oplus M$ and $(a', m') \in A \oplus M$

$$(a, m).(a', m') = (a.a', am' + (-1)^{|a'||m|}a'm).$$

With this definition, a derivation on $A$ with values in $M$ is a section of the natural projection $A \oplus M \to A$ inside the category of commutative dg-algebras (see [TV08, §1.2.1]). Such a section is of the form $(id_A, u) : A \to (A \oplus M)$, with $u : A \to M$ a morphism of complexes. The morphism $u$ must also satisfy the usual condition

$$u(a.b) = u(a).b + a.u(b) = a.u(b) + (-1)^{|a||b|}b.u(a)$$

for homogeneous elements $a$ and $b$ in $A$.

For any commutative dg-algebra $A$, there exists a universal derivation

$$d : A \to \Omega_{A/k}^1.$$
Here $\Omega_{A/k}^1$ is the dg-module of Kähler differentials on the commutative dg-algebra $A$ (relative to $k$), which can be explicitly presented as $I/I^2$ where $I \subset A \otimes A$ is the kernel of the multiplication map $A \otimes A \to A$. The universal property here states that for any left $A$-dg-module $M$ and any derivation $u : A \to M$, there exists a unique morphisme of $A$-dg-modules $f : \Omega_{A/k}^1 \to M$ such that $f \circ d = u$

<figure><img src="image_6.png" /></figure>

The construction $A \mapsto \Omega_{A/k}^1$ is functorial in $A$, and can be seen to be a left adjoint of a Quillen adjunction as follows. We form the category $CAlgMod(k)$, whose objects are pairs $(A, M)$, consisting of a commutative dg-algebra $A$ and a left $A$-dg-module $M$. The morphisms from $(A, M)$ to $(A', M')$ are pairs $(u, f)$, where $u : A \to A'$ is a morphism of commutative dg-algebras and $f : M \to M'$ is a morphism of $A$-dg-modules (where $M'$ is considered as a left $A$-dg-module via $u$). The category $CAlgMod(k)$ admits a model category structure for which a morphism $(u, f)$ as above is a weak equivalence (respectively a fibration) if $u$ and $f$ are quasi-isomorphisms of complexes (respectively an epimorphism of complexes). This model category is a particular example of a general construction of a relative model category structure, as explained in [HP15, Cor. 6.3.16].

The construction $(A, M) \mapsto A \oplus M$ defines a functor $CAlgMod(k) \to CAlg(k)$. This functor is a right Quillen functor and its left adjoint sends a commutative dg-algebra $A$ to the pair $(A, \Omega_{A/k}^1)$. We obtain this way an induced adjunction on the corresponding $\infty$-categories obtained by inverting quasi-isomorphisms

$$ \mathrm{cdga}_k \stackrel{\sim}{\hookrightarrow} \mathrm{cdgamod}_k. $$

The left adjoint sends a commutative dg-algebra $A$ to an object of the form $(A, \mathbb{L}_{A/k})$, where $\mathbb{L}_{A/k}$ is a left $A$-dg-module called the *cotangent complex of $A$*. An explicit model for the $A$-dg-module $\mathbb{L}_{A/k} \in \mathrm{dg}A$ is given by

$$ \mathbb{L}_{A/k} \simeq A' \otimes_A \Omega_{A'/k}^1, $$

where $A' \xrightarrow{\sim} A$ is a cofibrant model of $A$.

Being a left adjoint, the $\infty$-functor $A \mapsto (A, \mathbb{L}_{A/k})$ commutes with arbitrary colimits. If $A = \mathrm{Colim}_i A_i$ inside the $\infty$-category $\mathrm{cdga}_k$, then we have a canonical equivalence of $A$-dg-modules

$$ \mathbb{L}_{A/k} \simeq \mathrm{Colim}_i (A \otimes_{A_i} \mathbb{L}_{A_i/k}). $$

**1.4.2. The derived de Rham complex as a left adjoint.** We consider the functor $E \mapsto E^{(0)}$, from graded mixed complexes to complexes, defined by keeping the sole weight 0 piece. This functor possesses a natural symmetric lax monoidal structure and therefore naturally extends to a functor $A \mapsto A^{(0)}$ from the category of graded mixed cdga's to the
category of cdga's. This functor $Alg(\epsilon - C(k)^{gr}) \to Alg(C(k))$ is the right adjoint of a Quillen adjunction whose left adjoint is called the de Rham functor

$$DR(-/k) : Alg(C(k)) \to Alg(\epsilon - C(k)^{gr}).$$

Explicitly, the functor $DR$ sends a commutative dg-algebra $A$ to the graded mixed cdga whose weight $n$ piece is $Sym_A^n(\Omega_{A/k}^1[1])$. The multiplicative structure on $DR(A)$ is the standard multiplication induced on a symmetric algebra. The mixed structure is itself induced from the usual de Rham differential $dR : A \to \Omega_{A/k}^1$, and its natural multiplicative extension to the whole symmetric algebra $Sym_A(\Omega_{A/k}^1[1])$. In short we write $DR(A/k) = Sym_A(\Omega_{A/k}^1[1])$. We leave to the reader to check that this construction $A \mapsto DR(A/k)$ does indeed define a left adjoint to $(-)^0$, by observing that for any graded mixed cdga $B$ the morphism $\epsilon : B^{(0)} \to B(1)$ is derivation on $B^{(0)}$ with coefficients in the $B^{(0)}$-dg-module $B^{(1)}$, and using the universal property of $\Omega_{A/k}^1$.

The functor $A \mapsto DR(A/k)$ being left Quillen it induces an $\infty$-functor on the corresponding $\infty$-categories (see §A.1)

$$DR : cdga_k \to \epsilon - cdga_k^{gr}.$$

This $\infty$-functor is left adjoint to $A \mapsto A^{(0)}$, sending a graded mixed cdga $A$ to its piece of weight 0. The cotangent complexes being obtained as the left derived functor of $A \mapsto \Omega_{A/k}^1$ we see that on the level of $\infty$-categories we have canonical equivalences of graded cdga's

$$DR(A) \simeq Sym_A(\mathbb{L}_{A/k}[1])$$

where the symmetric algebra is understood in the symmetric monoidal $\infty$-category $dg_k^{gr}$ of graded complexes. In more concrete terms a model for $DR(A/k)$ is $DR(A'/k)$ where $A'$ is a cofibrant model of $A$.

**DEFINITION 1.4.2.1.** (1) The (derived) de Rham algebra of a commutative dg-algebra $A$ (over $k$) is the graded mixed commutative dg-algebra

$$DR(A) \simeq Sym_A(\mathbb{L}_{A/k}^1[1]) \in \epsilon - dg_k^{gr}$$

defined above.

(2) The (Hodge-completed and derived) de Rham complex of a commutative dg-algebra $A$ (over $k$) is the filtered complex

$$\hat{C}_{DR}(A/k) := |DR(A/k)|^t \in dg_k^{fil}.$$

By definition the derived de Rham complex $\hat{C}_{DR}(A/k)$ is a complete filtered complex, and its filtration is the Hodge filtration (hence the use of the expression *Hodge-completed* in the definition above). Its associated graded object is the graded cdga $Sym_A(\mathbb{L}_{A/k}[-1])$, where $\mathbb{L}_{A/k}$ sits in weight $-1$. It also comes equipped with a canonical commutative multiplicative structure, because the functor $| - |^t$ is endowed with a canonical symmetric lax monoidal structure
(see corollary 1.3.4.4). Therefore, $\hat{C}_{DR}(A/k)$ can be promoted to a filtered and complete commutative dg-algebra over $k$, but we will use this highly structured version rarely in the sequel.

In the particular case where $A$ is a smooth $k$-algebra, considered as a dg-algebra concentrated in degree 0, $\hat{C}_{DR}(A/k)$ turns out to be canonically equivalent to the usual algebraic de Rham complex of $A$ relative to $k$

$$ \hat{C}_{DR}(A/k) \simeq ( A \longrightarrow \Omega_{A/k}^1 \longrightarrow \Omega_{A/k}^2 \longrightarrow \dots \longrightarrow \Omega_{A/k}^i \longrightarrow \dots ) $$

Indeed, when $A$ is smooth the natural morphism $\mathbb{L}_{A/k} \to \Omega_{A/k}^1$ is a quasi-isomorphism. Therefore, for a cofibrant replacement $A' \dashrightarrow A$, the canonical projection of graded mixed cdga's

$$ \mathrm{Sym}_{A'}(\mathbb{L}_{A'/k}[1]) \to \mathrm{Sym}_A(\Omega_{A/k}^1[1]) $$

is a quasi-isomorphism. Passing to the Tate realizations, we get an equivalence of filtered and complete cdga's

$$ \hat{C}_{DR}(A/k) \simeq C_{DR}^*(A/k) $$

where the right hand side is the usual, *underived* de Rham complex of $A$ over $k$. This is a bounded complex concentrated in degrees $[0, d]$, where $d$ is the relative dimension of $A$ over $k$. This complex is filtered by the usual Hodge filtration, and is complete because the filtration is finite. As, by convention, all our filtrations are *increasing*, this Hodge filtration is indexed in a non-standard way by non-positive integers $p \le 0$ as follows

$$ F^p \hat{C}_{DR}(A/k) = ( \Omega_{A/k}^{-p} \longrightarrow \Omega_{A/k}^{-p+1} \longrightarrow \dots ) \subset \hat{C}_{DR}(A/k). $$

The associated graded $Gr^*(\hat{C}_{DR}(A/k))$ is thus the usual Hodge cohomology $\oplus \Omega_{A/k}^i[-i]$, with $\Omega_{A/k}^i$ of weight $-i$.

To finish this part we mention below two well known important properties of derived de Rham cohomology.

**PROPOSITON 1.4.2.2.** Let $X = \mathbf{Spec} A$ be a derived affine scheme of essentially of finite presentation over $\mathbb{C}$ and $X(\mathbb{C})$ its space of $\mathbb{C}$-points endowed with its transcendent topology. Then there exists a natural equivalence of cdga's

$$ \hat{C}_{DR}^*(A/\mathbb{C}) \simeq C^*(X(\mathbb{C}), \mathbb{C}), $$

where the right hand side denotes sheaf cohomology with coefficients in the constant sheaf $\mathbb{C}$.

*Proof.* This results is proven in [Bha12]. We can provide a direct proof as follows. As a first statement, the right hand side is well known to be functorially equivalent to algebraic de Rham cohomology of $X$. Algebraic de Rham cohomology can be computed by choosing a closed embedding $X \to Y = \mathrm{Spec} B$ for $Y$ a smooth affine scheme over $\mathbb{C}$, and then considering the
formal de Rham complex $C_{DR}^*(\hat{Y}_X)$ of $Y$ along $X$. The formal de Rham complex is obtained by definition from $DR(Y)$ by completion along $X$

$$C_{DR}^*(\hat{Y}_X) = \hat{B} \longrightarrow \hat{\Omega}_B^1 \longrightarrow \dots \longrightarrow \hat{\Omega}_B^r \longrightarrow \dots$$

where $\hat{B} = \lim_n B/I^n$ with $I \subset B$ the ideal of definition of $X$, and $\hat{\Omega}_B^r := \hat{B} \otimes_B \Omega_B^r$.

From the results reminded in A.6 the de Rham complex $C_{DR}^*(\hat{Y}_X)$ is also given by

$$C_{DR}^*(\hat{Y}_X) \simeq \lim_n \hat{C}_{DR}(Y_n)$$

where $Y_n = B/I^n$. But the Hodge completed derived de Rham cohomology is invariant by infinitesimal thickening (this can be see along the same lines as A.8.0.3, see A.8.0.4), so we have

$$C_{DR}^*(\hat{Y}_X) \simeq \hat{C}_{DR}(t_0(X)) \simeq \hat{C}_{DR}(X)$$

via the pull-backs along the closed immersions $t_0(X) \to X \to Y$. $\square$

For the next Proposition, recall that for any connective cdga $A$ over $k$, the structural morphism $k \to \mathbf{DR}(A/k)$ is a morphism of graded mixed cdga (where $k$ is endowed with the zero mixed structure). By passing to realizations we get this way a natural morphism of cdga

$$k \to |\mathbf{DR}^*(A/k)| \simeq \hat{C}_{DR}^*(A/k).$$

Suppose now that $A$ is a discrete commutative $\mathbb{Q}$-algebra and $I \subset A$ an ideal. We can apply the previous morphism to the augmentation $A \to A/I$, and get this way a natural morphism of cdga's

$$A \to \hat{C}_{DR}^*((A/I)/A).$$

Moreover, as this is obtained by realization from a morphism of graded mixed cdgas, this morphism is compatible with the augmentations to $A/I$. In particular, this morphism is compatible with the filtrations: the $I$-adic filtration on $A$ and the natural filtration on $\hat{C}_{DR}^*((A/I)/A)$. As $\hat{C}_{DR}^*((A/I)/A)$ is already known to be complete it induces a morphism of completed filtered cdga's

$$\hat{A} \to \hat{C}_{DR}^*((A/I)/A)$$

where $\hat{A}$ is the $I$-adic completion of $A$.

**PROPOSITON 1.4.2.3.** Let $A$ be a commutative $k$-algebra of finite type and $I \subset A$ an ideal. Then, the canonical morphism

$$\hat{A} \to \hat{C}_{DR}^*((A/I)/A)$$

is an equivalence. In particular

$$H^0(\hat{C}_{DR}^*((A/I)/A)) \simeq \hat{A} \qquad H^i(\hat{C}_{DR}^*((A/I)/A)) \simeq 0 \quad \forall i \neq 0.$$

*Proof.* See [Bha12, Thm. 4.10]. $\square$
1.5. The geometry of graded mixed objects

We recall here some results from [MRT22] and [Alf25] which provide a geometric interpretation of graded mixed complexes. These will be used for some specific arguments and are sometimes very useful (see for instance §2.5).

We consider the multiplicative group scheme $\mathbb{G}_m$. It acts by standard weight $-1$ on the additive group scheme $\mathbb{G}_a$, and thus on the group stack $B\mathbb{G}_a$. Here the weight of the $\mathbb{G}_m$ action is understood geometrically, the induced action of $\mathbb{G}_m$ on functions $\mathcal{O}(\mathbb{G}_a) = k[T]$ is such that $T$ has weight 1.

**DEFINITION 1.5.0.1.** We define the group stack $\mathcal{H}_0$ as the semi-direct product

$$ \mathcal{H}_0 := B\mathbb{G}_a \times \mathbb{G}_m $$

We can now consider complexes of $k$-modules endowed with an action of $\mathcal{H}_0$. They form an $\infty$-category which is formally defined as $\mathbf{QCoh}(B\mathcal{H}_0)$, the $\infty$-category of quasi-coherent complexes over the classifying stack $B\mathcal{H}_0$ of $\mathcal{H}_0$. The canonical projection $\mathcal{H}_0 \to \mathbb{G}_m$ possesses a group section $s: \mathbb{G}_m \to \mathcal{H}_0$, which provides a natural morphism $s: B\mathbb{G}_m \to B\mathcal{H}_0$. The fiber of the projection $\mathcal{H}_0 \to \mathbb{G}_m$ identifies with $B\mathbb{G}_a$, which is also naturally identified with the fiber of the section $s$. Therefore, $s$ exhibits $B\mathbb{G}_a$ as a $\mathcal{H}_0$-equivariant stack.

By writing $B\mathcal{H}_0$ as the standard colimit $B\mathcal{H}_0 = \mathrm{colim}_{[n] \in \Delta^{op}} \mathcal{H}_0^n$, we can write

$$ \mathbf{QCoh}(B\mathcal{H}_0) \simeq \lim_{[n] \in \Delta} \mathbf{QCoh}(\mathcal{H}_0^n). $$

Now, for each $n$, the stack $\mathcal{H}_0^n$ is an affine stack with cdga of functions $\mathcal{O}(\mathcal{H}_0^n) \simeq \mathcal{O}(\mathcal{H}_0)^{\otimes n} \simeq (k[t, t^{-1}] \oplus k[t, t^{-1}][{-1}](1))^{\otimes n}$. The category of graded mixed complexes $\epsilon - C(k)^{gr}$, before localization by quasi-isomorphisms, can be understood as the category of $(k[t, t^{-1}] \oplus k[t, t^{-1}][{-1}](1))$-dg-comodules, where $(k[t, t^{-1}] \oplus k[t, t^{-1}][{-1}](1))$ is the commutative and dg-Hopf algebra obtained by semi-direct product of the multiplicative Hopf algebra $k[t, t^{-1}]$ with $k \oplus k[{-1}](1) = \mathcal{O}(B\mathbb{G}_a)$. In other words, we can write down an equivalence of genuine categories (where the limit is computed in the $2$-category of categories, ad is taken before localization by quasi-isomorphisms)

$$ \epsilon - C(k)^{gr} \simeq \lim_{[n] \in \Delta} (k[t, t^{-1}] \oplus k[t, t^{-1}][{-1}](1))^{\otimes n} - dg - mod. $$

When we localize along quasi-isomorphisms, we get a functor $\infty$-functor

$$ \epsilon - \mathrm{dg}_{k}^{gr} \longrightarrow \mathbf{QCoh}(B\mathcal{H}_{0}). $$

By observation, this $\infty$-functor is also endowed with a canonical symmetric monoidal structure. In general localizations does not preserve limits, and tere are no obvious reasons why the previous $\infty$-functor is an equivalence. However the following result is proven in [MRT22, Prop. 4.2.3].
**PROPOSITON 1.5.0.2.** The $\infty$-functor constructed above induces a symmetric monoidal equivalence

$$ \epsilon - \mathrm{dg}_k^{gr} \simeq \mathrm{QCoh}(BH_0). $$

A simple corollary of the above Proposition, combined with the rectification results of [Lur22a, Prop. 4.5.4.6], is the following interpretation of graded mixed cdga's.

**COROLLARY 1.5.0.3.** There exists an natural equivalence of $\infty$-categories

$$ \epsilon - \mathrm{cdga}_k^{gr} \simeq \mathrm{CAlg}(\mathrm{QCoh}(BH_0)), $$

where the right hand side is the $\infty$-category of commutative algebras in $\mathrm{QCoh}(BH_0)$ in the sense of [Lur22a].

**Proof of the Corollary.** The equivalence of Proposition 1.5.0.2 being symmetric monoidal, we get an equivalence on the corresponding $\infty$-categories of commutative algebra objects $\mathrm{CAlg}(\epsilon - \mathrm{dg}_k^{gr}) \simeq \mathrm{CAlg}(\mathrm{QCoh}(BH_0))$. To finish the proof we invoke the rectification result of [Lur22a, Prop. 4.5.4.6], which can be applied here as $k$ is assumed of characteristic zero (and thus $\epsilon - C(k)^{gr}$ is a well powered symmetric monoidal model category). We thus get that the natural $\infty$-functor $L(\mathrm{CAlg}(\epsilon - C(k)^{gr})) = \epsilon - \mathrm{cdga}_k^{gr} \to \mathrm{CAlg}(\epsilon - \mathrm{cdga}_k^{gr})$ is an equivalence. Combining these two equivalences of $\infty$-categories finishes the proof of the Corollary. $\square$

**REMARK 1.5.0.4.** Using Proposition 1.5.0.2 and the group section $s: G_m \to \mathcal{H}$ introduced at the beginning of this subsection, we may realize geometrically the forgetful functor $\epsilon - \mathrm{dg}_k^{gr} \to \mathrm{dg}_k^{gr}$ as the pullback on quasi-coherent complexes along the map $Bs: BG_m \to BH$.

The previous description allows for the following easy generalization where $\mathrm{Spec} k$ is replaced by an arbitrary derived (pre)stack $X$

$$ (6) \qquad \epsilon - \mathrm{dg}_X^{gr} := \mathrm{QCoh}(BH_0 \times X) \qquad \epsilon - \mathrm{cdga}_X^{gr} \simeq \mathrm{CAlg}(\mathrm{QCoh}(BH_0 \times X)) $$

(where the indicated products are fibered products over $\mathrm{Spec} k$). This equivalence for generally derived stacks follows directly from corollary 1.5.0.3 by descent.

## 1.6. The classifying spaces of graded mixed structures

In this section we study the *space of graded mixed structures* on a *fixed graded cdga*. We first provide a precisely definition of such a space.

We consider the forgetful $\infty$-functor

$$ \mathrm{oub}_\epsilon : \epsilon - \mathrm{cdga}_k^{gr} \to \mathrm{cdga}_k^{gr} $$

obtained by forgetting the mixed structure. As $\mathrm{oub}_\epsilon$ is conservative, the fiber $\infty$-category $\mathrm{oub}_\epsilon^{-1}(B)$ taken at any point $B \in \mathrm{cdga}_k^{gr}$ is an $\infty$-groupoid, whose geometric realization will be denoted by $|\mathrm{oub}_\epsilon^{-1}(B)|$.
DEFINITION 1.6.0.1. For a given graded cdga $B \in \mathbf{cdga}_k^{gr}$, the classifying space of compatible graded mixed structures on $B$ is the simplicial set

$$ \mathcal{M}_{\epsilon}(B) := |\mathbf{oub}_{\epsilon}^{-1}(B)| \in \mathbb{T}. $$

Such spaces of algebraic structures have been heavily studied in the literature. We start by recalling the well known fact that they can be computed, in our specific case, as mapping spaces between graded dg-Lie algebras. For this, let $B \in \mathbf{cdga}_k^{gr}$ be a fixed graded cdga over $k$. We can form $Der^{gr}(B)$ the graded dg-Lie algebra of graded derivations from $B$ with values in $B$. It can be defined using a strict cofibrant model $\tilde{B}$ of $B$, and considering strict graded derivations of $\tilde{B}$ into $\tilde{B}$ as a model for $Der^{gr}(B)$. More explicitly, the piece of weight $n$ of $Der^{gr}(B)$ is the complex of derivations $\tilde{B} \to \tilde{B}(n)$, and the Lie bracket is the usual commutator of derivations. The space $\mathcal{M}_{\epsilon}(B)$ can then be understood as a mapping space in the $\infty$-category $\mathbf{dgLie}_k^{gr}$ of graded deg-Lie algebras (i.e. Lie algebra object in $\mathbf{dg Lie}_k^{gr}$).

PROPOSITON 1.6.0.2. There is a canonical equivalence $\mathrm{Map}_{\mathbf{dgLie}^{gr}}(k(1)[1], Der^{gr}(B)) \to \mathcal{M}_{\epsilon}(B)$.

*Proof.* The proposition is a particular case of a more general result. Let $L$ be any graded dg-Lie algebra. We have a symmetric monoidal model category $L - \mathrm{Mod}^{gr}$ of graded $L$-dg-modules (for which equivalences and fibrations are defined on the underlying graded complexes), and a corresponding symmetric monoidal $\infty$-category $L - \mathbf{dg Lie}_k^{gr}$. Commutative algebras in $L - \mathbf{dg Lie}_k^{gr}$ form a new $\infty$-category $L - \mathbf{cdga}_k^{gr}$, whose objects can be called graded $L$-cdga's. It comes equipped with a forgetful $\infty$-functor

$$ \mathbf{oub}_L : L - \mathbf{cdga}_k^{gr} \to \mathbf{cdga}_k^{gr}. $$

We claim that, for any fixed $B \in \mathbf{cdga}_k^{gr}$, the two spaces

$$ \mathbf{oub}_L^{-1}(B) \qquad \mathrm{Map}_{\mathbf{dgLie}_k^{gr}}(L, Der^{gr}(B)) $$

are functorially equivalent. The proposition would follow from the special case where $L = k(1)[1]$, as $k(1)[1] - \mathbf{cdga}_k^{gr}$ is nothing else that the $\infty$-category of graded mixed cdga's.

We fix $B \in \mathbf{cdga}_k^{gr}$, and chose cofibrant model for it. We thus have two $\infty$-functors

$$ L \mapsto \mathbf{oub}_L^{-1}(B) \qquad L \mapsto \mathrm{Map}_{\mathbf{dgLie}_k^{gr}}(L, Der^{gr}(B)). $$

These $\infty$-functor can be represented as strict $\infty$-functors from the category of cofibrant graded dg-Lie algebras to the category of spaces. The first one sends $L$ to the nerve of the category $C(L, B)$ whose objects are pairs $(B', u)$, with $B' \in L - \mathbf{cdga}_k^{gr}$ and $u : B \to \mathbf{oub}_L(B')$ is a quasi-isomorphism. The second functor sends $L$ to the simplicial set $\mathrm{Hom}(c^*(L), Der^{gr}(B))$, where $c^* : \mathbf{dgLie}_k^{gr} \to (\mathbf{dgLie}_k^{gr})^{\Delta}$ is a left framing in the sense of [Hov99, Def. 5.2.7].

We can construct a natural transformation between these two $\infty$-functors. Indeed, there is a canonical map from the set $\mathrm{Hom}(L, Der^{gr}(B))$ to the nerve of $C(L, B)$, sending a morphism $\rho : L \to Der^{gr}(B)$ to the pair $(B_{\rho}, \mathrm{id})$, where $B_{\rho}$ is the object in $L - \mathbf{cdga}_k^{gr}$ defined by the
morphism $\rho$ (and whose underlying object equals $B$). This map is functorial in $L$, and thus it defines a morphism of simplicial sets

$$ \mathrm{Hom}(c^*(L), \mathrm{Der}^{gr}(B)) \to |[n] \to N(C(c^n(L), B))| $$

Now, as quasi-isomorphisms between graded dg-Lie induces Quillen equivalences between the corresponding $\infty$-categories of graded cdga's, we have that all the morphism in the simplicial diagram $[n] \mapsto N(C(c^n(L), B))$ are weak equivalences, and thus the natural morphism

$$ N(C(L, B)) \to |[n] \to N(C(c^n(L), B))| $$

is also a weak equivalence. We have thus defined a well defined natural transformation of $\infty$-functors

$$ \psi_L : \mathrm{Hom}(c^*(L), \mathrm{Der}^{gr}(B)) \to |[n] \to N(C(c^n(L), B))| \simeq N(C(L, B)). $$

It remains to show that $\psi_L$ is an equivalence for all $L$. For this we use the following steps.

(1) The induced map

$$ \psi_L : \pi_0(\mathrm{Hom}(c^*(L), \mathrm{Der}^{gr}(B))) \to \pi_0(\mathrm{N}(C(L, B))) $$

is bijective for all $L$.

(2) Both $\infty$-functors $L \mapsto \mathrm{Hom}(c^*(L), \mathrm{Der}^{gr}(B))$ and $L \mapsto N(C(L, B))$ sends colimis (in the argument $L$) to limits.

As a first observation, sets (1) and (2) above finishes the proof of the proposition. Indeed, we can apply (1) to graded dg-Lie algebras of the form $K \otimes L$, for $K$ an arbitrary simplicial set, which by (2) will imply that the morphism $\psi_L$ induces a bijection

$$ [K, \mathrm{Hom}(c^*(L), \mathrm{Der}^{gr}(B))] \to [K, \mathrm{N}(C(L, B))]. $$

As this is true for all $K$, Yoneda implies that $\psi_L$ is an isomorphism in $\mathrm{Ho(Top)}$ and thus is an equivalence.

The point (1) is a direct exercise in model category theory, and is left to the reader. For point (2), we use that if $L = \mathrm{colim}_i L_i$ is a colimit in the $\infty$-category $\mathrm{dgLie}_k^{gr}$, then the natural $\infty$-functor

$$ L - \mathrm{cdga}_k^{gr} \to \mathrm{lim}_i L_i - \mathrm{cdga}_k^{gr} $$

is an equivalence of $\infty$-categories. Indeed, this can be seen easily by first considering the case of graded dg-modules

$$ L - \mathrm{dg}_k^{gr} \to \mathrm{lim}_i L_i - \mathrm{dg}_k^{gr}. $$

The fact that this $\infty$-functor is an equivalence follows directly from the equivalence between dg-modules over $L$ and dg-modules over the universal enveloping algebra $U(L)$, and the fact that $L \mapsto U(L)$ preserves colimits. Indeed, for a colimit in graded dg-algebras $U = \mathrm{colim}_i U_i$, the $\infty$-category of graded $U$-dg-modules is equivalent to the limit of the $\infty$-categories of graded $U_i$-dg-modules as shown in the next lemma.
LEMMA 1.6.0.3. Let $U = \mathrm{colim}_i U_i$ be a colimit in the $\infty$-category $\mathbf{dga}_k^{gr}$ graded (associative and unital) dg-algebras over $k$. The canonical $\infty$-functor on $\infty$-categories of graded dg-modules

$$U - \mathbf{dg}_k^{gr} \to \mathrm{lim}_i U_i - \mathbf{dg}_k^{gr}$$

is an equivalence.

Proof of the lemma. We have a fibration sequence

$$\mathbf{Map}_{\mathbf{dga}_k^{gr}}(U, T(x, x)) \to \mathbf{Map}_{\mathbf{grdgcat}_k}(BU, T) \to \mathbf{Map}_{\mathbf{grdgcat}_k}(\mathbb{K}, T),$$

where here $T$ is a graded dg-category, the fiber is taken over the unique morphism $\mathbb{K} \to T$ point an object $x \in T$, and $BU$ is the graded dg-category with a unique object determined by $U$. This is the graded version of the fibration sequence appearing in the proof of [TV07, Lem. 2.11], and can proven the exact same manner using the homotopy theory of graded dg-categories. This fibration sequence implies the lemma as $\mathbf{Map}_{(\mathbf{dga}_k^{gr})(-, T(x, x))}$ transforms colimits in limits. $\square$

The lemma finishes the proof of the point (2) above and thus of the proposition. $\square$

Proposition 1.6.0.2 is useful as the mapping space $\mathrm{Map}_{\mathbf{dgLie}^{gr}}(k(1)[1], \mathrm{Der}^{gr}(B))$ can be described using an explicit cofibrant model for the graded dg-Lie algebra $k(1)[1]$, introduced in [CPT+17]. We let $\mathcal{L}$ be the dg-Lie algebra freely generated by elements $p_i$, for $i > 0$, such that $p_i$ sits in cohomological degree $1 - 2i$ and in weight $i$, with the relations

$$d(p_n) + \sum_{a+b=n} [p_q, p_b] = 0.$$

We consider the natural projection $\mathcal{L} \to k(1)[1]$ sending $p_1$ to $\epsilon$ and $p_i$ to zero for $i > 1$. As explained in [PTVV13], this projection is a trivial fibration. As $\mathcal{L}$ is clearly cofibrant (as being a cell object), this shows that $\mathcal{L}$ is a cofibrant model for $k(1)[1]$. Therefore, we have a diagram of spaces

$$\mathrm{Hom}_{\mathbf{dgLie}^{gr}}(\mathcal{L}, \mathrm{Der}^{gr}(B)) \longrightarrow \mathrm{Map}_{\mathbf{dgLie}^{gr}}(\mathcal{L}, \mathrm{Der}^{gr}(B)) \stackrel{\sim}{\longrightarrow} \mathrm{Map}_{\mathbf{dgLie}^{gr}}(k(1)[1], \mathrm{Der}^{gr}(B))$$

where $\mathrm{Hom}_{\mathbf{dgLie}^{gr}}(\mathcal{L}, \mathrm{Der}^{gr}(B))$ is the set of morphisms of graded dg-Lie algebras (which depends on the choice of a cofibrant model for $B$). Applying $\pi_0$ provides a natural surjective map $\mathrm{Hom}_{\mathbf{dgLie}^{gr}}(\mathcal{L}, \mathrm{Der}^{gr}(B)) \to \pi_0(\mathrm{Map}_{\mathbf{dgLie}^{gr}}(k(1)[1], \mathrm{Der}^{gr}(B)))$. Combined with the Proposition 1.6.0.2 we obtain the following corollary.

COROLLARY 1.6.0.4. Let $B$ be a fixed cofibrant graded cdga. Then, there exists a natural surjective map

$$\mathrm{Hom}_{\mathbf{dgLie}^{gr}}(\mathcal{L}, \mathrm{Der}^{gr}(B)) \to \pi_0(M_{\epsilon}(B)).$$

The previous results can be made a bit more explicit. Indeed, by the definition of $\mathcal{L}$ we have that an element in $\mathrm{Hom}_{\mathbf{dgLie}^{gr}}(\mathcal{L}, \mathrm{Der}^{gr}(B))$ is given by a sequence of $k$-linear graded
derivations

$$d_i : B \to B(i)[1-2i]$$

satisfying the system of equations

$$d(d_n) + \sum_{a+b=n} [d_q, d_b] = 0.$$

Suppose moreover that $B$ is explicitly given by $Sym_A(E[1])$, where $A$ is a cofibrant connective cdga, and $E$ is a cofibrant perfect $A$-dg-module. Then, each $d_i$ consists itself as a pair

$$\alpha_i : E \to \wedge_A^{i+1} E[1-i] \quad \beta_i : A \to \wedge_A^i E[1-i],$$

such that $\beta_i$ is a $k$-linear derivation, and $\alpha_i$ is a morphism which is $A$-linear with respect to $\beta_i$

$$\alpha_i(a.m) = a\alpha_i(m) + (-1)^i \beta_i(a) \wedge m$$

(where ? is the adequate sign). This description provides an explicit way to write down elements in the set $Hom_{dgLie}^{gr}(\mathcal{L}, Der^{gr}(B))$, and thus also an explicit manner to write down all the elements in $\pi_0(M_c(B))$ by Corollary 1.6.0.4. We also refer to [Mon22] for another closely related, but slightly different, way to describe these classifying spaces of graded mixed structures.
# CHAPTER 2

## Derived foliations

In this chapter we give the basic definitions and study the general properties of derived foliations. We provide examples, and a comparison with the theory of dg-Lie algebroids. One of the main result, in the third section, states that derived foliations are always formally integrable, which is a major difference with the situation of underived possibly singular foliations. We provide two versions of this result, a general version for derived foliations on general derived Artin stacks, and a more specific version for a class of derived foliations called quasi-smooth and rigid on smooth varieties. This second version is useful to study existence of derived enhancement of underived singular foliations, and to relate them to the classical notion of Godbillon-Vey sequences.

As before, unless other is specified, all structures (algebras, modules, schemes ...) are over our base commutative ring $k$. We assume that $k$ is furthermore a $\mathbb{Q}$-algebra. The notion of derived foliations in arbitrary characteristics situations will be treated in a separate later chapter of this book, as it requires some extra care and more evolved notions (see §7).

### 2.1. First definitions and examples

We are now ready to define the notion of derived foliations. We will first consider the situation for an affine derived scheme, and then proceed by gluing in order to define derived foliations on more global objects such as derived schemes and derived Artin stacks.

#### 2.1.1. Definition in the affine setting.
Let $X = \text{Spec } A$ be a derived affine scheme over $k$. Recall that this implicitly means that $A$ is a connective cdga over $k$, and thus it is cohomologically concentrated in non-positive degrees (see appendix §A.2). We will allow ourselves to implicitly identify the $\infty$-categories of $A$-dg-modules and of quasi-coherent complexes of $\mathcal{O}_X$-modules. We will also use the derived de Rham algebra $\text{DR}(A/k)$, that will also be denoted by $\text{DR}(X/k)$.

A derived foliation $\mathcal{F}$ on $X$ will be given, by definition, a graded mixed cdga $\text{DR}(\mathcal{F})$, together with a morphism $u: \text{DR}(A/k) \to \text{DR}(\mathcal{F})$ of graded mixed cdga's over $k$, satisfying some extra conditions. In order to phrase these conditions, note that the morphism $u$ induces a morphism of graded cdga's

$$A \to \text{DR}(A/k) \to \text{DR}(\mathcal{F}),$$
where the first of this morphism is the inclusion of the weight 0 part. This endows $\mathbf{DR}(\mathcal{F})$ with a canonical structure of a graded $A$-linear cdga structure. In particular, the inclusion of the weight 1 part $\mathbf{DR}(\mathcal{F})^{(1)} \subset \mathbf{DR}(\mathcal{F})$ induces a canonical morphism of graded $A$-linear cdga's

$$ \phi_u : \mathrm{Sym}_A(\mathbf{DR}(\mathcal{F})^{(1)}) \to \mathbf{DR}(\mathcal{F}). $$

Note that, the morphism $\phi_u$ above, of course, depends on the original morphism $u$.

**DEFINITION 2.1.1.1.** Let $X = \mathbf{Spec} A$ be a derived affine scheme over $k$. A derived foliation $\mathcal{F}$ over $X$ relative to $k$ is a pair $(\mathbf{DR}(\mathcal{F}), u)$, which consists of a graded mixed cdga $\mathbf{DR}(\mathcal{F})$, and a morphism of graded mixed cdga's

$$ u : \mathbf{DR}(A/k) \to \mathbf{DR}(\mathcal{F}), $$

such that the induced morphism

$$ \phi_u : \mathrm{Sym}_A(\mathbf{DR}(\mathcal{F})^{(1)}) \to \mathbf{DR}(\mathcal{F}) $$

is a quasi-isomorphism.

As a first comment, it is important to note that the condition that $\phi_u$ being a quasi-isomorphism includes in particular the condition that the morphism $u$ induces a quasi-isomorphism in weight 0 $A \simeq \mathbf{DR}(\mathcal{F})^{(0)}$. As a second observation, the universal property of the derived de Rham algebra $\mathbf{DR}(A/k)$ (see Definition 1.4.2.1), the morphism $u$ is completely characterized by the morphism of graded cdga's (without any mixed structure)

$$ A \to \mathbf{DR}(A/k) \to \mathbf{DR}(\mathcal{F}). $$

Therefore, the definition of derived foliations above could also be phrased as follows: a derived foliation $\mathcal{F}$ on $X$ over $k$ consists of a graded mixed cdga $\mathbf{DR}(\mathcal{F})$, together with a quasi-isomorphism of cdga's $A \to \mathbf{DR}(\mathcal{F})^{(0)}$, and such that the canonical morphism

$$ \mathrm{Sym}_A(\mathbf{DR}(\mathcal{F})^{(1)}) \to \mathbf{DR}(\mathcal{F}) $$

is a quasi-isomorphism.

The idea behind Definition 2.1.1.1 is that $\mathbf{DR}(\mathcal{F})$ is the de Rham algebra of forms along the leaves of $\mathcal{F}$. Therefore, this $A$-linear dg-module $\mathbf{DR}(\mathcal{F})^{(1)}$ should be understood as the (shifted by $-1$) cotangent sheaf, or rather cotangent complex, classifying differential forms along the leaves of $\mathcal{F}$. This complex is one of the fundamental data contained in a derived foliation, and we will therefore give a specific name, and introduce a specific notation for it.

**DEFINITION 2.1.1.2.** Let $\mathcal{F}$ be a derived foliation on a derived affine scheme $X = \mathbf{Spec} A$. The cotangent complex of $\mathcal{F}$ relative to $k$ is defined to be

$$ \mathbb{L}_{\mathcal{F}/k} := \mathbf{DR}(\mathcal{F})^{(1)}[-1]. $$

When the ground ring $k$ is unambiguous it is simple denoted by $\mathbb{L}_{\mathcal{F}}$.
An important remark is that the cotangent complex $\mathbb{L}_{\mathcal{F}/k}$ of a derived foliation $\mathcal{F}$ always comes equipped with a canonical morphism of $A$-dg-modules

$$ \mathbb{L}_{A/k} \to \mathbb{L}_{\mathcal{F}/k}, $$

which is induced by the (shifted) weight 1 part of the morphism $u : \mathbf{DR}(A/k) \to \mathbf{DR}(\mathcal{F})$. This is often referred to as the *anchor map*.

Using the notation and the terminology above, a derived foliation $\mathcal{F}$ on $X$, can be thought as a graded mixed structure $\epsilon$ on $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}/k}[1])$. This graded mixed structure is itself the de Rham differential on differential forms along $\mathcal{F}$. However, one important point in Definition 2.1.1.1 is that $\epsilon$ does not strictly exists on $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}/k}[1])$, but rather on $\mathbf{DR}(\mathcal{F})$ which is only quasi-isomorphic $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}/k}[1])$. Trying to transport the mixed structure along this quasi-isomorphism provides a derivation $\epsilon$ on $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}/k}[1])$, but $\epsilon^2$ will now only be homotopic to zero (as opposed to strictly equal to zero). We can use the quasi-isomorphism to show that there exists a canonical such homotopy, which itself should satisfy some higher coherences and so on and so forth. Despite its naive form, the reader must be warned that Definition 2.1.1.1 hides in fact an infinite number of rather evolved homotopy coherences, very similar to the one being found in shifted symplectic and Poisson structures of [CPT+17]. We will come back to this point later on in this book and will make these coherence explicit. The reader can already notice that by construction $\epsilon$ is a point in the moduli of compatible graded mixed structures on $\mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}/k}[1])$ in the sense of Definition 1.6.0.1, and thus can be described by the coherence of homotopies given by corollary 1.6.0.4.

Keeping this in mind, a derived foliation $\mathcal{F}$ has a cotangent complex $\mathbb{L}_{\mathcal{F}/k}$, which comes equipped with de Rham differentials

$$ A \to \mathbb{L}_{\mathcal{F}/k} \to \wedge_A^2 \mathbb{L}_{\mathcal{F}/k} \to \dots $$

These differential satisfy all the required formula (such as $d^2 = 0$, multiplicativity ect), but only up to natural homotopy coherences. As far as we know, it is in general not possible to make these identities to hold strictly on the noze.

Before showing several basic examples of derived foliations, we introduce some terminology. Derived foliations will be characterized according to the *size* of their cotangent complexes.

**DEFINITION 2.1.1.3.** Let $\mathcal{F}$ be a derived foliation on a derived affine scheme $X = \mathrm{Spec} A$.

(1) The derived foliation $\mathcal{F}$ is perfect (resp. almost perfect) if $\mathbb{L}_{\mathcal{F}/k}$ is a perfect (resp. almost perfect) $A$-dg-module.

(2) The derived foliation $\mathcal{F}$ is $n$-connective, for some integer $n \in \mathbb{Z}$, if $H^i(\mathbb{L}_{\mathcal{F}/k}) = 0$ for all $i > n$ (i.e. if $\mathbb{L}_{\mathcal{F}/k}[n]$ is connective).

(3) The derived foliation $\mathcal{F}$ is smooth if it is perfect and if $\mathbb{L}_{\mathcal{F}/k}$ is of Tor amplitude contained in $[0, \infty[$.

(4) The derived foliation $\mathcal{F}$ is quasi-smooth if it is perfect and if $\mathbb{L}_{\mathcal{F}/k}$ is of Tor amplitude contained in $[-1, \infty[$.
**2.1.2. Basic examples of derived foliations.** We gather in this subsection the very first examples of derived foliations. More evolved examples will be given later all along this book.

**EXAMPLE 2.1.2.1. Classical foliations on smooth affine schemes.** We start by the most basic examples of how usual foliations define derived foliations in our sense. For this, we let $X = \mathbf{Spec} A$ be a smooth affine scheme over $k$. Let $V \subset T_X$ be a subbundle of the tangent bundle of $X$, which is stable by the Lie bracket of $T_X$. We consider the dual bundle $V^\vee$ and $\mathrm{Sym}_A(V^\vee[1])$. The inclusion $V \subset T_X$ defines a quotient map $\Omega_{X/k}^1 \to V^\vee$, and thus a morphism of graded cdga's (with zero cohomological differential)

$$ \mathbf{DR}(A/k) = \mathrm{Sym}_A(\Omega_{A/k}^1[1]) \to \mathrm{Sym}_A(V^\vee[1]). $$

As $V$ is stable by the Lie bracket of vector fields, the de Rham differential $dR : A \to \Omega_{A/k}^1$ descends to a de Rham differential $dR : A \to V^\vee$, which can be extended multiplicatively to a graded mixed structure on the graded mixed cdga $\mathrm{Sym}_A(V^\vee[1])$. We thus have defined a morphism of graded mixed cdga's

$$ \mathbf{DR}(A/k) \to \mathrm{Sym}_A(V^\vee[1]). $$

This morphism defines a derived foliation $\mathcal{F}$ on $X$ relative to $k$ according to the Definition 2.1.1.1, such that $\mathbf{DR}(\mathcal{F}) = \mathrm{Sym}_A(V^\vee[1])$ endowed with its induced de Rham differential. This derived foliation has $V^\vee$ as its cotangent complex, and thus is smooth in the sense of Definition 2.1.1.3.

**EXAMPLE 2.1.2.2. Lie algebroids on smooth affine schemes.** The previous example possesses a slight generalization. We let $X = \mathbf{Spec} A$ be a smooth affine scheme over $k$. Let now $V$ be a vector bundle on $X$ endowed with the structure of a Lie algebroid (see §2.2 for more on the relations to Lie algebroids). We remind that this consists of a pair $(a, [., ])$, where $a : V \to T_X$ is a morphism of vector bundles on $X$, and $[., ]$ is a $k$-linear Lie bracket on sections of $V$. These data are furthermore assumed to satisfy two compatibility conditions. First we have the standard Liebniz rule: for any sections $s$ and $t \in V$ and any function $f \in A$ on $X$, we have

$$ [f.s, t] = f.[s, t] + a(t)(f).s, $$

where $a(t)(f)$ is the value of the derivation $a(t) \in T_X$ on the function $f$. The second condition states that the morphism $a$ is compatible with the Lie brackets on $T_X$: $a([s, t]) = [a(s), a(t)]$ for any pair of section $(s, t)$ of $V$. Classical foliations on $X$ are therefore Lie algebroids $(V, a, [., ])$ such that $a$ identifies $V$ with a subbundle of $T_X$.

For a Lie algebroid $(V, a, [., ])$ as above, we can form its de Rham algebra $\mathrm{Sym}_A(V^\vee[1])$. This is a graded cdga which comes equipped with an extension of the de Rham differential

$$ A \to V^\vee \to \wedge_A^2(V^\vee) \to \dots \to \wedge_A^r(V^\vee) \to \dots $$
The morphism $A \to V^\vee$ is given by composing the univeral derivation $A \to \Omega_{A/k}^1$ with the $A$-linear dual of the map $a$. The differential $dR : V^\vee \to \wedge_A^2(V^\vee)$ is itself dual to the Lie bracket $[,]$ on $V$ in the following sense: for $a$ and $b$ sections of $V$, and $v \in V^\vee$, we have

$$dR(v)(s \wedge t) = a(s)(v(t)) - a(t)(v(s)) + v([s, t]).$$

This differential extends to a canonical graded mixed structure on $Sym_A(V^\vee[1])$, and defines a graded mixed cdga, which we denote by $DR(V)$. By construction, the $A$-linear dual of $a$ defines a morphism of graded mixed cdga's $DR(A/k) \to DR(V)$. The conditions of Definition 2.1.1.1 are satisfied and therefore this construction defines a derived foliation on $X$. We refer to §2.2 for a more general construction as well as for a precise comparison between derived foliations and objects of Lie-types.

**EXAMPLE 2.1.2.3. Tautological derived foliations.** For a given derived affine scheme $X = \text{Spec } A$ there are two tautological derived foliations on $X$ of particular importance. The first one is called the 0-foliation, or the pointwise foliation, as morally its leaves are points of $X$. In terms of graded mixed cdga's it is given by the augmentation morphism

$$DR(A/k) \to A,$$

inducing the identity on $A$ and zero on all piece of higher weights. Here $A$ is considered as a graded mixed cdga with zero mixed structure, and $A$ is considered in weight 0. This augmentation morphism obviously satisfies the conditions of Definition 2.1.1.1, and here the cotangent complex is simply given by 0. This derived foliation will be denoted by $0_X$, and is the initial object of the $\infty$-category of derived foliations on $X$ (see §2.1.4.7 for a more general statement).

The second tautological foliation is the extreme opposite, and is called the *de Rham foliation*, because of its direct relation with the de Rham complex. Morally it has only one leave, namely $X$ itself. In terms of graded mixed cdga's it is given by the identity morphism

$$DR(A/k) \to DR(A/k).$$

Here the cotangent complex of this derived foliation is $\text{L}_{X/k}$ and the anchor map is the identity map. It is denoted by $*_X$ and we will see later on that it is the final object in the $\infty$-category of derived foliations over $X$.

**EXAMPLE 2.1.2.4. Derived foliations induced by a morphism.** We finish by a very last basic example, the relative de Rham derived foliation. This is again the derived foliation $*_X$ described above, except that the base ring $k$ is now replaced by a base cdga $B$ mapping to $A$.

Let $f : X = \text{Spec } A \to Y = \text{Spec } B$ be a morphism of derived affine schemes. This morphism induces a morphism of cdga $B \to A$ and thus of graded mixed cdga's $DR(B/k) \to DR(A/k)$. We define the graded mixed cdga's $DR(A/B)$, of $A$ relative to $B$, by

$$DR(A/B) := DR(A/k) \otimes_{DR(B/k)} B.$$
The tensor product above is taken inside the $\infty$-category $\epsilon-\mathrm{cdga}_k^{gr}$ of graded mixed cdga's over $k$, where the morphism $\mathrm{DR}(B/k) \to B$ is the augmentation morphism. The natural induced morphism $\mathrm{DR}(A/k) \to \mathrm{DR}(A/B)$ defines a derived foliation on $X$ in the sense of Definition 2.1.1.1. The cotangent complex of this derived foliation is here $\mathbb{L}_{A/B}$ the relative cotangent complex of $A$ over $B$, which sits in an exact triangle of $A$-dg-modules

$$ \mathbb{L}_{B/k} \otimes_B A \longrightarrow \mathbb{L}_{A/k} \longrightarrow \mathbb{L}_{A/B}. $$

This is the relative de Rham foliation on $X$ over $Y$, and is denoted by $*_X/Y$. This is the very basic example of algebraically integrable derived foliations, and we will see later on that it is the pull-back along $f$ of the tautological pointwise foliation $\mathcal{O}_Y$ (see §2.1.3).

**2.1.3. Global definition.** We now turn to the global notion of derived foliations on derived Artin stacks in the sense of §A.2. For this, we will use the standard procedure by performing a left Kan extension from derived affine schemes to all derived stacks.

We start by defining an $\infty$-functor

$$ \mathcal{Fol}(-/k) : \mathrm{dAff}_k^{op} \to \mathrm{Cat}_\infty, $$

from the $\infty$-category of derived affine schemes to the opposite of the $\infty$-category of $\infty$-categories. For an affine derived scheme $X = \mathrm{Spec} A$, $F(X/k)$ is defined to be the $\infty$-category of graded mixed $\mathrm{DR}(X/k)$-cdga's which satisfies the conditions of Definition 2.1.1.1. For an object $\mathcal{F} \in \mathcal{Fol}(X/k)$, we will use the notation $\mathrm{DR}(\mathcal{F})$ for the corresponding graded mixed $\mathrm{DR}(X/k)$-cdga.

For a morphism of derived affine schemes $f : Y = \mathrm{Spec} B \to X = \mathrm{Spec} A$, we define an $\infty$-functor $f^* : \mathcal{Fol}(X/k) \to \mathcal{Fol}(Y/k)$, by sending a graded mixed $\mathrm{DR}(X/k)$-cdga $\mathrm{DR}(\mathcal{F})$ (corresponding to an object $\mathcal{F} \in \mathcal{Fol}(Y/k)$) to $\mathrm{DR}(Y/k) \otimes_{\mathrm{DR}(X/k)} \mathrm{DR}(\mathcal{F})$, which is a graded mixed $\mathrm{DR}(Y/k)$-cdga. We note here that $\mathrm{DR}(Y/k) \otimes_{\mathrm{DR}(X/k)} \mathrm{DR}(\mathcal{F})$ also satisfies the conditions of Definition 2.1.1.1 and thus defines an object $f^*(\mathcal{F}) \in \mathcal{Fol}(Y/k)$: if as a graded $A$-linear cdga we have $\mathrm{DR}(\mathcal{F}) \simeq \mathrm{Sym}_A(\mathbb{L}[1])$, then $\mathrm{DR}(Y/k) \otimes_{\mathrm{DR}(X/k)} \mathrm{DR}(\mathcal{F})$ is of the form $\mathrm{Sym}_B(\mathbb{L}'[1])$, where $\mathbb{L}'$ sits in a push-out diagram of $B$-dg-modules

$$ \begin{tikzcd} B \otimes_A \mathbb{L}_{A/k}[1] \arrow[r] & B \otimes_A \mathbb{L}[1] \arrow[d] \\ \mathbb{L}_{B/k}[1] \arrow[u] \arrow[r] & \mathbb{L}'[1]. \arrow[u] \arrow[r] \arrow[d] \arrow[d] \end{tikzcd} $$
Indeed, after applying the $Sym_B \infty$-functor, the above push-out diagram of $B$-dg-modules is transformed to the push-out diagram of graded cdgas defining $\mathbf{DR}(Y/k) \otimes_{\mathbf{DR}(X/k)} \mathbf{DR}(\mathcal{F})$

<figure><img src="image_7.png" /></figure>

We thus have defined an $\infty$-functor

$$ \mathcal{Fol}(-/k) : \mathbf{dAff}_k^{op} \to \mathbf{Cat}_\infty, $$

which sends $X = \mathbf{Spec} A$ to $\mathcal{Fol}(X/k)$ and $f : Y = \mathbf{Spec} B \to X = \mathbf{Spec} A$ to $f^* : \mathcal{Fol}(X/k) \to \mathcal{Fol}(Y/k)$. It is important to notice here that if $\mathcal{F} \in \mathcal{Fol}(X/k)$ is a derived foliation with $\mathbb{L}_F$ as a cotangent complex, then $f^*(\mathcal{F})$ is a derived foliation on $Y$ with a cotangent complex $\mathbb{L}_{f^*(\mathcal{F})}$ defined by the push-out of $B$-dg-modules

<figure><img src="image_8.png" /></figure>

In particular, even thought there exists a canonical map $f^*(\mathbb{L}_F) \to \mathbb{L}_{f^*(\mathcal{F})}$, of quasi-coherent complexes on $Y$, this map is not an equivalence in general (unless the map $f$ is itself formally etale). In short: cotangent complexes of derived foliations are not stable by pull-backs. This fact will make more complicated the definition of a global cotangent complex when working with global derived stacks, as the gluing procedure can not be straightforward. Note that we will also sometimes use the notation $\mathcal{Fol}(A/k)$ for $\mathcal{Fol}(X/k)$ (when $X = \mathbf{Spec} A$).

Our first observation is that the $\infty$-functor $\mathcal{Fol}(-/k)$ is a derived stack for the étale topology, showing that derived foliations is a local notion for the étale topology.

**PROPOSITON 2.1.3.1.** The $\infty$-functor $\mathcal{Fol}(-/k) : \mathbf{dAff}_k^{op} \to \mathbf{Cat}_\infty$ is a (hypercomplete) stack for the étale topology.

*Proof.* We start by noticing that $X \mapsto \mathbf{DR}(X/k)$ defines a hypercomplete stack for the étale topology on $\mathbf{dAff}_k^{op}$.

**LEMMA 2.1.3.2.** The $\infty$-functor $\mathbf{DR}(-/k) : \mathbf{dAff}_k^{op} \to \epsilon - \mathbf{cdga}_k^{qr}$ is a hypercomplete stack for the étale topology.

*Proof of the lemma.* We consider the forgetful $\infty$-functor $G : \epsilon - \mathbf{cdga}_k^{qr} \to \mathbf{dg}_k^{qr}$, which forgets the algebra and the mixed structures and only retains the underlying graded complex. This $\infty$-functor is conservative and commutes with all limits. Therefore, in order to prove the lemma it is enough to check that the composed $\infty$-functor

$$ G \circ \mathbf{DR}(-/k) : \mathbf{dAff}_k^{op} \to \mathbf{dg}_k^{qr} $$
is a hypercomplete stack. This last statement is equivalent to the fact that for all $n \in \mathbb{Z}$, the weight $n$ part $X \mapsto \mathbf{DR}(X/k)^{(n)}$ is a hypercomplete stack of complexes. Moreover, this last $\infty$-functor sends, by definition, a derived affine scheme $X = \mathbf{Spec} A$ to the complex $\wedge_A^n \mathbb{L}_{A/k}[n]$ (where $\wedge^n$ is declared zero for $n < 0$ by convention).

For fixed $n \in \mathbb{Z}$, and fixed $X = \mathbf{Spec} A$, the $\infty$-functor $(\mathbf{Spec} A' \to X) \mapsto \wedge_{A'}^n \mathbb{L}_{A'/k}[n]$ is a quasi-coherent complex of $\mathcal{O}_X$-modules when restricted to the small étale site of $X$. Indeed, for $\mathbf{Spec} A' \to \mathbf{Spec} A$ any étale morphism, the canonical morphism of $A'$-dg-modules

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

is a quasi-isomorphism, and thus so are all the induced morphisms on wedge powers

$$ (\wedge_A^n \mathbb{L}_{A/k}) \otimes_A A' \to \wedge_{A'}^n \mathbb{L}_{A'/k}. $$

However, it is well known that quasi-coherent $\mathcal{O}_X$-modules satisfies hypercomplete étale descent (see [TV08, Prop. 2.2.2.13]). We thus have proven that $G \circ \mathbf{DR}(-/k)$ is indeed an étale hyperstack, and thus have proven the lemma. $\square$

To prove that $\mathcal{Fol}(-/k)$ is a derived stack we use the criterion from [TV08, Cor. 1.3.2.4], and thus we have to check that $\mathcal{Fol}(-/k)$ preserves finite products and étale hypercoverings of connective cdga's.

**Compatibility with finite products.** Let $A$ be a connective cdga and suppose that it can be written as a product of cdga's $A = \coprod_{i \in I} A_i$, with $I$ a finite set. We can write $X = \mathbf{Spec} A \simeq \coprod_i X_i$ with $X_i = \mathbf{Spec} A_i$. We have a canonical $\infty$-functor

$$ \phi : \mathcal{Fol}(X/k) \to \coprod_{i \in I} \mathcal{Fol}(X_i/k), $$

simply obtained as the product of the pull-backs $\infty$-functors $j_i^* : \mathcal{Fol}(X/k) \to \mathcal{Fol}(X_i/k)$, where $j_i : X_i \hookrightarrow X$ is the canonical inclusion. The $\infty$-functor $\phi$ possesses an inverse defined as follows. For a family $\mathcal{F}_i \in \mathcal{Fol}(X_i)$, corresponding to graded mixed $\mathbf{DR}(A_i)$-cdgas $\mathbf{DR}(\mathcal{F}_i)$, we can consider the graded mixed cdga $\coprod_{i \in I} \mathbf{DR}(\mathcal{F}_i)$. It weight 0 part is $A = \coprod_{i \in I} A_i$, and thus it is naturally equipped with a structure of a graded mixed $\mathbf{DR}(A/k)$-cdga. Moreover, as each graded cdga $\mathbf{DR}(\mathcal{F}_i)$ is of the form $\text{Sym}_{A_i}(\mathbb{L}_i)$, we have an equivalence of graded cdga's

$$ \coprod_{i \in I} \mathbf{DR}(\mathcal{F}_i) \simeq \text{Sym}_A(\coprod_{i \in I} \mathbb{L}_i). $$

Therefore, we have defined an $\infty$-functor sending a family $\mathcal{F}_i \in \mathcal{Fol}(X_i)$ to the derived foliation on $X$ defined by the graded mixed $\mathbf{DR}(X/k)$-cdga $\coprod_{i \in I}(\mathbf{DR}(\mathcal{F}_i))$. Let us denote by

$$ \psi : \coprod_{i \in I} \mathcal{Fol}(X_i/k) \to \mathcal{Fol}(X/k) $$

this $\infty$-functor.
We claim that $\psi$ and $\phi$ are inverse to each others. Indeed, for $\mathcal{F} \in \mathcal{F}ol(X/k)$, the canonical morphism of graded mixed $\mathbf{DR}(X/k)$-cdga's

$$ \mathbf{DR}(\mathcal{F}) \to \prod_{i \in I} (\mathbf{DR}(X_i/k) \otimes_{\mathbf{DR}(X/k)} \mathbf{DR}(\mathcal{F})) $$

induces a natural morphism $\psi\phi(\mathcal{F}) \to \mathcal{F}$ in $\mathcal{F}ol(X/k)$. Because of the lemma 2.1.3.2 we know that $\mathbf{DR}(X/k) \simeq \prod_{i \in I} \mathbf{DR}(X_i/k)$, and thus this natural morphism is in fact an equivalence of derived foliations over $X$. This shows that $\psi\phi$ is equivalent to the identity $\infty$-functor.

In the other direction, for $\{\mathcal{F}_i\} \in \prod_{i \in I} \mathcal{F}ol(X_i/k)$, we have a morphism $\{\mathcal{F}_i\} \to \phi\psi(\{\mathcal{F}_i\})$ corresponding to the canonical projections

$$ (\prod_{i \in I} \mathbf{DR}(\mathcal{F}_i)) \otimes_{\mathbf{DR}(X/k)} \mathbf{DR}(X_j/k) \to \mathbf{DR}(\mathcal{F}_j) $$

for $j \in I$. Again, as $\mathbf{DR}(X/k) \simeq \prod_{i \in I} \mathbf{DR}(X_i/k)$ this morphism is an equivalence in $\prod_{i \in I} \mathcal{F}ol(X_i/k)$. This shows that $\phi\psi$ is also equivalent to the identity $\infty$-functor.

**Compatibility with hypercoverings.** Let now $A \to B_*$ be an etale hypercovering of connective cdga's as in [TV08, Cor. 1.3.4]. This defines a coaugmented cosimplicial graded mixed cdga $\mathbf{DR}(A/k) \to \mathbf{DR}(B_*/k)$, which by lemma 2.1.3.2 induces an equivalence $\mathbf{DR}(A/k) \simeq \mathrm{Lim}_{i \in \Delta} \mathbf{DR}(B_i/k)$. By base change we get an $\infty$-functor on the $\infty$-categories of graded mixed modules

$$ \phi : \mathbf{DR}(A/k) - \epsilon - \mathbf{dg}^{gr} \to \mathrm{Lim}_{i \in \Delta} \mathbf{DR}(B_i/k) - \epsilon - \mathbf{dg}^{gr}. $$

This $\infty$-functor possesses a right adjoint

$$ \psi : \mathrm{Lim}_{i \in \Delta} \mathbf{DR}(B_i/k) - \epsilon - \mathbf{dg}^{gr} \to \mathbf{DR}(A/k) - \epsilon - \mathbf{dg}^{gr}. $$

which sends a cosimplicial graded mixed $\mathbf{DR}(B_*/k)$-dg-module $M_*$, to

$$ \psi(M_*) = \mathrm{Lim}_{i \in \Delta} M_i \in \mathbf{DR}(A/k) - \epsilon - \mathbf{dg}^{gr}, $$

where each $M_i$ is considered as a graded mixed module over $\mathbf{DR}(A/k)$ via the coaugmentation $\mathbf{DR}(A/k) \to \mathbf{DR}(B_i/k)$.

Let $\mathcal{F} \in \mathcal{F}ol(A/k)$ and let us consider the adjunction morphism

$$ \mathbf{DR}(\mathcal{F}) \to \mathrm{Lim}_{i \in \Delta} (\mathbf{DR}(B_i/k) \otimes_{\mathbf{DR}(A/k)} \mathbf{DR}(\mathcal{F})). $$

To prove that this morphism is an equivalence, we can forget the mixed and algebra structures, as the corresponding forgetful $\infty$-functor commutes with limits and is conservative, and thus only consider the above morphism as a morphism of graded complexes. Moreover, as each map $A \to B_i$ is étale, we have $\mathbf{DR}(B_i/k) \simeq B_i \otimes_A \mathbf{DR}(A/k)$, and thus $\mathbf{DR}(B_i/k) \otimes_{\mathbf{DR}(A/k)} \mathbf{DR}(\mathcal{F}) \simeq B_i \otimes_A \mathbf{DR}(\mathcal{F})$. Therefore, the above adjunction morphism, when considered as a morphism of graded complexes, is equivalent to the natural morphism

$$ \mathbf{DR}(\mathcal{F}) \to \mathrm{Lim}_{i \in \Delta} (B_i \otimes_A \mathbf{DR}(\mathcal{F})). $$
By étale descent for modules (see [TV08, Lem. 2.2.2.13]), this is an equivalence. This shows that the counit of the adjunction $\psi\phi \Rightarrow id$ is an equivalence.

We consider now an object $\mathcal{F}_* \in \text{Lim}_i \mathcal{F}ol(B_i/k)$, and the unit of the adjunction $\phi\psi(\mathbf{DR}(\mathcal{F}_*)) \to \mathbf{DR}(\mathcal{F}_*)$, which is a morphism of cosimplicial graded mixed $\mathbf{DR}(B_*)$-cdga's. It is a general fact that the projection to the degree 0 piece $\text{Lim}_{i \in \Delta} C_i \to C_0$ is conservative for any limits of $\infty$-categories along $\Delta$. As a consequence, to prove that the unit morphism is an equivalence it is enough to show that the natural map $\phi\psi(\mathbf{DR}(\mathcal{F}_0)) \to \mathbf{DR}(\mathcal{F}_0)$ is an equivalence of graded mixed cdga's. This morphism is given by the canonical map

$$B_0 \otimes_A (\text{Lim}_{i \in \Delta} \mathbf{DR}(\mathcal{F}_i)) \to \mathbf{DR}(\mathcal{F}_0).$$

We can, as above, forget the mixed structures, and consider this as a morphism of graded cdga's. But, as a graded cosimplicial $B_*\text{-cdga}$, $\mathbf{DR}(\mathcal{F}_*)$ is of the form $\text{Sym}_{B_*(\mathbb{L}_{\mathcal{F}_*})}$, where $\mathbb{L}_{\mathcal{F}_*}$ is the cosimplicial $B_*\text{-module}$ defined by the cotangent complex of $\mathcal{F}_*$. Because all transition maps in $B_*$ are formally étale, $\mathbb{L}_{\mathcal{F}_*}$ is a cartesian cosimplicial $B_*\text{-dg-module}$, and thus by étale descent (see [TV08, Lem. 2.2.2.13]) comes from a $A$-dg-module $\mathbb{L}$. Moreover, because the descent equivalence $A - \mathbf{dg} \simeq \text{Lim}_i B_i - \mathbf{dg}$ is a symmetric monoidal equivalence, we have that the graded $B_*\text{-dg-module}$ $\mathbf{DR}(\mathcal{F}_*)$ comes from $\text{Sym}_A(\mathbb{L})$ by base change along the coaugmentation $A \to B_*$. Moreover, we have $\text{Sym}_A(\mathbb{L}) \simeq \text{Lim}_i(\mathbf{DR}(\mathcal{F}_*))$. In particular, we see that the morphism $B_0 \otimes_A (\text{Lim}_{i \in \Delta} \mathbf{DR}(\mathcal{F}_i)) \to \mathbf{DR}(\mathcal{F}_0)$ becomes equivalent to the canonical map

$$B_0 \otimes A \text{Sym}_A(\mathbb{L}) \to \text{Sym}_{B_0}(B_0 \otimes_A \mathbb{L})$$

which is an equivalence. This shows that $\mathbf{DR}(\mathcal{F}_*)$ comes by pull-back from a graded mixed $\mathbf{DR}(A/k)$-cdga $\mathbf{DR}(\mathcal{F})$, but also that as a graded $A$-linear cdga $\mathbf{DR}(\mathcal{F})$ is of the form $\text{Sym}_A(\mathbb{L})$.

To conclude, we have shown that the adjunction $(\phi, \psi)$ restricts to an equivalence of $\infty$-categories of derived foliations

$$\mathcal{F}ol(A/k) \simeq \text{Lim}_{i \in \Delta} \mathcal{F}ol(B_i/k).$$

□

The Proposition 2.1.3.1 implies that $\mathcal{F}ol$ can be considered itself as a derived stack in $\infty$-categories. As such, for any derived stack $X \in \mathbf{d}\text{St}_k$, we can form the $\infty$-category of morphisms from $X$ to $\mathcal{F}ol(-/k)$

$$(7) \qquad \mathcal{F}ol(X/k) := \text{Map}_{\mathbf{d}\text{St}_k}(X, \mathcal{F}ol(-/k)) \in \text{Cat}_{\infty}.$$

In more concrete terms, we have

$$\mathcal{F}ol(X/k) \simeq \text{Lim}_{\text{Spec } A \to X} \mathcal{F}ol(A/k),$$

where the limit is taken over the (opposite) $\infty$-category $(\mathbf{d}\text{Aff}_k/X)^{\text{op}}$ of derived affine schemes over $X$.
**DEFINITION 2.1.3.3.** For a derived stack $X$ over $k$, the $\infty$-category of derived foliations of $X$ (relative to $k$) is $\mathcal{F}ol(X/k)$ constructed above. When the base ring $k$ is clear from the context, we will simply write $\mathcal{F}ol(X)$ for $\mathcal{F}ol(X/k)$.

When a derived foliation $\mathcal{F} \in \mathcal{F}ol(X/k)$ lives over a derived affine scheme $X = \mathbf{Spec} A$, its cotangent complex $\mathbb{L}_{\mathcal{F}}$ is the $A$-dg-module such that $\mathbf{DR}(\mathcal{F}) \simeq \mathrm{Sym}_A(\mathbb{L}_{\mathcal{F}}[1])$ as graded cdga's. Using the equivalence between quasi-coherent modules over $X$ and $A$-dg-modules (see [TV08, Cor. 1.3.7.3]), $\mathbb{L}_{\mathcal{F}}$ will be considered an object in $\mathbf{QCoh}(X)$, the $\infty$-category of quasi-coherent complexes over $X$.

When $X$ is a general derived stack, and $\mathcal{F} \in \mathcal{F}ol(X/k)$, the global cotangent complex of $\mathcal{F}$ still makes sense but is slightly more complicated to express. We first define an $\mathcal{O}_X$-dg-module as follows. For each $u : \mathbf{Spec} A \to X$, we have $u^*(\mathcal{F})$, the induced derived foliation on $\mathbf{Spec} A$, and thus a quasi-coherent complex $\mathbb{L}_{u^*(\mathcal{F})} \in \mathbf{QCoh}(\mathbf{Spec} A)$. For two morphisms

$$ \mathbf{Spec} B \xrightarrow{f} \mathbf{Spec} A \xrightarrow{u} X, $$

there exists a canonical morphism of quasi-coherent complexes over $\mathbf{Spec} B$

$$ f^*(\mathbb{L}_{u^*(\mathcal{F})}) \to \mathbb{L}_{(u f)^*(\mathcal{F})}. $$

In general, the above morphism is not an equivalence, unless the map $f$ is formally étale. Therefore, the rule that associate to $\mathbf{Spec} A \to X$ the $A$-dg-module $\mathbb{L}_{u^*(\mathcal{F})}$ defines a complex of $\mathcal{O}_X$-dg-modules in the sense reminded in §A.4, but which is not quasi-coherent in general. This will be called the *big cotangent complex* of $\mathcal{F}$, or the *fake cotangent complex*. We will denote it by

$$ \mathbb{L}_{\mathcal{F}}^{\text{big}} \in \mathcal{O}_X - \text{dg}. $$

As reminded in §A.4, the $\infty$-category of quasi-coherent complexes over $X$ sits inside $\mathcal{O}_X - \text{dg}$ as a full sub-$\infty$-category, and the natural embedding

$$ \mathbf{QCoh}(X) \hookrightarrow \mathcal{O}_X - \text{dg} $$

possesses a canonical right adjoint

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

called the *quasi-coherator*. Note that the natural morphism $\mathbb{L}_{X/k}^{\text{big}} \to \mathbb{L}_{\mathcal{F}}^{\text{big}}$ induces a morphism on the associated quasi-coherent complexes. The quasi-coherent complex $(\mathbb{L}_{X/k}^{\text{big}})^{qcoh}$ identifies naturally with the global cotangent complex $\mathbb{L}_{X/k}$, as reminded in theorem A.5.0.1. Therefore, when $X$ is a derived Artin stack and $\mathcal{F} \in \mathcal{F}ol(X/k)$, the cotangent complex $\mathbb{L}_{\mathcal{F}}$ always comes equipped with a natural map

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

**DEFINITION 2.1.3.4.** For any derived Artin stack $X$ and $\mathcal{F} \in \mathcal{F}ol(X)$, the cotangent complex of $\mathcal{F}$ is defined to be

$$ \mathbb{L}_{\mathcal{F}} := (\mathbb{L}_{\mathcal{F}}^{\text{big}})^{qcoh} \in \mathbf{QCoh}(X). $$
The conormal complex of $\mathcal{F}$ is defined to be the fiber of the natural morphism $\mathbb{L}_X \to \mathbb{L}_{\mathcal{F}}$, and is denoted by $\mathcal{N}_{\mathcal{F}/X}^*$.

We will see examples of global cotangent complexes in our next paragraph. Before this we can mention that there is a special case for which the cotangent complex defined above is easy to understand, precisely when $X$ is a Deligne-Mumford derived stack (see A.2). Indeed, under this extra assumption, $\mathrm{QCoh}(X)$ can be described using quasi-coherent $\mathcal{O}_X$-dg-modules on the small étale site $(\mathrm{dAff}/X)_{et}$, of derived affine schemes étale over $X$. When restricted to this small étale site, $\mathbb{L}_{\mathcal{F}}^{\mathrm{big}}$ is already quasi-coherent, because all the morphisms in $(\mathrm{dAff}/X)_{et}$ are automatically étale. Thus, $\mathbb{L}_{\mathcal{F}}^{\mathrm{big}} = \mathbb{L}_{\mathcal{F}}$ as a quasi-coherent $\mathcal{O}_X$-dg-module over the small étale site of $X$.

Finally, we introduce the global version of the de Rham algebra $\mathrm{DR}(\mathcal{F}/k)$ of a derived foliation $\mathcal{F}$. For a derived stack $X$ we have a stack of graded mixed cdga's on the big étale site of $F$

$$ \mathrm{DR}_{X/k} : (\mathrm{dAff}_k/X)^{op} \to \epsilon - \mathrm{cdga}_k^{gr}, $$

which sends $u : \mathrm{Spec} A \to X$ to $\mathrm{DR}(A/k)$. More generally, for $X$ as above and $\mathcal{F} \in \mathrm{Fol}(X/k)$, we have a stack

$$ \mathrm{DR}_{\mathcal{F}/k} : (\mathrm{dAff}_k/X)^{op} \to \epsilon - \mathrm{cdga}_k^{gr}, $$

which sends $u : \mathrm{Spec} A \to X$ to $\mathrm{DR}(u^*(\mathcal{F})/k)$. This stack comes moreover equipped with a canonical morphism $\mathrm{DR}_{X/k} \to \mathrm{DR}_{\mathcal{F}/k}$ of stacks of graded mixed cdga's, and thus will be referred to as a *graded mixed DR*$_{X/k}$-algebra.

**DEFINITION 2.1.3.5.** Let $X \in \mathrm{dSt}_k$ and $\mathcal{F} \in \mathrm{Fol}(X/k)$.

(1) The stack of de Rham algebras of $\mathcal{F}$ (relative to $k$) is the stack of graded mixed cdga's $\mathrm{DR}_{\mathcal{F}/k}$ defined above.

(2) The (global) de Rham algebra of $\mathcal{F}$ (relative to $k$) is

$$ \mathrm{DR}(\mathcal{F}/k) := \Gamma(\mathrm{DR}_{\mathcal{F}/k}) \in \epsilon - \mathrm{cdga}_k^{gr}. $$

With the definition above, it is straightforward to reinterpret derived foliations on $X$ as sheaves of graded mixed $\mathrm{DR}_{X/k}$-algebras. This reinterpretation simply associates to a derived foliation $\mathcal{F}$ the sheaf $\mathrm{DR}_{\mathcal{F}/k}$ of the above definition. We get this way a full embedding of $\mathrm{Fol}(X/k)$ into the $\infty$-category $\mathrm{DR}_{X/k} - \epsilon - \mathrm{cdga}^{gr}$ of sheaves of graded mixed $\mathrm{DR}_{X/k}$-algebras

$$ \mathrm{Fol}(X/k) \hookrightarrow \mathrm{DR}_{X/k} - \epsilon - \mathrm{cdga}^{gr}. $$

The essential image of this full embedding is easy to describe by simply unwiding the definitions.

**COROLLARY 2.1.3.6.** The above full embedding identifies $\mathrm{Fol}(X/k)$ with the full sub-$\infty$-category of sheaves of graded mixed $\mathrm{DR}_{X/k}$-algebras $\mathcal{D}$ which are quasi-coherent over $\mathrm{DR}_{X/k}$: for any diagram of derived stacks $\mathrm{Spec} B \to \mathrm{Spec} A \to X$ the natural morphism

$$ \mathrm{DR}_{X/k}(\mathrm{Spec} B) \otimes_{\mathrm{DR}_{X/k}(\mathrm{Spec} B)} \mathcal{D}(\mathrm{Spec} A) \to \mathcal{D}(\mathrm{Spec} B) $$

is an equivalence of graded mixed complexes.
**REMARK 2.1.3.7.** We warn the reader that in the statement above $\mathbf{DR}_{X/k}$ is itself not quasi-coherent over $\mathcal{O}_X$, even as merely a sheaf of graded $\mathcal{O}_X$-cdga's (by forgetting the mixed structure). In particular, $\mathcal{D}$ is also not quasi-coherent as a graded $\mathcal{O}_X$-algebra over $X$. It is rather of the form $\mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}[1])$ for an $\mathcal{O}_X$-dg-module $\mathbb{L}$ which is not necessarily quasi-coherent.

When the mixed structure is forgotten, the sheaf $\mathbf{DR}_{\mathcal{F}/k}$ becomes, as a graded $\mathbf{DR}_{X/k}$-cdga, equivalent to $\mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\mathcal{F}/k}^{\mathrm{big}}[1])$. It is important to note that the graded cdga $\mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\mathcal{F}/k}[1])$ does not carry any natural compatible mixed structure, at least not before passing to global sections (see theorem. 2.1.3.8). This is already true for the tautological derived foliation $*_X$ whose de Rham algebra is $\mathbf{DR}_{X/k}$, as the de Rham differential does not exists as a morphism $\mathcal{O}_X \to \mathbb{L}_{X/k}$ but only as a morphism to $\mathbb{L}_{X/k}^{\mathrm{big}}$.

We finish by the important descent statement showing that even thought the de Rham differential of a derived foliation does not exists as a morphism $\mathcal{O}_X \to \mathbb{L}_{\mathcal{F}/k}$, it does so after having taken global sections. This result can be seen as an extension of the descent statement for closed forms of [PTVV13, Prop. 1.14] to the derived foliation setting.

**THEOREM 2.1.3.8.** Let $\mathcal{F} \in \mathcal{Fol}(X/k)$ be a derived foliation over a derived Artin stack $X$. We assume that the conormal complex $\mathcal{N}_{\mathcal{F}}^*$ is a perfect complex on $X$. Then, the natural morphism $\mathbb{L}_{\mathcal{F}/k} \to \mathbb{L}_{\mathcal{F}/k}^{\mathrm{big}}$ induces an equivalence $\mathcal{O}_X$-dg-modules

$$ \wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k} \to (\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k}^{\mathrm{big}})^{q\mathrm{coh}}. $$

In particular, there exists a natural equivalence of graded (non-mixed) cdga's

$$ \Gamma(X, \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\mathcal{F}/k}[1])) \simeq \Gamma(X, \mathrm{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\mathcal{F}/k}^{\mathrm{big}}[1])) \simeq \mathbf{DR}(\mathcal{F}/k). $$

*Proof.* The last statement of the theorem is a direct consequence of the fact that the quasi-coherator, by its definition as the right adjoint of the natural inclusion $\infty$-functor, does not change global sections. Also, for $i=0$ the statement is trivial, as for $i=1$ for which it is true by definition of $\mathbb{L}_{\mathcal{F}/k}$.

We first show the following lemma, for which the assumption that $\mathcal{N}_{\mathcal{F}/k}^*$ is perfect is not needed.

**LEMMA 2.1.3.9.** The commutative diagram of $\mathcal{O}_X$-dg-modules

<figure><img src="image_9.png" /></figure>

is homotopy cartesian.

*Proof of the lemma.* In the diagram under consideration, the vertical morphisms are the adjunction morphism coming from the fact that top horizontal morphism is obtained from the
bottom horizontal morphism by applying $(-)^{qcoh}$ (see theorem A.5.0.1). Therefore, as $(-)^{qcoh}$ is a stable $\infty$-functor, it is enough to show that the cofiber of $\mathbb{L}_{X/k}^{big} \to \mathbb{L}_{\mathcal{F}/k}^{big}$ is already a quasi-coherent $\mathcal{O}_X$-dg-module.

For $u : U = \text{Spec } A \to X$, the morphism of $A$-dg-modules $\mathbb{L}_{X/k}^{big}(U) \to \mathbb{L}_{\mathcal{F}/k}^{big}(U)$ is equivalent, by definition, to the natural morphism

$$ \mathbb{L}_{A/k} \to \mathbb{L}_{u^*(\mathcal{F})/k}. $$

For any $f : V = \text{Spec } B \to U$, with $v = uf : V \to X$, the commutative diagram

$$ \begin{tikzcd} B \otimes_A \mathbb{L}_{A/k} \arrow[r] \arrow[d] & B \otimes_A \mathbb{L}_{u^*(\mathcal{F})/k} \arrow[d] \\ \mathbb{L}_{B/k} \arrow[r, "\cong"] & \mathbb{L}_{f^*u^*(\mathcal{F})/k} \arrow[d] \end{tikzcd} $$

is cartesian by definition of pull-backs of derived foliations over affine derived schemes. Therefore, if $C$ denotes the cofiber of $\mathbb{L}_{X/k}^{big} \to \mathbb{L}_{\mathcal{F}/k}^{big}$, then the natural morphism of $B$-dg-modules $B \otimes_A C(U) \to C(V)$ is an equivalence. This is precisely the statement that $C$ is a quasi-coherent $\mathcal{O}_X$-dg-module. $\square$

According to the lemma, we have a commutative diagram with exact rows

$$ \begin{tikzcd} \mathbb{L}_{X/k} \arrow[r] \arrow[d] & \mathbb{L}_{\mathcal{F}/k} \arrow[r] \arrow[d] & C \arrow[d] \\ \mathbb{L}_{X/k}^{big} \arrow[r, "\cong"] & \mathbb{L}_{\mathcal{F}/k}^{big} \arrow[r, "\cong"] & C. \end{tikzcd} $$

For all $i > 1$ fixed, these exact rows induces finite filtrations on $\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k}$ and $\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k}^{big}$ whose graded pieces of weight $p$ are respectively

$$ \wedge_{\mathcal{O}_X}^p \mathbb{L}_{X/k} \otimes_{\mathcal{O}_X} \wedge_{\mathcal{O}_X}^{i-p} C \quad \text{and} \quad \wedge_{\mathcal{O}_X}^p \mathbb{L}_{X/k}^{big} \otimes_{\mathcal{O}_X} \wedge_{\mathcal{O}_X}^{i-p} C. $$

These filtrations are moreover compatible with the vertical morphism in the middle. Therefore, in order to show that the morphism $\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k} \to (\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k}^{big})^{qcoh}$ is an equivalence, it is enough to show that it induces an equivalence of those graded pieces.

**LEMMA 2.1.3.10.** For any perfect complex $E$ on $X$ and any $E' \in \mathcal{O}_X$-dg, the natural morphism

$$ E \otimes (E')^{qcoh} \to (E \otimes E')^{qcoh} $$

is an equivalence.
**Proof of the lemma.** We use here that perfect complexes are dualizable objects in $\mathbf{QCoh}(X)$, and thus also in the bigger symmetric monoidal $\infty$-category $\mathcal{O}_X - \mathrm{dg}$. Therefore, for any quasi-coherent complex $E_0$, we have functorial equivalences

$$ \mathbf{Map}(E_0, E \otimes (E')^{qcoh}) \simeq \mathbf{Map}(E_0 \otimes E^\vee, (E')^{qcoh}) \simeq \mathbf{Map}(E_0 \otimes E^\vee, E') \simeq \mathbf{Map}(E_0, E \otimes E'). $$

By observation, the composed equivalence $\mathbf{Map}(E_0, E \otimes (E')^{qcoh}) \to \mathbf{Map}(E_0, E \otimes E')$ is induced by the canonical morphism $E \otimes (E')^{qcoh} \to (E \otimes E')^{qcoh}$, and the lemma thus follows from Yoneda. $\square$

To finish the proof of the theorem, we invoke theorem A.5.0.1 and the lemma 2.1.3.10. They imply that the natural morphisms

$$ \wedge_{\mathcal{O}_X}^p \mathbb{L}_{X/k} \otimes_{\mathcal{O}_X} \wedge_{\mathcal{O}_X}^{i-p} C \to \wedge_{\mathcal{O}_X}^p \mathbb{L}_{X/k}^{big} \otimes_{\mathcal{O}_X} \wedge_{\mathcal{O}_X}^{i-p} C $$

are equivalences (note that $C$ is perfect by assumption), and thus, by what we have already seen before, that the morphism $\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k} \to (\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}/k}^{big})^{qcoh}$ is also an equivalence. $\square$

### 2.1.4. The $\infty$-categories of derived foliations.
By construction, for a derived stack $X \in \mathrm{dSt}_k$, derived foliations on $X$ (relative to $k$) form an $\infty$-category $\mathcal{Fol}(X/k)$. We study here two basic properties of these $\infty$-categories: existence of finite limits and pull-backs functionality along arbitrary morphisms.

Let $f : X \to Y$ be a morphism of derived stacks. It induces an $\infty$-functor on the $\infty$-categories of maps to the derived stack $\mathcal{Fol}(-/k)$

$$ f^* : \mathcal{Fol}(Y/k) = \mathbf{Map}(Y, \mathcal{Fol}(-, k)) \to \mathcal{Fol}(X/k) = \mathbf{Map}(X, \mathcal{Fol}(-, k)). $$

This defines an $\infty$-functor

$$ \mathcal{Fol}(-/k) : \mathrm{dSt}_k^{op} \to \mathrm{Cat}_\infty, $$

which is the left Kan extension of $\mathcal{Fol}(-/k) : \mathrm{dAff}_k^{op} \to \mathrm{Cat}_\infty$ along the Yoneda embedding $\mathrm{dAff}_k \hookrightarrow \mathrm{dSt}_k$ (see [Toe14, §2.1]).

Note that the big cotangent complex construction of Definition 2.1.3.4 defines an $\infty$-functor

$$ \mathbb{L}^{big} : \mathcal{Fol}(X/k) \to (\mathcal{O}_X - \mathrm{dg})^{op}. $$

Composed with the quasi-coherator, it defines another $\infty$-functor

$$ \mathbb{L} : \mathcal{Fol}(X/k) \to (\mathcal{O}_X - \mathrm{dg})^{op} \to (\mathbf{QCoh}(X))^{op}. $$

the $\infty$-functors $\mathbb{L}^{big}$ and $\mathbb{L}$ are not functorial in $X$ and rather defines a lax natural transformations between derived stacks of $\infty$-categories $\mathcal{Fol}(-/k) \to (\mathcal{O}_X - \mathrm{dg})^{op} \to (\mathbf{QCoh}(-))^{op}$. In other words, for a morphism of derived stacks $f : X \to Y$, and $\mathcal{F} \in \mathcal{Fol}(Y/k)$, there exists a natural morphism of $\mathcal{O}_X$-dg-modules $u : f^*(\mathbb{L}_{\mathcal{F}}^{big}) \to \mathbb{L}_{f^*(\mathcal{F})}^{big}$. This morphism $u$ depends functorially in $f$ and $\mathcal{F}$ in some obvious manner, with all the required homotopy coherences. The formal way
to express these coherences is by stating the existence of an $\infty$-functor on the Grothendieck's integral which commutes with the natural projections to $\mathbf{dSt}_k$ (see [TV15, §1.3])

<figure><img src="image_10.png" /></figure>

The next Proposition shows that $\mathcal{Fol}(X/k)$ admits arbitrary limits for any derived stack $X \in \mathbf{dSt}_k$.

**PROPOSITON 2.1.4.1.** (1) For any derived stack $X \in \mathbf{dSt}_k$, the $\infty$-category $\mathcal{Fol}(X/k)$ admits small limits.

(2) For any morphism of derived stacks $f : X \to Y$, the pull-back $\infty$-functor $f^* : \mathcal{Fol}(Y/k) \to \mathcal{Fol}(X/k)$ preserves limits.

*Proof.* (1) The $\infty$-category $\mathcal{Fol}(X/k)$ can be written as a limit

$$ \mathcal{Fol}(X/k) \simeq \mathrm{Lim}_{\mathbf{Spec} A \to X} \mathcal{Fol}(A/k). $$

In order to show that $\mathcal{Fol}(X/k)$ admits small limits it is therefore enough to show the following two facts.

(1) For all $\mathbf{Spec} A \in \mathbf{dAff}_k$, the $\infty$-category $\mathcal{Fol}(A/k)$ admits small limits.

(2) For any morphism $f : \mathbf{Spec} B \to \mathbf{Spec} A$ in $\mathbf{dAff}_k$ the induced $\infty$-functor $f^* : \mathcal{Fol}(B/k) \to \mathcal{Fol}(A/k)$ preserves small limits.

For the first of these assertions, let $I \to \mathcal{Fol}(A/k)$ be a small diagram. It corresponds to a diagram $D : I^{op} : \mathbf{DR}(A/k) - \epsilon - \mathbf{cdga}^{qr}$, of graded mixed $\mathbf{DR}(A/k)$-cdga's. We consider $C := \mathrm{Colim}_{i \in I^{op}} D(i) \in \mathbf{DR}(A/k) - \epsilon - \mathbf{cdga}^{qr}$. We claim that $C$ is a graded mixed $\mathbf{DR}(A/k)$-cdga satisfying the conditions of Definition 2.1.1.1. Indeed, $D$ induces a diagram on cotangent complexes $i \mapsto \mathbb{L}_{D(i)/k}$, from $I^{op}$ to $A - \mathbf{dg}$, simply by considering the $\infty$-functor obtained by the weight one piece. This is a diagram of $A$-dg-modules under $\mathbb{L}_{A/k}$. We know that the forgetful $\infty$-functor, from graded mixed cdga's to graded cdga's commutes, and even reflects, limits and colimits. Therefore, $C$ is, as a graded $\mathbf{DR}(A/k)$-cdga, of the form

$$ C \simeq \mathrm{Colim}_{i \in I^{op}} \mathrm{Sym}_A(\mathbb{L}_{D(i)/k}[1]) \simeq \mathrm{Sym}_A(\mathbb{L}[1]), $$

where $\mathbb{L} = \mathrm{Colim}_{i \in I^{op}} \mathbb{L}_{D(i)/k}$ is the colimit in the $\infty$-category $\mathbb{L}_{A/k}/A - \mathbf{dg}$ of $A$-dg-modules under $\mathbb{L}_{A/K}$. Finally, by definition $\mathcal{Fol}(A/k)$ is a full sub-$\infty$-category of the complete $\infty$-category $(\mathbf{DR}(A/k) - \epsilon - \mathbf{cdga}^{qr})^{op}$, and we just have seen that this sub-$\infty$-category is stable by arbitrary limits. In particular, $\mathcal{Fol}(A/k)$ possesses limits.

The second of the above assertions is also true, as $f^*$ is defined, on the level of graded mixed cdga's, by the base change $-\mathbf{DR}(A/k)\mathbf{DR}(B/k)$, and thus sends colimits of graded mixed $\mathbf{DR}(A/k)$-cdga's to colimits of graded mixed $\mathbf{DR}(B/k)$-cdga's.
(2) As before, the statement is reduced to morphisms between affine derived schemes, for which we already have seen that pull-backs preserve limits. □

As a corollary of the proof of Proposition 2.1.4.1 we have the following formula for the cotangent complex of a limit of derived foliations. In the statement below we denote by $\mathbb{L}_{X/k}^{\text{big}}$ the big or fake cotangent complex of a derived stack $X$, defined as an $\mathcal{O}_X$-dg-module on $\text{dAff}_k/\text{dSt}_k$ by sending $\text{Spec } A \to X$ to $\mathbb{L}_{A/k}$. Note that its image by the quasi-coherator $\mathcal{O}_X - \text{dg} \to \text{QCoh}(X)$ is the usual cotangent complex $\mathbb{L}_{X/k}$ (see theorem A.5.0.1).

**COROLLARY 2.1.4.2.** Let $X$ be a derived stack, and $\mathcal{F}$ a derived foliation that is expressed as a finite limit $\mathcal{F} \simeq \lim_{i \in I} \mathcal{F}_i$ in $\text{Fol}(X/k)$. Then, the canonical morphisms

$$ \text{Colim}_{i \in I^{\text{op}}} (\mathbb{L}_{\mathcal{F}_i}^{\text{big}}) \to \mathbb{L}_{\mathcal{F}}^{\text{big}} \qquad \text{Colim}_{i \in I^{\text{op}}} (\mathbb{L}_{\mathcal{F}_i}) \to \mathbb{L}_{\mathcal{F}} $$

are equivalences in the $\infty$-categories $\mathbb{L}_{X/k}^{\text{big}}/\mathcal{O}_X - \text{dg}$ and $\mathbb{L}_{X/k}/\text{QCoh}(X)$ of objects under $\mathbb{L}_{X/k}^{\text{big}}$ and $\mathbb{L}_{X/k}$.

It is also important to note that cotangent complexes of pull-backs of derived foliations can be computed explicitly.

**COROLLARY 2.1.4.3.** Let $f : X \to Y$ be a morphism of derived Artin stacks, and $\mathcal{F} \in \text{Fol}(Y/k)$. Then, there exists a cocartesian square of quasi-coherent complexes on $X$

$$ \begin{tikzcd} f^*(\mathbb{L}_{Y/k}) \arrow[r] \arrow[d] & f^*(\mathbb{L}_{\mathcal{F}/k}) \arrow[d] \\ \mathbb{L}_{X/k} \arrow[r, "\simeq"] & \mathbb{L}_{f^*(\mathcal{F})/k} \arrow[d] \arrow[d] \end{tikzcd} $$

*Proof.* We have tautological equivalences of $\mathcal{O}_X$-dg-modules

$$ f^*(\mathbb{L}_{Y/k}^{\text{big}}) \simeq \mathbb{L}_{X/k}^{\text{big}} \qquad f^*(\mathbb{L}_{\mathcal{F}/k}^{\text{big}}) \simeq \mathbb{L}_{f^*(\mathcal{F})/k}^{\text{big}} $$

By Lemma 2.1.3.9 applied to $\mathcal{F}$ we have a cocartesian square of $\mathcal{O}_Y$-dg-modules

$$ \begin{tikzcd} \mathbb{L}_{Y/k} \arrow[r] \arrow[d] & \mathbb{L}_{\mathcal{F}/k} \arrow[d] \\ \mathbb{L}_{Y/k}^{\text{big}} \arrow[r, "\simeq"] & \mathbb{L}_{\mathcal{F}/k}^{\text{big}} \arrow[d] \arrow[d] \end{tikzcd} $$

Applying $f^*$ we get another cocartesian square

$$ \begin{tikzcd} f^*(\mathbb{L}_{Y/k}) \arrow[r] \arrow[d] & f^*(\mathbb{L}_{\mathcal{F}/k}) \arrow[d] \\ f^*(\mathbb{L}_{Y/k}^{\text{big}}) \arrow[r, "\simeq"] & f^*(\mathbb{L}_{\mathcal{F}/k}^{\text{big}}) \arrow[d] \arrow[d] \end{tikzcd} $$
The corollary follows by the tautological identifications

$$f^*(\mathbb{L}_{Y/k}^{\text{big}}) = \mathbb{L}_{X/k}^{\text{big}} \qquad f^*(\mathbb{L}_{\mathcal{F}/k}^{\text{big}}) = \mathbb{L}_{f^*(\mathcal{F})/k}^{\text{big}}$$

and an application of the $\infty$-functor $(-)^{qcoh}$.

<figure><img src="image_11.png" /></figure>

We finish this part with the specific case of derived foliations over Deligne-Mumford derived stacks. We remind that $X \in \text{dSt}_k$ is called Deligne-Mumford if it is $n$-geometric for some $n$ in the sense of [TV08, Def. 1.3.3.1], and if there are étale morphisms $\text{Spec } A_i \to X$ such that $\coprod_i \text{Spec } A_i \to X$ is an epimorphism. Derived schemes are in particular Deligne-Mumford, as each of the morphism $\text{Spec } A_i \to X$ can be chosen to be a Zariksi open immersion. For such a Deligne-Mumford derived stack $X$, we can define the small étale site $X_{et}$, which is the full sub-$\infty$-category of $\text{dAff}_k/X$ consisting of all étale morphisms $\text{Spec } A \to X$, endowed with the induced étale topology. Stacks on the $\infty$-site $X_{et}$ form a sub-$\infty$-topos of $(\text{dAff}_k/X)^{\sim,et}$, called the small étale $\infty$-topos of $X$. One important feature of the small étale site is the inclusion of the truncation $j : t_0 X \to X$ induces an equivalence of $\infty$-topos $j^* : X_{et} \simeq (t_0 X)_{et}$. In particular the small étale topos does not depend on derived structures, in a similar manner than the small étale topos of a scheme does not depend on non-reduced scheme structures.

It is immediate to check that the big cotangent complex $\mathbb{L}_{X/k}^{\text{big}}$ of a derived Deligne-Mumford stack $X$ is quasi-coherent when restricted to the small étale site. More precisely, the adjunction morphism of $\mathcal{O}_X$-dg-modules $\mathbb{L}_{X/k} \to \mathbb{L}_{X/k}^{\text{big}}$ restricts on $X_{et}$ to an equivalence. Similarly, if $\mathcal{F} \in \text{Fol}(X/k)$ is a derived foliation on $X$, then the canonical morphism $\mathbb{L}_{\mathcal{F}/k} \to \mathbb{L}_{\mathcal{F}/k}^{\text{big}}$ restricts to an equivalence on $X_{et}$. Implementing on this, we can prove the following proposition, which describes derived foliations in terms of sheaf theory on the small étale sites. For this we introduce the following notation: if $\mathcal{F} \in \text{Fol}(X/k)$ we denote by $\text{DR}_{\mathcal{F}/k,et}$ the restriction of the sheaf of graded mixed cdga's $\text{DR}_{\mathcal{F}/k}$ of Definition 2.1.3.5 to $X_{et}$.

**PROPOSITON 2.1.4.4.** Let $X \in \text{dSt}_k$ be a derived Deligne-Mumford stack. The $\infty$-functor

$$\text{Fol}(X/k) \to (\text{DR}_{X/k,et} - \epsilon - \text{cdga}^{gr})^{op}$$

from derived foliations to stacks of graded mixed $\text{DR}_{X/k,et}$-cdga's on the small étale site of $X$, is fully faithful. Its essential image consists of stacks of graded mixed $\text{DR}_{X/k,et}$-cdga $D$ satisfying the following two properties.

(1) The weight 1 piece $D(1)$ of $D$ is a quasi-coherent $\mathcal{O}_X$-dg-module, for the natural $\mathcal{O}_X$-module structure induced by the graded $\text{DR}_{X/k,et}$-algebra structure on $D$.

(2) The natural morphism of stacks of graded $\text{DR}_{X/k,et}$-cdga's $\text{Sym}_{\mathcal{O}_X}(D(1)) \to D$ is an equivalence.

*Proof.* This is a formal exercise left to the reader.

<figure><img src="image_12.png" /></figure>

The importance of Proposition 2.1.4.4 is that $\text{DR}(\mathcal{F}/k)_{et}$ is, as a stack of graded cdga's, of the form $\text{Sym}_{\mathcal{O}_X}(\mathbb{L}_{\mathcal{F}/k}[1])$. In other words, the de Rham differential for $\mathcal{F}$ exists as a morphism
For two quasi-coherent complexes $E$ and $E'$ on $X$, we can form the derived stack of morphism $\underline{\mathrm{Map}}(E, E') \in \mathrm{dSt}_k$. By definition, it sends a connective cdga $A$ to the simplicial set $\mathrm{Map}_{\mathrm{QCoh}(X)}(E, A \otimes E')$, where $A \otimes_k E'$ is the external tensor product of $E'$ by the $k$-linear complex underlying $A$. It is also the $\infty$-functor sending $A$ to $\mathrm{Map}_{\mathrm{QCoh}(X \times \mathrm{Spec} A)}(p^*(E), p^*(E'))$, where $p : X \times \mathrm{Spec} A \to X$ is the natural projection. The stack $\underline{\mathrm{Map}}(E, E')$ possesses a natural linear structure (i.e. is a stack of $\mathcal{O}$-dg-modules), but we will consider it merely as a space valued stacks and will not keep track of its linear structure in what follows. When $E$ and $E'$ both sits under a given object $a : E_0 \to E$, and $b : E_0 \to E'$, we also have the mapping stack under $E_0 \underline{\mathrm{Map}}_{E_0/}(E, E')$, of morphisms from $E$ to $E'$ that commute with the morphism $a$ and $b$. By definition, we have a fibration sequence of derived stacks

$$ \underline{\mathrm{Map}}_{E_0/}(E, E') \to \underline{\mathrm{Map}}(E, E') \to \underline{\mathrm{Map}}(E_0, E') $$

where the fiber is taken at the point corresponding to $b$.

Finally, we remind (§1.5) that $\mathcal{H}_0$ is the group stack $\mathbb{G}_m \times B\mathbb{G}_a$, semi-direct product of $B\mathbb{G}_a$ with $\mathbb{G}_m$ (for the natural induced weight 1 action of $\mathbb{G}_m$ on $\mathbb{G}_a$).

**PROPOSITON 2.1.4.6.** Let $X$ be a derived stack and $\mathcal{F}, \mathcal{F}' \in \mathrm{Fol}(X/k)$ two derived foliations on $X$. Then, there exists a stack $\pi : \mathcal{M}(\mathcal{F}, \mathcal{F}') \to B\mathcal{H}_0$ with the following properties.

(1) The base change $\mathcal{M}(\mathcal{F}, \mathcal{F}') \times_{B\mathcal{H}_0} B\mathbb{G}_m$, is equivalent to

$$ \underline{\mathrm{Map}}_{\mathbb{L}_{X/k}^{gr}}^{gr}(\mathbb{L}_{\mathcal{F}'/k}, \oplus_{i \ge 1} \wedge_{\mathcal{O}_X}^{i} \mathbb{L}_{\mathcal{F}}^{big}[i-1])^{qcoh}) $$

the derived stack of graded morphisms from $\mathbb{L}_{\mathcal{F}'/k}$ to $\oplus_{i \ge 0} \wedge_{\mathcal{O}_X}^{i} \mathbb{L}_{\mathcal{F}}^{big}[i-1]$ under $\mathbb{L}_{X/k}$, with the natural gradings where $\mathbb{L}_X$ and $\mathbb{L}_{\mathcal{F}'}$ have weight one, and $\wedge_{\mathcal{O}_X}^{i} \mathbb{L}_{\mathcal{F}}^{big}$ has weight $i$.

(2) There is a natural equivalence of spaces

$$ \mathbb{H}(B\mathcal{H}_0, \mathcal{M}(\mathcal{F}, \mathcal{F}')) := p_*(\mathcal{M}(\mathcal{F}, \mathcal{F}'))(k) \simeq \mathrm{Map}_{\mathcal{F}ol(X)}(\mathcal{F}, \mathcal{F}') $$

where $p : B\mathcal{H}_0 \to *$ is the structural morphism, and $p_* : \mathrm{dSt}_k/B\mathcal{H}_0 \to \mathrm{dSt}_k$ is the direct image $\infty$-functor.

*Proof.* We use the equivalence

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

between commutative algebras in the symmetric monoidal $\infty$-category $\mathrm{QCoh}(B\mathcal{H}_0)$ and graded mixed cdga's (see §1.5). Using this equivalence, for two graded mixed cdga $D$ and $D'$ the mapping space $\mathrm{Map}_{\epsilon-\mathrm{cdga}_k^{gr}}(D, D')$ can be written as in the Proposition

$$ \mathrm{Map}_{\epsilon-\mathrm{cdga}_k^{gr}}(D, D') \simeq p_*(\mathcal{M}(D, D'))(k). $$

Here the derived stack $\mathcal{M}(D, D')$ lives over $B\mathcal{H}_0$ and can be defined functorially as follows. For $u : \mathrm{Spec} A \to B\mathcal{H}_0$, the space $\mathcal{M}(D, D')(A)$ is given by $\mathrm{Map}_{\mathrm{cdga}_A}(u^*(D), u^*(D'))$, where $D$ and $D'$ are here considered as quasi-coherent cdga's over $B\mathcal{H}_0$ via the equivalence mentioned
above. Note that a similar formula remains valid when $D$ and $D'$ are $D_0$-graded mixed cdga, for a given graded mixed cdga $D_0$, with $M(D, D')$ being replace by $M_{D_0}(D, D')$ the derived stack of morphisms of $D_0$-cdga's. Assuming further that $D$ (resp. $D'$ and $D_0$) are, as graded cdga's, of the form $Sym_A(E)$ (resp. $Sym_A(E')$ and $Sym_A(E_0)$) for some $A$-dg-module $E$, we can instead of $M_{D_0}(D, D')$ consider its reduced version

$$ \mathcal{M}_{D_0}^*(D, D'), $$

that is the fiber of $M_{D_0}(D, D') \to M_{D_0}(D, A)$, taken at the natural augmentation $D \to A$. Now, the mapping space $\mathbf{Map}_{\mathcal{F}ol(X/k)}(\mathcal{F}, \mathcal{F}')$ is given by the limit

$$ \mathbf{Map}_{\mathcal{F}ol(X/k)}(\mathcal{F}, \mathcal{F}') \simeq \lim_{\mathbf{Spec} A \to X} \mathbf{Map}_{\mathbf{DR}(A/k)-\epsilon-\mathbf{cdga}^{gr}}(\mathbf{DR}_{\mathcal{F}'}(A), \mathbf{DR}_{\mathcal{F}}(A)). $$

Using the previous identification for mapping spaces of graded mixed cdga's, we see that the Proposition (2) is true for a certain derived stack $\mathcal{M}(\mathcal{F}, \mathcal{F}') \to BH_0$, namely

$$ \mathcal{M}(\mathcal{F}, \mathcal{F}') \simeq \lim_{\mathbf{Spec} A \to X} \mathcal{M}_{\mathbf{DR}(A/k)-\epsilon-\mathbf{cdga}^{gr}}^*(\mathbf{DR}_{\mathcal{F}'}(A), \mathbf{DR}_{\mathcal{F}}(A)). $$

The base change along $BG_m \to BH_0$ is then given by the derived stack

$$ \lim_{\mathbf{Spec} A} \mathbf{Map}_{\mathbf{DR}(A/k)-\mathbf{cdga}^{gr}}^*(\mathbf{DR}_{\mathcal{F}'}(A), \mathbf{DR}_{\mathcal{F}}(A)), $$

limits of the derived stacks $\mathbf{Map}_{\mathbf{DR}(A/k)-\mathbf{cdga}^{gr}}^*(\mathbf{DR}_{\mathcal{F}'}(A), \mathbf{DR}_{\mathcal{F}}(A))$ of morphisms of $\mathbf{DR}(A/k)$-linear morphisms of graded cdga's from $\mathbf{DR}_{\mathcal{F}'}(A)$ to $\mathbf{DR}_{\mathcal{F}}(A)$ (obtained by forgetting the mixed structures) compatible with the augmentations to $A$. Because these cdga's are graded free, we see that this base change can also be written in terms of graded morphisms of complexes

$$ \lim_{\mathbf{Spec} A} \mathbf{Map}_{\mathbb{L}_{A/k}/\mathbf{cdga}^{gr}}^{gr}(\mathbb{L}_{\mathcal{F}'}^{big}(A), \oplus_{i \ge 1} \wedge_A^i \mathbb{L}_{\mathcal{F}'}^{big}(A)[i-1])) \simeq \mathbf{Map}_{\mathbb{L}_X^{big}}^{gr}(\mathbb{L}_{\mathcal{F}'}^{big}, \oplus_{i \ge 1} \wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}'}^{big}[i-1]). $$

(note that the sum does not involve the term $i=0$ because we use the reduced version of the derived stacks of morphisms of graded cdga's). To finish the proof of the Proposition it thus simply remains to prove that the quasi-coherator construction induces for each $\mathcal{O}_X$-dg-module $E$ with a morphism $\mathbb{L}_X^{big} \to E$, an equivalence

$$ \mathbf{Map}_{\mathbb{L}_X^{big}}(\mathbb{L}_{\mathcal{F}'}^{big}, E) \to \mathbf{Map}_{\mathbb{L}_X}(\mathbb{L}_{\mathcal{F}'}^{big}, E^{qcoh}), $$

as the result will be obtained by the special cases where $E = \wedge^i \mathbb{L}_{\mathcal{F}'}^{big}$.

To see this last step, by replacing $X$ with $X \times \mathbf{Spec} A$ we see that it is enough to check this simply on mapping spaces rather than mapping derived stacks. To check this last step, we consider the commutative diagram

$$ \begin{tikzcd} \mathbf{Map}_{\mathbb{L}_X^{big}}(\mathbb{L}_{\mathcal{F}'}^{big}, E) \arrow[r] \arrow[d] & \mathbf{Map}(\mathbb{L}_{\mathcal{F}'}^{big}, E) \arrow[r] \arrow[d] & \mathbf{Map}(\mathbb{L}_X^{big}, E) \arrow[d] \\ \mathbf{Map}_{\mathbb{L}_X}(\mathbb{L}_{\mathcal{F}'}^{big}, E^{qcoh}) \arrow[r] & \mathbf{Map}(\mathbb{L}_{\mathcal{F}'}^{big}, E^{qcoh}) \arrow[r] & \mathbf{Map}(\mathbb{L}_X, E^{qcoh}). \arrow[d] \end{tikzcd} $$
Each row of this diagram is a fibration sequence, for the base point in $\mathbf{Map}(\mathbb{L}_X^{\text{big}}, E)$ corresponding to the fixed morphism $\mathbb{L}_X^{\text{big}} \to E$. However, we know that the square

<figure><img src="image_13.png" /></figure>

is cocartesian, and thus the square

<figure><img src="image_14.png" /></figure>

is cartesian. This implies that the natural morphism $\mathbf{Map}_{\mathbb{L}_X^{\text{big}}}(\mathbb{L}_{\mathcal{F}'}^{\text{big}}, E) \to \mathbf{Map}_{\mathbb{L}_X}(\mathbb{L}_{\mathcal{F}'}^{\text{big}}, E^{\text{qcoh}})$ is indeed an equivalence as wanted. $\square$

An important corollary of the previous Proposition is obtained when $X$ is moreover a derived Artin stack, as we know from A.5.0.1 that $\wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}}^{\text{big}}[i-1])^{\text{qcoh}} \simeq \wedge_{\mathcal{O}_X}^i \mathbb{L}_{\mathcal{F}}[i-1]$. In particular we get the following important consequence.

**COROLLARY 2.1.4.7.** Let $X$ be a derived Artin stack and let $\mathcal{F}$ be a derived foliation such that $\mathbb{L}_{\mathcal{F}} \simeq 0$. Then $\mathcal{F}$ is an initial object in $\mathcal{F}ol(X)$.

*Proof of Corollary.* In this case, for any $\mathcal{F}' \in \mathcal{F}ol(X)$, the natural projection $\mathcal{M}(\mathcal{F}, \mathcal{F}') \to \times_{B_{\mathcal{H}_0}} B\mathbb{G}_m$ must be an equivalence and thus $p_*(\mathcal{M}(\mathcal{F}, \mathcal{F}')) = *$ is the punctual derived stack. It follows from the proposition 2.1.4.6 that $\mathbf{Map}_{\mathcal{F}ol(X)}(\mathcal{F}, \mathcal{F}') \simeq *$. $\square$

**2.1.5. More examples.** We have given several examples of derived foliations on derived affine schemes in Section 2.1.2. We now provide several global examples on derived schemes and stacks.

**EXAMPLE 2.1.5.1. Tautological derived foliations.** For any derived stack $X$, the $\infty$-category possesses a final and and sometimes an initial object. The final object is denoted by $*_X$, whose cotangent complex is $\mathbb{L}_{X/k}$. As an $\infty$-functor $(\mathbf{dAff}_k/X)^{\text{op}} \to \epsilon - \text{cdga}^{\text{gr}}$ it is given by sending $\text{Spec } A \to X$ to $\mathbf{DR}(A/k)$.

The initial object of $\mathcal{F}ol(X/k)$, when it exists, is denoted by $0_X$. When $X$ is representable by a derived Artin stack, such an initial object always exists and moreover its cotangent complex is 0. As an $\infty$-functor $(\mathbf{dAff}_k/X)^{\text{op}} \to \epsilon - \text{cdga}^{\text{gr}}$, $0_X$ is given by sending $\text{Spec } A \to X$ to $\mathbf{DR}(A/X)$, the relative de Rham algebra of $\text{Spec } A$ over $X$. We need here to provide more
definitional details, as X is not assumed to be affine and thus DR(A/X) has not been defined yet. We define this notion by using descent over X, by the standard formula

$$ \mathrm{DR}(A/X) = \lim_{\mathbf{Spec} B \to X} \mathrm{DR}(\mathbf{Spec} A \times_{\mathbf{Spec},B} X/B). $$

In more general terms, for a morphism of derived stacks $f: Y \to Z$, we set

$$ \mathrm{DR}(Y/Z) := \lim \mathrm{DR}(A/B) \in \epsilon - \mathrm{cdga}_k^{qr}, $$

where the limit is taken over the $\infty$-category of commutative diagrams of the form

<figure><img src="image_15.png" /></figure>

As the diagonal of $X$ is representable by a derived Artin stack, all the morphisms $\mathbf{Spec} A \to X$ are also representable, and thus the relative cotangent complex $\mathbb{L}_{A/X}$ always exists as an $A$-dg-module. Moreover, by the descent statement theorem 2.1.3.8 we have an equivalence of graded cdga's

$$ \mathrm{DR}(A/X) \simeq \mathrm{Sym}_A(\mathbb{L}_{A/X}[1]). $$

This shows that $0_X$ is indeed a stack of graded mixed cdga's on $\mathbf{dAff}/X$ which corresponds to an object in $\mathcal{Fol}(X/k)$. Moreover, the big cotangent complex of $0_X$ is the $\mathcal{O}_X$-dg-module sending $\mathbf{Spec} A \to X$ to $\mathbb{L}_{A/X}$, and we thus have an exact triangle of $\mathcal{O}_X$-dg-modules

$$ \mathbb{L}_{X/k} \to \mathbb{L}_{X/k}^{\mathrm{big}} \to \mathbb{L}_{0_X}^{\mathrm{big}}. $$

By our theorem A.5.0.1 we have $\mathbb{L}_{0_X} = (\mathbb{L}_{0_X}^{\mathrm{big}})^{qcoh} \simeq 0$ and thus by Corollary 2.1.4.7 $0_X$ is indeed an initial object.

**EXAMPLE 2.1.5.2. Globally integrable derived foliations.** As in the affine case (see 2.1.2.4, we will say that a derived foliation $\mathcal{F} \in \mathcal{Fol}(X/k)$ is globally integrable (or simply integrable) if there is a morphism $f: X \to Y$ with $Y$ a derived Artin stack such that $\mathcal{F} \simeq f^*(0_Y)$. We note that $Y$ is not uniquely defined, as any factorization $f: X \xrightarrow{g} Y' \xrightarrow{p} Y$ with $p$ being étale has the property that $g^*(0'_{Y'}) \simeq f^*(0_Y)$ (simply because $p^*(0_Y) = 0'_{Y'}$ by étaleness and the corollary 2.1.4.7).

**EXAMPLE 2.1.5.3. Relative global derived foliations.** Let $f: X \to Y$ be a morphism in $\mathbf{dSt}_k$ with $Y$ being a derived Artin stack. We define the $\infty$-category of derived foliations on $X$ relative to $Y$ to be the comma $\infty$-category

$$ \mathcal{Fol}(X/Y) := \mathcal{Fol}(X/k)/f^*(0_Y), $$

of derived foliations over $f^*(0_Y)$ (we know $0_Y$ exists by our previous example). Informally, a derived foliations on $X$ relative to $Y$ is a family of derived foliations in the fibers of $f$. When $X = \mathbf{Spec} A \to Y = \mathbf{Spec} B$ is a morphism of derived affine schemes, $\mathcal{Fol}(X/Y)$ is