# Formal Integration of Derived Foliations

Lukas Brantner, Kirill Magidson, Joost Nuiten

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Lukas Brantner, Kirill Magidson, Joost Nuiten. Formal Integration of Derived Foliations. 2025. [hal-04952766](https://hal.science/hal-04952766v1)

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Distributed under a Creative Commons Attribution 4.0 International License
# FORMAL INTEGRATION OF DERIVED FOLIATIONS

LUKAS BRANTNER, KIRILL MAGIDSON, AND JOOST NUITEN

ABSTRACT. Frobenius' theorem in differential geometry asserts that every involutive subbundle of the tangent bundle of a manifold $M$ integrates to a decomposition of $M$ into smooth leaves.

We prove an infinitesimal analogue of this result for locally coherentqcqs schemes $X$ over coherent rings. More precisely, we integrate partition Lie algebroids on $X$ to formal moduli stacks $X \to \tilde{S}$, where $\tilde{S}$ is the formal leaf space and the fibres of $X \to \tilde{S}$ are the formal leaves. We deduce that deformations of $X$-families of algebro-geometric objects are controlled by partition Lie algebroids on $X$. Combining our integration equivalence with a result of Fu, we deduce that Toën-Vezzosi's infinitesimal derived foliations (under suitable finiteness hypotheses) are formally integrable.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Affine Koszul duality</td><td>7</td></tr><tr><td>3. Partition Lie algebroids</td><td>20</td></tr><tr><td>4. Infinitesimal deformations of families</td><td>33</td></tr><tr><td>5. Formally integrating partition Lie algebroids</td><td>48</td></tr><tr><td>Appendix A. Derived algebra recollections</td><td>67</td></tr><tr><td>Appendix B. Derived algebraic geometry recollections</td><td>78</td></tr><tr><td>References</td><td>89</td></tr></table>

## 1. INTRODUCTION

A smooth algebraic foliation $\mathcal{F}$ on a smooth complex variety $X$ is a subbundle

$$E_{\mathcal{F}} \subseteq T_{X}$$

which is closed under the commutator bracket of vector fields, i.e. a locally free Lie algebroid whose anchor map is the inclusion of a summand. We call $\mathcal{F}$ algebraically integrable if there is a decomposition

$$X = \coprod_{\alpha} \mathcal{F}_{\alpha}$$

of $X$ into smooth complex subvarieties $\mathcal{F}_{\alpha}$ – the leaves – whose tangent vectors span $E_{\mathcal{F}} \subset T_{X}$. Note that algebraic integrability is a highly nontrivial condition. One can approach the problem of integrating a given smooth foliation $\mathcal{F}$ in three steps.

First, we integrate $\mathcal{F}$ formally, which means that for every point $x \in M$, we find power series

$$f_1, \dots, f_k \in \mathbb{C}[[z_1, \dots, z_n]]$$
in local coordinates $z_1, \dots, z_n$ such that a local vector field $\alpha$ lies in $E_{\mathcal{F}}$ if and only if $\alpha.df_i = 0$ for all $i$. The formal leaf passing through $x$ is then encoded by the functor

$$ \mathcal{F}_x^{\wedge}: A \mapsto \{a_1, \dots, a_n \in A \mid \forall i: f_i(a_1, \dots, a_n) = 0\} $$

from local Artinian $\mathbb{C}$-algebras to sets.

Varying $x \in X$, we obtain a 'quotient' map of prestacks

$$ p: X \to \hat{S} $$

to the formal leaf space such that $\mathcal{F}_x^{\wedge}$ sends $A$ to the set of (isomorphism classes of) dotted lifts

<figure><img src="image_2.png" /></figure>

Next, we check the local convergence of these power series to obtain complex manifolds $\mathcal{F}_x$ passing through all points of $X$, thereby verifying Frobenius theorem for complex manifolds.

Finally, we examine whether or not the various leaves $\mathcal{F}_x$ are algebraic subvarieties of $X$. If this is the case, we obtain a submersion of Deligne–Mumford stacks $p: X \to S$ such that $X \to \hat{S}$ is the formal completion of $X$ along $p$.

<figure><img src="image_3.png" /><figcaption>FIGURE 1. A foliation (depicted by tangent vectors) along with its formal leaves (in gray) and leaves (in black).</figcaption></figure>

In this article, we generalise the first step to more general Lie algebroids on more general schemes. More precisely, we replace complex varieties by locally coherent qcqs schemes $X$ over coherent rings $R$, both possibly derived, and formally integrate *partition Lie algebroids*

$$ \rho: E \to T_{X/R}[1] $$

with $\rho$ not necessarily injective and $E$ not necessarily locally free. The result of our integration procedure is a *formal moduli stack on X* – around each point $x$ in $X$, we obtain a formal moduli problem $\mathcal{F}_x^{\wedge}$ encoding the formal leaf passing through $x$, which can be non-smooth, stacky, and derived.
Formal moduli stacks $X \to \hat{S}$ also arise in deformation theory, where they capture infinitesimal derived deformation functors of $X$-indexed families of algebro-geometric objects

$$ \begin{array}{ccc} M & & \\ & \downarrow & \\ & & X. \end{array} $$

Our integration equivalence then implies that every such deformation functor is governed by an essentially unique partition Lie algebroid on $X$, generalising various earlier equivalences in characteristic 0 ([Dri, Del, Hin01, Lur11, Pri10, Hen18, ?, Nui19b]) and $p$ ([BM, BW20]).

Foliations on smooth complex varieties $X$ can also be defined as quotient bundles

$$ \Omega_X \to \Omega_F $$

inducing maps of commutative differential graded algebras

$$ (\Lambda^*\Omega_X, d_{dR}) \to (\Lambda^*\Omega_F, \bar{d}_{dR}). $$

This definition was generalised by Toën-Vezzosi [TV23] to the non-smooth derived setting – they call the resulting structure an infinitesimal derived foliation. Fu [Fu24] recently constructed an equivalence between almost perfect infinitesimal derived foliations and dually almost perfect partition Lie algebroids. Combining Fu's equivalence with our main result, we deduce that almost perfect infinitesimal derived foliations (in the sense of Toën-Vezzosi) are formally integrable.

1.1. **Statement of results.** A Lie algebroid on a smooth manifold $X$ consists of an anchor map

$$ \rho : E \to T_X $$

of vector bundles together with a bilinear Lie bracket $[-, -]$ on the space $\Gamma(X, E)$ of global sections such that for all global sections $s_1, s_2 \in \Gamma(M, E)$ and all smooth functions $f$, we have

$$ [s_1, fs_2] = \rho(s_1)(f) \cdot s_2 + f[s_1, s_2]. $$

Lie algebroids were first introduced by Pradines [Pra67] and have since found numerous applications.

In this article, we will first generalise the theory of Lie algebroids the setting of algebraic geometry over very general bases, and then formally integrate them.

*Setup.* Let us fix a coherent base ring $R$, i.e. a commutative ring whose finitely generated ideals are also finitely presented. Let $X$ be a quasi-compact quasi-separated (‘qcqs’) derived $R$-scheme which is locally coherent, meaning that it admits an open cover by affine derived schemes $\text{Spec}(B)$ with $\pi_0(B)$ coherent and $\pi_i(B)$ finitely presented over $\pi_0(B)$.

Every such scheme admits a quasi-coherent cotangent complex $L_{X/R}$ and a tangent complex

$$ T_{X/R} = L_{X/R}^{\vee}. $$

This tangent complex belongs to the $\infty$-category $\text{QC}_X^{\vee} = \text{Ind}(\text{Coh}_X^{\text{op}})$ of pro-coherent sheaves on $X$, which is the natural home of duals of quasi-coherent sheaves. Pro-coherent sheaves were first introduced by Deligne [Har66, Appendix]; we refer to Appendix B.2 for further details.
Partition Lie algebroids. In Section 3, we construct the $\infty$-category of partition Lie algebroids

$$ \mathrm{LieAlgd}^{\pi}_{\Delta, X/R} $$

on the scheme $X$. These homotopical objects exhibit many expected properties:

(1) (Forgetful functor) There is a monadic forgetful functor $\mathrm{LieAlgd}^{\pi}_{\Delta, X/R} \to (\mathrm{QC}^{\vee}_{X})_{/T_{X/R}[1]}$; hence partition Lie algebroids are maps of pro-coherent sheaves

$$ \mathfrak{g} \xrightarrow{\rho} T_{X/R}[1] $$

equipped with additional Lie algebraic structure.

(2) (Global sections) Taking derived global sections gives rise to a functor

$$ \mathrm{LieAlgd}^{\pi}_{\Delta, X/R} \to (\mathrm{LieAlg}^{\pi}_{\Delta, R})_{/R\Gamma(X, T_{X/R}[1])}, $$

where $\mathrm{LieAlg}^{\pi}_{\Delta, R} = \mathrm{LieAlgd}^{\pi}_{\Delta, \mathrm{Spec}(R)/R}$ denotes the $\infty$-category of $R$-partition Lie algebras and

$$ R\Gamma(X, T_{X/R}[1]) $$

is the Kodaira-Spencer partition Lie algebra encoding infinitesimal deformations of the $R$-scheme $X$ (cf. Section 4.4).

(3) (Descent) Every étale map

$$ Y \to X $$

induces a functor $\mathrm{LieAlgd}^{\pi}_{\Delta, X/R} \to \mathrm{LieAlgd}^{\pi}_{\Delta, Y/R}$ lifting the pullback functor on the level of pro-coherent sheaves. Moreover, there is a canonical equivalence of $\infty$-categories

$$ \mathrm{LieAlgd}^{\pi}_{\Delta, X/R} \xrightarrow{\approx} \lim_{U \subseteq X \text{ affine open}} \mathrm{LieAlgd}^{\pi}_{\Delta, U/R}. $$

Before proceeding to formal integration, let us highlight various special cases of our construction:

(a) For $k$ a field of characteristic 0 and $X = \mathrm{Spec}(k)$, the $\infty$-category $\mathrm{LieAlgd}^{\pi}_{\Delta, X/k} = \mathrm{LieAlg}^{\pi}_{\Delta, k}$ is modelled by (shifted) differential graded Lie algebras over $k$. For $X = \mathrm{Spec}(A)$ an affine $k$-scheme, $\mathrm{LieAlgd}^{\pi}_{\Delta, X/k}$ arises from the category of differential graded Lie algebroids on $X$ with the tame semi-model structure, see [Nui19a, Variant 3.12].

(b) For $k$ a field of characteristic $p$ and $X = \mathrm{Spec}(k)$, we recover the $\infty$-category of partition Lie algebras introduced in [BM]. These can be modelled by cosimplicial-simplicial $k$-vector spaces with additional operations parametrised by nested chains of partitions, see [BCN, Definition 5.43]. For $X = \mathrm{Spec}(F)$ with $F/k$ a finite purely inseparable field extension, $\mathrm{LieAlgd}^{\pi}_{\Delta, X/k}$ recovers the $\infty$-category of $F/k$-partition Lie algebroids appearing in the purely inseparable Galois correspondence [BW20]. For $R = A$ complete local Noetherian with residue field $k$ and $X = \mathrm{Spec}(k)$, the $\infty$-category $\mathrm{LieAlgd}^{\pi}_{\Delta, X/A}$ is equivalent to the mixed partition Lie algebras appearing in [BM, Definition 6.22].

(c) For $R$ a coherent ring and $X = \mathrm{Spec}(R)$, the $\infty$-category $\mathrm{LieAlgd}^{\pi}_{\Delta, X/R} = \mathrm{LieAlg}^{\pi}_{\Delta, R}$ appears in [BCN, Section 3]. It admits an explicit model in terms of simplicial-cosimplicial objects, given in [BCN, Theorem 5.42].

Formal moduli stacks. Given $X$ and $R$ as above, we will now introduce formal moduli stacks

$$ X \to \hat{S} $$

under $X$. Informally, these can be thought of in at least two different ways:

(1) as decompositions of $X$ into of formal leaves $\mathcal{F}^{\wedge}_{x}$ parametrised by the points $x$ of $\hat{S}$;

(2) as infinitesimal deformation functors of $X$-indexed families of algebro-geometric objects.
To formalise the notion of a formal moduli stack, we will use *animated rings*.

**Notation 1.1.** We write $\mathrm{CAlg}^{\mathrm{an}} = \mathcal{P}_{\Sigma}(\mathrm{Poly})$ for the $\infty$-category of animated rings, which is obtained by freely adding sifted colimits to the category of finitely generated polynomial rings. Note that $\mathrm{CAlg}^{\mathrm{an}}$ is also sometimes denoted by Ani, $\mathrm{DAlg}_{\ge 0}$, $\mathrm{CAlg}^{\Delta}$, or SCR in the literature. The last notation emphasises that animated rings can be modelled by simplicial commutative rings.

Formal moduli stacks will belong to the $\infty$-category of $R$-prestacks

$$ \mathrm{PrStk}_R $$

i.e. the $\infty$-category of (accessible) functors from animated $R$-algebras to spaces. Every derived $R$-scheme $X$ gives a prestack $R \mapsto \mathrm{Map}_{\mathrm{Sch}^{\mathrm{der}}}(\mathrm{Spec}(R), X)$, which we denote by the same name.

**Definition 1.2 (Formal moduli stacks).** A formal moduli stack $\mathcal{F}$ under an $R$-prestack $X$ is a map of $R$-prestacks

$$ X \to \hat{S} $$

satisfying the following conditions:

(1) $\hat{S}$ has deformation theory, which means that it preserves limits of Postnikov towers and pullbacks along nilpotent extensions of animated $R$-algebras, see Definition B.6;

(2) $X \to \hat{S}$ is locally almost of finite presentation, see Definition B.3;

(3) $X \to \hat{S}$ restricts to an equivalence on all reduced affine schemes.

Write $\mathrm{ModuliStk}_{X/R} \subset (\mathrm{PrStk}_R)_X$ for the full subcategory of formal moduli stacks under $X$.

**Remark 1.3 (Formal leaves of formal moduli stacks).** Given an $R$-point $x$ in $X$ and a formal moduli stack $X \to \hat{S}$, the formal leaf $\mathcal{F}_x^{\wedge}$ passing through $x$ is the formal moduli problem sending an augmented local Artinian animated $R$-algebra $A$ to the space of dotted lifts

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

**Remark 1.4 (Deformation functors and formal moduli stacks).** Let $B$ be a coherent eventually coconnective animated $R$-algebra. Formal deformation functors of (suitably nice) $B$-indexed families of algebro-geometric objects over $R$ are then parametrised by functors

$$ F: \mathrm{Art}_{R/B} \to \mathcal{S} $$

from the $\infty$-category

$$ \mathrm{Art}_{R/B} \subset \mathrm{CAlg}_{R/B}^{\mathrm{an}} $$

of Artinian extensions of $B$ (cf. Definition 4.1) to the $\infty$-category $\mathcal{S}$ of spaces satisfying

(1) $F(B) \simeq *$;

(2) $F$ preserves pullbacks along nilpotent extensions.

In Section 5, we show that the $\infty$-category of such functors is equivalent to the $\infty$-category of formal moduli stacks under $X = \mathrm{Spec}(B)$; this statement is essentially contained in [GR17b, 5.1.2].
Indeed, if $X \to \hat{S}$ is a formal moduli stack, then the corresponding functor $\text{Art}_{R/B} \to S$ sends an Artinian extension $A \to B$ over $R$ to the space of dotted lifts in the following diagram:

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

**The main theorem.** Given a formal moduli stack $X \to \hat{S}$, we can consider the sheaf of 'derived tangent vectors' to the formal leaves $\mathcal{F}_X^\wedge$. To make this idea precise, we use that the $\infty$-category $\text{QC}_X^\vee$ of pro-coherent sheaves can be identified with the $\infty$-category of functors $\text{Coh}_{X, \ge 0} \to S$ preserving terminal objects and pullbacks along $\pi_0$-surjections. Here $\text{Coh}_{X, \ge 0}$ denotes the $\infty$-category of connective coherent sheaves on $X$.

The pro-coherent tangent sheaf

$$T_{X/\hat{S}}$$

then corresponds to the functor sending $I \in \text{Coh}_{X, \ge 0}$ to the space of dotted arrows in the following square:

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

Here $\text{sqz}_X(I)$ denotes the derived scheme $(X, \mathcal{O}_X \oplus I)$, where $\mathcal{O}_X \oplus I$ is the trivial square-zero extension of the structure sheaf $\mathcal{O}_X$ by $I$. There is a natural map

$$T_{X/\hat{S}} \to T_{X/R},$$

which captures the derived tangents to the formal leaves. We then prove:

**Theorem 1.5 (Main theorem).** Let $X$ be a locally coherent $\text{qcqs}$ derived $R$-scheme over a coherent animated ring $R$. The shifted tangent fibre functor

$$\text{ModuliStk}_{X/R} \to (\text{QC}_X^\vee)_{T_{X/R}[1]} \\ (X \to Y) \mapsto (T_{X/Y}[1] \to T_{X/R}[1])$$

lifts to an equivalence of $\infty$-categories

$$T_{X/-}[1] : \text{ModuliStk}_{X/R} \xrightarrow{\sim} \text{LieAlgd}_{\Delta, X/R}^{\pi}.$$

Hence every partition $\text{Lie algebroid}$ on $X$ integrates uniquely to a formal moduli stack under $X$. This equivalence is functorial with respect to almost finitely presented maps.

A key ingredient in the proof of Theorem 1.5 is a Koszul equivalence for animated rings. Before stating it, we introduce some notation.

**Definition 1.6.** An animated $R$-algebra $A$ over $B$ is said to be *complete almost finitely augmented* if

(1) the induced map $\pi_0(A) \to \pi_0(B)$ is surjective on $\pi_0$ with kernel $I$;

(2) the animated ring $A$ is derived $I$-complete;

(3) the $A$-module $B$ is almost perfect as an $A$-module.

Write $\text{CAlg}_{R/B}^{\text{an,aft,\wedge}} \subset \text{CAlg}_{R/B}^{\text{an}}$ for the full subcategory spanned by those animated $R$-algebras which are complete almost finitely augmented over $B$.
**Example 1.7.** For $R = B$ a complete local Noetherian animated ring, we can identify $\mathrm{CAlg}_{R/B}^{\mathrm{an},\mathrm{aft},\wedge}$ with the $\infty$-category of augmented animated $R$-algebras which are complete, local, and Noetherian.

In Section 3, we lift the assignment $A \mapsto (L_{B/A}^{\vee}[1] \to L_{B/R}^{\vee}[1])$ to a functor

$$ \mathcal{D} : \mathrm{CAlg}_{R/B}^{\mathrm{an}} \to \mathrm{Lie}_{\Delta, X/R}^{\pi} $$

for $X = \mathrm{Spec}(B)$ and establish the following result:

**Theorem 1.8.** The functor $\mathcal{D}$ restricts to an equivalence between the full subcategory

$$ \mathrm{CAlg}_{R/B}^{\mathrm{an},\mathrm{aft},\wedge} \subset \mathrm{CAlg}_{R/B}^{\mathrm{an}} $$

of complete almost finitely augmented objects and the full subcategory

$$ \mathrm{Lie}_{\Delta, X/R}^{\pi, \mathrm{dap}, \le 0} \subset \mathrm{Lie}_{\Delta, X/R}^{\pi} $$

of partition $\mathrm{Lie}$ algebroids $\mathfrak{g} \to L_{B/R}^{\vee}[1]$ with $\mathfrak{g}$ dually almost perfect of Tor-amplitude $\le 0$.

Theorem 1.5 and Theorem 1.8 generalise various earlier results:

(a) Let $k$ be a field of characteristic 0. For $R = k$ and $X = \mathrm{Spec}(k)$ a point, our results specialise to Lurie-Pridham's equivalence between formal moduli problems and differential graded Lie algebras [Lur11, Pri10], which is in turn extends earlier work by Deligne [Del], Drinfel'd [Dri], Hinich [Hin01], Manetti [Man09], and others.

Now assume $A$ is a Noetherian and eventually coconnective animated $k$-algebra. For $R = k$ and $X = \mathrm{Spec}(A)$, our result is contained in [Nui19b], whereas for $R = A$ and $X = \mathrm{Spec}(A)$, it was proven by Hennion [Hen18].

In characteristic 0, the monadicity of the tangent fibre functor is also proven in the work of Gaitsgory-Rozenblyum [GR17b, Chapter 8]. Note, however, that their proof crucially relies on the fact that looped Lie algebras in characteristic 0 are abelian — this fails away from characteristic 0 due to the presence of power operations.

(b) Let $k$ be a field of characteristic $p$. For $X = \mathrm{Spec}(k)$ and $R = k$ or, more generally, $R = A$ a complete local Noetherian ring with residue field $k$, our results are contained in [BM]. For $X = \mathrm{Spec}(L)$ with $L$ a finite purely inseparable field extension over $R = k$, our Koszul statement Theorem 1.8 recovers [BW20, Theorem 3.15].

1.2. Acknowledgements. We are grateful to Jiaqi Fu, Sofía Marlasca Aparicio Bertrand Toën, and Nikola Tomić for various discussions related to the content of this paper.

L.B. was supported as a Royal Society University Research Fellow at Oxford University through grant URF\R1\211075 and by the Centre national de la recherche scientifique (CNRS) at Orsay. K.M. was supported by NSF grants DMS-2152235 and DMS-21002010. J.N. was supported by the CNRS, under the programme PEPS JCJC, and the ANR project ANR-24-CE40-5367-01 ("LieDG").

## 2. AFFINE KOSZUL DUALITY

We write $\mathrm{CAlg}_R^{\mathrm{der}}$ for the $\infty$-category of derived commutative rings and $\mathrm{CAlg}_R^{\mathrm{an}}$ for the $\infty$-category of animated $R$-algebras; a summary of their theory is given in Appendix A. Note that animated rings are the same as connective derived rings:

$$ \text{`animated} = \text{derived} + \text{connective}. $$

The goal of this section will be to provide some results about derived rings that are derived $I$-adically complete for ideals $I$ that are of “sufficiently finite type”; most importantly, we prove a comonadicity result (Theorem 2.25) generalising results from [BM].
We will start by recalling a derived analogue of endowing a ring with the I-adic filtration with respect to an ideal I. Instead of working directly with ideals $I \subseteq A$, it is more convenient to work with the corresponding closed immersion $A \to A/I = B$. We will refer to objects $A \to A/I = B$ as arrows and use the notation $(A \to B)$.

**2.1. Derived filtered and graded rings.** To describe the adic filtration associated to an arrow $(A \to B)$, we first recall how one can obtain the $\infty$-category of derived filtered rings as the $\infty$-category of $\mathbb{LSym}$-algebras in the $\infty$-category of filtered modules.

**Notation 2.1.** We will consider the following $\infty$-categories:

(a) The $\infty$-category $\mathrm{Mod}_Z^{\Delta^1} := \mathrm{Fun}(\Delta^1, \mathrm{Mod}_Z)$ of maps of $\mathbb{Z}$-modules $(X \to Y)$.

(b) The $\infty$-category $\mathrm{FilMod}_Z = \mathrm{Fun}((\mathbb{Z}_{\ge 0}, \le)^{\mathrm{op}}, \mathrm{Mod}_Z)$ of non-negatively filtered $\mathbb{Z}$-modules, i.e. diagrams in $\mathrm{Mod}_Z$ of the form

$$ \dots \to F^1 V \to F^0 V = V. $$

We will denote objects of $\mathrm{FilMod}_Z$ as $F^*V$, and often denote the underlying object $F^0V$ of the filtration simply as $V$.

(c) The $\infty$-category $\mathrm{GrMod}_Z = \mathrm{Fun}((\mathbb{Z}_{\ge 0}, =), \mathrm{Mod}_Z)$ of non-negatively graded $\mathbb{Z}$-modules. We use the notation $V^\bullet$ for objects of $\mathrm{GrMod}_Z$ to emphasise the grading.

We will be interested in the following three functors relating these $\infty$-categories:

$$ \mathrm{GrMod}_Z \leftarrow \mathrm{FilMod}_Z \xleftarrow{\mathrm{ker}} \mathrm{Mod}_Z^{\Delta^1}. $$

Here $\mathrm{Gr}^\bullet(V)$ is the associated graded of a filtered module $F^*V$, i.e. $\mathrm{Gr}^i(V) = F^i V / F^{i+1}V$. The functor $\mathrm{aug}$ sends a filtered module $F^*V$ to the natural augmentation $(V \to \mathrm{Gr}^0(V))$ to its 0-th graded piece. The functor $\mathrm{ker}$ is its left adjoint and sends an arrow $(p: V \to V_0)$ to the filtered module $\dots \to 0 \to 0 \to \mathrm{fib}(p) \to V$.

We endow each of these $\infty$-categories with a $t$-structure making it a stably projectively generated $\infty$-category in the sense of Definition A.1, as well as a closed symmetric monoidal structure:

(a) We endow $\mathrm{Mod}_Z^{\Delta^1}$ with the *surjective t-structure*, in which an object $X \to Y$ is connective if $Y$ and $\mathrm{fib}(X \to Y)$ (and hence also $X$) are connective $\mathbb{Z}$-modules. In other words, $X \to Y$ is connective if $X$ and $Y$ are connective and $\pi_0(X) \to \pi_0(Y)$ is surjective. An object $X \to Y$ is coconnective if $X$ and $\mathrm{fib}(X \to Y)$ are coconnective $\mathbb{Z}$-modules. The monoidal structure is the levelwise tensor product

$$ (X \to Y) \otimes (X' \to Y') = (X \otimes X' \to Y \otimes Y'). $$

(b) We endow $\mathrm{FilMod}_Z$ with the Day convolution product and the $t$-structure in which $F^*V$ is (co)connective if each $F^iV$ is (co)connective.

(c) Likewise, $\mathrm{GrMod}_Z$ is equipped with the Day convolution product and the $t$-structure in which $V^\bullet$ is (co)connective if each $V^\bullet$ is (co)connective.

The generators of $\mathrm{Mod}_Z^{\Delta^1}$ are the surjections $\mathbb{Z}^{\oplus m+n} \to \mathbb{Z}^{\oplus n}$ and the generators of $\mathrm{FilMod}_Z$ and $\mathrm{GrMod}_Z$ are the finitely generated free abelian groups, where each generator is of a certain (filtration) weight. Each of these $\infty$-categories is then a derived algebraic context in the sense of Definition A.8 (see [Rak20, Section 4.2]) and hence comes with a notion of derived commutative algebra: these are algebras over the monad obtained by right-left extending the (strict) symmetric algebra monad on the categories of surjections of flat abelian groups and of flat filtered/graded abelian groups.
**Definition 2.2.** We define the $\infty$-categories of *animated* (resp. *derived*) *filtered rings*, *animated* (resp. *derived*) *graded rings* and *animated* (resp. *derived*) *surjections* to be the $\infty$-categories

$$ \mathrm{FilCAlg}^{\mathrm{an}} = \mathrm{CAlg}^{\mathrm{an}}(\mathrm{FilMod}_{\mathbb{Z}, \ge 0}) $$

$$ \mathrm{GrCAlg}^{\mathrm{an}} = \mathrm{CAlg}^{\mathrm{an}}(\mathrm{GrMod}_{\mathbb{Z}, \ge 0}) $$

$$ \mathrm{CAlg}^{\mathrm{an}, \Delta^1} = \mathrm{CAlg}^{\mathrm{an}}(\mathrm{Mod}_{\mathbb{Z}, \ge 0}^{\Delta^1}) $$

$$ \mathrm{FilCAlg}^{\mathrm{der}} = \mathrm{CAlg}^{\mathrm{der}}(\mathrm{FilMod}_{\mathbb{Z}}) $$

$$ \mathrm{GrCAlg}^{\mathrm{der}} = \mathrm{CAlg}^{\mathrm{der}}(\mathrm{GrMod}_{\mathbb{Z}}) $$

$$ \mathrm{CAlg}^{\mathrm{der}, \Delta^1} = \mathrm{CAlg}^{\mathrm{der}}(\mathrm{Mod}_{\mathbb{Z}}^{\Delta^1}) $$

of animated and derived commutative algebras in the sense of Definition A.9.

**Remark 2.3.** An animated surjection is simply a map of animated rings $A \to B$ which is surjective on $\pi_0$; these have been introduced by Z. Mao in [Mao21], where they are called animated pairs.

**Remark 2.4.** By Remark A.13, every derived filtered or graded ring has an underlying $\mathbb{E}_\infty$-algebra with respect to Day convolution, and modules are modules over this underlying $\mathbb{E}_\infty$-algebra.

Taking associated graded objects and augmentations commutes with the $\mathbb{L}$Sym-monads, so that we obtain functors

$$ \mathrm{GrCAlg}^{\mathrm{der}} \leftarrow \mathrm{Gr} \xrightarrow{\mathrm{aug}} \mathrm{FilCAlg}^{\mathrm{der}} \xrightarrow{\mathrm{aug}} \mathrm{CAlg}^{\mathrm{der}, \Delta^1}. $$

Both functors preserve limits, colimits and connective objects, and the functor aug sends a filtered algebra $F^*A$ to the augmentation arrow $F^0A \to \mathrm{Gr}^0(A)$.

**Definition 2.5.** The *adic filtration* functor adic: $\mathrm{CAlg}^{\mathrm{der}, \Delta^1} \to \mathrm{FilCAlg}^{\mathrm{der}}$ is the left adjoint to aug.

Since the augmentation functor is left $t$-exact, taking adic filtrations preserves connective objects. We therefore obtain a diagram of the form

$$ \mathrm{GrCAlg}^{\mathrm{an}} \leftarrow \mathrm{Gr} \xrightarrow{\mathrm{adic}} \mathrm{FilCAlg}^{\mathrm{an}} \xrightarrow{\mathrm{aug}} \mathrm{CAlg}^{\mathrm{an}, \Delta^1}. $$

where all functors are left adjoints. Note that taking the associated graded does not preserve limits at the connective level.

We give a familiar description of the adic filtration on the level of 0-th homotopy groups.

**Proposition 2.6.** Let $(A \to B)$ be an animated surjection, and $I = \ker(\pi_0 A \to \pi_0 B)$. Then $\pi_0(\mathrm{adic}(A \to B))$ is given by the commutative algebra in $\mathrm{Fun}((\mathbb{Z}, \ge), \mathrm{Mod}_{\mathbb{R}}^{\mathbb{Z}})$

$$ \dots \to \mathrm{Sym}_{\pi_0(A)}^3(I) \to \mathrm{Sym}_{\pi_0(A)}^2(I) \to I \to \pi_0(A). $$

In the situation where $I$ is flat as a $\pi_0(A)$-module, we have that $\mathrm{Sym}_{\pi_0(A)}^n(I) \simeq I^n$. Each map is then an inclusion and $\pi_0(\mathrm{adic}(A \to B))$ is given by the classical $I$-adic filtration on $\pi_0(A)$.

*Proof.* We can assume that both $A$ and $B$ are discrete. The filtered algebra $\pi_0\mathrm{adic}(A \to B)$ has the universal property that given any discrete filtered algebra $F^*R'$, any map $(A \to B) \to (R' \to \mathrm{Gr}^0(R'))$ extends uniquely to a filtered map $\pi_0\mathrm{adic}(A \to B) \to F^*R'$. Note that the category of discrete filtered algebras is equivalent to the category of commutative algebras in $\mathrm{FilMod}_{\mathbb{R}}^{\mathbb{Z}} = \mathrm{Fun}((\mathbb{Z}, \ge), \mathrm{Mod}_{\mathbb{R}}^{\mathbb{Z}})$; in other words, $F^{n+1}R' \to F^nR'$ need not be injective.

The data of a map of surjections $(A \to B) \to (R' \to \mathrm{Gr}^0(R'))$ is the same as the data of a map $(A, I) \to (R', F^1R')$ of algebras equipped with an ideal. Let $f_1: I \to F^1R'$ be induced map on ideals. This extends uniquely to a map of discrete filtered algebra $\mathrm{Sym}_A^*I \to F^*R'$, given in weight $n$ by

$$ \mathrm{Sym}_A^n(f): \mathrm{Sym}_A^n(I) \longrightarrow F^nR; \quad f^{\otimes n}(x_1 \otimes \cdots \otimes x_n) = f_1(x_1) \cdots f_1(x_n), $$
where the product is taken inside $F^*R$.

□

Finally, recall that a filtered module is complete if $\lim F^nV = 0$ and that a derived filtered commutative ring is complete if it is complete as a filtered module. Equivalently, a filtered module is complete if it is local with respect to all maps inducing equivalences on the associated graded. Since taking symmetric algebras preserves such graded equivalences, it follows that the completion of a derived filtered ring is again a derived filtered ring. In good situations, one can relate the completeness of a filtered derived algebra $F^*A$ to completeness of the underlying algebra $F^0A$ with respect to its augmentation ideal:

**Lemma 2.7.** Let $A$ be an animated filtered ring and let $I$ denote the kernel of the augmentation $\pi_0(F^0A) \to \pi_0(\mathrm{Gr}^0(A))$. Suppose that $I$ is a finitely generated ideal of $\pi_0(F^0A)$. Then the following hold:

(1) If $M$ is a complete filtered $A$-module, then the underlying $F^0A$-module $F^0M$ is $I$-complete [Lur16, Definition 7.3.1.1].

(2) Suppose that $A = \mathrm{adic}(F^0A \to \mathrm{Gr}^0(A))$ is the adic filtration of an animated surjection, and let $A^\wedge$ be its completion. If $F^0A$ is $I$-complete and $\mathrm{Gr}^0(A) \otimes_{F^0A} F^0A^\wedge \to \mathrm{Gr}^0(A)$ is an equivalence, then $A \simeq A^\wedge$.

The proof of Lemma 2.7 makes use of the following version of the “derived Nakayama lemma”:

**Lemma 2.8.** Let $A \to B$ be a map of connective $\mathbb{E}_2$-rings inducing a surjection on $\pi_0$ and suppose that $I = \ker(\pi_0(A) \to \pi_0(B))$ is a finitely generated ideal. If $M$ is a connective $I$-complete $A$-module such that $B \otimes_A M \simeq 0$, then $M \simeq 0$.

*Proof.* Suppose that there exists a smallest integer $n$ such that $\pi_n(M) \neq 0$. Then $\pi_n(M)$ is an $I$-complete $\pi_0(A)$-module [Lur16, Theorem 7.3.4.1] such that $\pi_n(M) \otimes_{\pi_0(A)} \pi_0(B) \simeq 0$. But [DG02, Proposition 6.5] would then imply that $\pi_n(M) = 0$. □

*Proof of Lemma 2.7.* If $M$ is a complete filtered module, then $F^0M \simeq \lim F^0M/F^nM$ is a limit of $I$-complete $F^0A$-modules: since each $x \in I$ is of weight $\geq 1$, $x^n: F^0M/F^nM \to F^0M/F^nM$ is null.

For (2), the map $A \to A^\wedge$ induces an equivalence on the associated graded, so that it suffices to verify that $F^0A \to F^0A^\wedge$ is an equivalence. To see this, let us start by observing that since $A$ arises as an adic filtration, Proposition 2.6 implies that

$$ \pi_0(F^0A^\wedge) \cong \lim_n \mathrm{coker}(\pi_0(F^nA) \to \pi_0(F^0A)) \cong \pi_0(F^0A)^\wedge $$

agrees with the (classical) $I$-adic completion of $\pi_0(F^0A)$. Since $\pi_0(F^0A)$ is $I$-complete, the map $\pi_0(F^0A) \to \pi_0(F^0A^\wedge)$ is surjective. Consequently, the kernel $I' = \ker(\pi_0(F^0A^\wedge) \to \pi_0(\mathrm{Gr}^0(A)))$ is the image of $I$; in particular, it is finitely generated. It follows from part (1) that $F^0A^\wedge$ is $I'$-complete and hence $I$-complete as an $A$-module. Lemma 2.8 then implies that $F^0A \to F^0A^\wedge$ is an equivalence if and only if $\mathrm{Gr}^0(A) \to \mathrm{Gr}^0(A) \otimes_{F^0A} F^0A^\wedge$, is an equivalence. This map is a section of the natural map $\mathrm{Gr}^0(A) \otimes_{F^0A} F^0A^\wedge \to \mathrm{Gr}^0(A)$. □

**2.2. Cotangent fibre.** The usual cotangent complex formalism for derived rings extends in an evident way to the graded/filtered/arrow setting; we refer to Appendix A.1 for a general discussion. We will be mainly interested in the relative cotangent complexes of augmentation maps:

**Definition 2.9 (Cotangent fibre).** For every $A$ in $\mathrm{GrCAlg}^{\mathrm{der}}$ or $\mathrm{FilCAlg}^{\mathrm{der}}$, we define the cotangent fibre to be the desuspended relative cotangent complex

$$ \cot(A) = L_{\mathrm{Gr}^0(A)/A}[-1]. $$
Here we view $Gr^0(A)$ as a derived graded/filtered algebra concentrated in weight 0. Note that $cot(A)$ is concentrated in weights $\ge 1$, i.e. $Gr^0(cot(A)) = 0$.

Likewise, for every derived arrow $(A \to B)$ we define the cotangent fibre to be the desuspended relative cotangent complex

$$cot(A \to B) = L_{B/A}[-1].$$

One can also identify this with the relative cotangent complex of the map of arrows $(A \to B) \to (B \to B)$, which is given by the $(B \to B)$-module $(cot(A \to B), 0)$.

Note that the cotangent fibre is connective for each animated graded/filtered ring and each animated surjection.

**Remark 2.10.** For any filtered derived ring $A$ and any derived arrow $(B' \to B)$, there are natural equivalences

$$ \begin{align*} \mathrm{Gr}(\cot(A)) &\simeq \cot(\mathrm{Gr}(A)) \\ \mathrm{aug}(\cot(A)) &\simeq \cot(\mathrm{aug}(A)) \\ F^{*\le 1} \cot(B' \to B) &\simeq \cot(\mathrm{adic}(B' \to B)) \end{align*} $$

where $F^{*\le 1} \cot(B' \to B)$ denotes the $B$-module $\cot(B' \to B)$ put in filtration weight 1. The first two equivalences follow from the fact that the functors $\mathrm{Gr}$ and $\mathrm{aug}$ preserve cotangent complexes (since they commute with taking derived symmetric algebras) and the last equivalence follows by adjunction from the fact that $\mathrm{aug}$ preserves square zero extensions.

**Lemma 2.11** ([BM, Remark 4.25]). Let $f: A \to B$ be a map between derived graded rings. Then the following are equivalent:

(1) $f$ is an equivalence.

(2) $A^0 \to B^0$ is an equivalence and $cot(A) \to cot(B)$ is an equivalence of graded modules.

Similarly, a map between complete animated filtered rings $f: A \to B$ is an equivalence if and only if $Gr^0(A) \to Gr^0(B)$ is an equivalence and $cot(A) \to cot(B)$ induces an equivalence on the associated graded.

*Proof.* The complete filtered case follows from the graded case since equivalences between complete derived animated rings are detected on the associated graded and because the associated graded commutes with taking cotangent complexes. For the graded case, suppose that $f$ satisfies condition (2) and that there exists a lowest weight $n \ge 1$ such that cofib$(A^n \to B^n)$ is not zero. Let $A^{\ge n}$ and $B^{\ge n}$ denote the weight $\ge n$ parts of $A$ and $B$, both of which are graded $A$-modules, and consider the natural map

$$B' = A \otimes_{\mathrm{LSym}_A(A^{\ge n})} \mathrm{LSym}_A(B^{\ge n}) \to B$$

induced by $f$ and the inclusion $B^{\ge n} \to B$. Using that $\mathrm{LSym}_A^{\ge 2}(B^{\ge n})$ and $\mathrm{LSym}_A^{\ge 2}(B^{\ge n})$ are concentrated in weights $\ge 2n$ (hence in weights $\ge n+1$), one sees that $B' \to B$ is an equivalence in weights $\le n$. It follows that $L_{B/B'}$ is zero in weights $\le n$; this can be seen for instance by using the bar resolution to resolve $B$ by free $B'$-algebras, each of which agrees with $B'$ in weight $\le n$. Consequently, in weight $n$ we find that $L_{B/A}^n \simeq L_{B'/A}^n \simeq \mathrm{cofib}(A^n \to B^n)$ is non-zero. By part (1) of Lemma A.21, this implies that $B^0 \otimes_B L_{B/A} \simeq \mathrm{cofib}(cot(A) \to cot(B))$ is non-zero, contradicting the assumption that $cot(A) \simeq cot(B)$. $\square$
**Lemma 2.12** ([BM, Proposition 4.26]). For any derived arrow $(A \to B)$, there is a natural equivalence of derived graded rings

$$ \mathbb{L}\mathrm{Sym}_B(\cot(A \to B)(1)) \xrightarrow{\sim} \mathrm{Gr}(\mathrm{adic}(A \to B)) $$

where $\cot(A \to B)(1)$ denotes the cotangent fibre concentrated in weight 1.

*Proof.* For any graded algebra $B^\bullet$, the map $\mathbb{L}\mathrm{Sym}_{B^0}(B^1) \to B^\bullet$ induces an equivalence in degrees $\le 1$ and hence induces a natural equivalence of $B^0$-modules $B^1 \to \cot(B)^1$. Since

$$ \cot(\mathrm{Gr}(\mathrm{adic}(A \to B))) \simeq \cot(A \to B)(1) $$

is concentrated in weight 1, we obtain an equivalence $\cot(A \to B) \simeq \mathrm{Gr}(\mathrm{adic}(A \to B))^1$ of $B$-modules. This induces a map of derived graded rings as in the lemma, which is an equivalence by Lemma 2.11. $\square$

**2.3. Almost finite type conditions.** We will use the cotangent fibre to study finiteness properties of animated (graded or filtered) rings. To this end, let us recall from Definition A.18 that a map of animated (graded or filtered) rings $A \to B$ is almost finitely presented if $B$ is an almost compact object in the $\infty$-category of animated (graded or filtered) $A$-algebras.

**Lemma 2.13** ([BM, Proposition 5.15]). The following are equivalent for an animated graded ring $A$:

(1) $\cot(A)$ is an almost perfect graded $A^0$-module.

(2) the map of graded algebras $A \to A^0$ is almost finitely presented.

(3) the map of graded algebras $A^0 \to A$ is almost finitely presented.

*Proof.* It is clear that (3) $\Rightarrow$ (2) $\Rightarrow$ (1), so assume that $\cot(A)$ is almost perfect. We can then define a sequence of animated graded algebras $A^0 = A_{(0)} \to A_{(1)} \to \dots$ with colimit $A$, where

$$ A_{(n)} = A_{(n-1)} \otimes_{\mathbb{L}\mathrm{Sym}_{A^0}(A_{(n-1)}^n)} \mathbb{L}\mathrm{Sym}_{A^0}(A^n) $$

and $A_{(0)} = A^0$. The natural map $A_{(n)} \to A$ is an equivalence in weight $\le n$, so that the colimit is indeed $A$. Furthermore, an inductive argument shows that $\cot(A_{(n)})$ is precisely the weight $\le n$ part of $\cot(A)$. Using this, there are equivalences of $A^0$-modules concentrated in weight $n$

$$ (A/A_{(n-1)})^n \simeq L_{A/A_{(n-1)}}^n \simeq (A^0 \otimes_A L_{A/A_{(n-1)}})^n \simeq \cot(A)^n. $$

Here the first two equivalences use that $A_{(n-1)} \to A$ is an equivalence in weight $\le n-1$. The last equivalence uses the cofibre sequence $\cot(A_{(n-1)}) \to \cot(A) \to A^0 \otimes_A L_{A/A_{(n-1)}}$ and the fact that $\cot(A_{(n-1)})$ coincides with the weight $\le n-1$ part of $\cot(A)$. By assumption, each $\cot(A)^n$ is an almost perfect $A^0$-module that becomes increasingly connective as $n$ tends to $\infty$. It follows that each $A_{(n-1)} \to A_{(n)}$ is almost finitely presented, with cofibre becoming increasingly connective as $n$ tends to $\infty$. This implies that the colimit $A$ is almost finitely presented as well. $\square$

**Proposition 2.14.** Let $A \to B$ be a map of animated filtered rings inducing a surjection on $\pi_0$. Then the following are equivalent:

(1) $B$ is an almost finitely presented filtered $A$-algebra.

(2) $L_{B/A}$ is an almost perfect filtered $B$-module and the map of filtered algebras $\pi_0(A) \to \pi_0(B)$ is of finite presentation.

(3) $B$ is almost perfect as a filtered $A$-module.

The same assertions apply to animated rings and animated graded rings.

*Proof.* This is a special case of Proposition A.19. $\square$
**Definition 2.15.** We will say that:

(1) an animated graded algebra $A$ is almost finitely augmented if it satisfies the equivalent conditions of Lemma 2.13.

(2) an animated filtered algebra $A$ is complete almost finitely augmented if it is complete and the (filtered) map $A \to \text{Gr}^0(A)$ satisfies the equivalent conditions of Proposition 2.14.

(3) an animated surjection $(A \to B)$ is complete almost finitely augmented if $\text{adic}(A \to B)$ is complete almost finitely augmented.

**Lemma 2.16.** Let $A$ be a complete filtered animated ring and let $M$ be a connective filtered $A$-module. Then $M$ is an almost perfect $A$-module if and only if it is complete and $\text{Gr}(M)$ is an almost perfect graded $\text{Gr}(A)$-module.

*Proof.* This is a special case of Lemma A.22. □

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

We now verify that this agrees with the notions introduced in [BM] (in the pointed setting):

**Lemma 2.17 ([BM, Proposition 5.28]).** Let $A$ be a complete animated filtered ring. Then $A$ is complete almost finitely augmented if and only if $\text{Gr}(A)$ is an almost finitely augmented graded ring.

*Proof.* By Proposition 2.14, $A$ is almost finitely augmented if and only $\text{Gr}^0(A)$ is almost perfect as a filtered $A$-module. By Lemma 2.16, this equivalent to $\text{Gr}^0(A)$ being an almost perfect graded $\text{Gr}(A)$-module, which in turn is equivalent to $\text{Gr}(A)$ being almost finitely augmented by Proposition 2.14. □

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

**Proposition 2.18.** Let $(A \to B)$ be an animated surjection and let $I = \ker(\pi_0(A) \to \pi_0(B))$. Then the following are equivalent:

(1) The map $(A \to B)$ is complete almost finitely augmented.

(2) The map $(A \to B)$ is almost finitely presented and $A$ is $I$-complete.

(3) $B$ is almost perfect as an $A$-module and $A$ is $I$-complete.

*Proof.* Write $A_{\text{ad}} = \text{adic}(A \to B)$. Assertions (2) and (3) are equivalent by Proposition 2.14. For (1) ⇒ (2), suppose that the adic filtration $A_{\text{ad}}$ is complete almost finitely augmented and note that $A \to B$ is equivalent to $F^0 A_{\text{ad}} \to \text{Gr}^0(A_{\text{ad}})$. To see that this map is almost finitely presented, note that $F^0$: $\text{FilCAlg}_{\text{der}} \to \text{CAlg}_{\text{der}}$ preserves almost finitely presented maps. Indeed, its right adjoint, sending a derived algebra $A'$ to the filtered algebra $\dots \to A' \to A' \to A'$, preserves filtered colimits and $n$-coconnective objects.

Since $A_{\text{ad}} \to \text{Gr}^0(A_{\text{ad}})$ is almost finitely presented, we conclude that $F^0 A_{\text{ad}} \to \text{Gr}^0(A_{\text{ad}})$ is almost finitely presented as well. Finally, $A$ is $I$-complete by Lemma 2.7.

Let us now assume that condition (2) holds and show that $A_{\text{ad}}$ is complete almost finitely augmented. The functor adic preserves almost finitely presented objects, since its right adjoint preserves filtered colimits and sends $n$-coconnective objects to $(n+1)$-coconnective objects. Since the map of animated surjections $(A \to B) \to (B \to B)$ is almost finitely presented, we conclude that $A_{\text{ad}} \to \text{Gr}^0(A_{\text{ad}})$ is almost finitely presented. It follows that $A_{\text{ad}}^{\wedge}$ is complete, connective (by the Milnor sequence) and $\text{Gr}(A_{\text{ad}}^{\wedge}) \simeq \text{Gr}(A_{\text{ad}})$ is almost finitely augmented. Lemma 2.17 therefore implies that $A_{\text{ad}}^{\wedge}$ is almost finitely augmented.

It therefore remains to show that $A_{\text{ad}} \to A_{\text{ad}}^{\wedge}$ is an equivalence. By Lemma 2.7 and the fact that $A_{\text{ad}}$ is $I$-complete, it suffices to verify that

$$f: \text{Gr}^0(A) \otimes_A F^0 A_{\text{ad}}^{\wedge} \to \text{Gr}^0(A)$$
is an equivalence. This map is an isomorphism on $\pi_0$ because $\pi_0(F^0A^\wedge) \cong \pi_0(A)^\wedge$ and $\pi_0(\mathrm{Gr}^0(A)) \cong \pi_0(A)/I$. It then suffices to verify that its cotangent complex vanishes. Unravelling the definitions, the cotangent complex of $f$ is the underlying module of the filtered module cofib(cot$(A_{\mathrm{ad}}) \to \mathrm{cot}(A_{\mathrm{ad}}^\wedge))$, which is indeed null: its associated graded is null and $\mathrm{cot}(A_{\mathrm{ad}})$ and $\mathrm{cot}(A_{\mathrm{ad}}^\wedge)$ are both complete, by Remark 2.10 and Proposition 2.18 (and since almost perfect $\mathrm{Gr}^0(A)$-modules are complete). $\square$

**Corollary 2.19** ([BM, Proposition 5.23]). Let $A$ be an animated filtered ring that is complete almost finitely augmented. Then the induced pair $F^0A \to \mathrm{Gr}^0A$ is complete almost finitely presented.

*Proof.* Since the functor aug preserves free algebras, it sends almost finitely presented maps of filtered animated rings to almost finitely presented maps of pairs. In particular, $F^0A \to \mathrm{Gr}^0A$ is almost finitely presented. Lemma 2.7 implies that $A$ is complete with respect to the kernel of $\pi_0(F^0A) \to \pi_0\mathrm{Gr}^0(A)$, and we conclude by Proposition 2.18. $\square$

**Lemma 2.20.** Let $B$ be an animated (filtered, graded) ring, let $M$ be an almost perfect connective (filtered, graded) $B$-module and let $f: B_\alpha = B \oplus_\alpha M \to B$ be a square zero extension of $B$ by $M$ classified by a map $\alpha: L_B[-1] \to M$. Then the following assertions hold:

(1) Let $\mathcal{K}$ be the thick subcategory of (filtered, graded) $B_\alpha$-modules generated by the $B_\alpha$-modules underlying an almost perfect $B$-module. Then $\mathcal{K} = \mathrm{APerf}_{B_\alpha}$.

(2) $f$ is almost of finite presentation.

*Proof.* Assuming (1), we see that $B$ is almost perfect as a $B_\alpha$-module, so that (2) follows from Proposition 2.14. To see that $\mathrm{APerf}_{B_\alpha} \subseteq \mathcal{K}$, notice that every almost perfect $B_\alpha$-module $N$ fits into a cofibre sequence where the outer terms arise from almost perfect $B$-modules

$$M \otimes_B (B \otimes_{B_\alpha} N) \cong M \otimes_{B_\alpha} N \longrightarrow N \longrightarrow B \otimes_{B_\alpha} N.$$

To see that $\mathcal{K} \subseteq \mathrm{APerf}_{B_\alpha}$, observe that any object $N \in \mathcal{K}$ is eventually connective and that $\pi_k(N)$ is a finitely generated module over $\pi_0(B_\alpha)$ if $N$ is $k$-connective. Indeed, these properties hold for almost perfect $B$-modules and are stable under (co)fibres and retracts. Consequently, every $k$-connective $N \in \mathcal{K}$ fits into a cofibre sequence $B_\alpha[k]^{\oplus n} \to N \to N'$ where $N'$ is $(k+1)$-connective. Notice that $B_\alpha$ is contained in $\mathcal{K}$, since it fits into a cofibre sequence $M \to B_\alpha \to B$ where both $M$ and $B$ are almost perfect $B$-modules. It follows that $N'$ defines a $(k+1)$-connective object in $\mathcal{K}$.

Given a module $N \in \mathcal{K}$ and $n \ge 0$, we can therefore apply this construction iteratively to obtain a cofibre sequence $K \to N \to N/K$ where $K$ is a perfect $B_\alpha$-module and $N/K$ is $n$-connective. This implies that $N$ is an almost perfect $B_\alpha$-module. $\square$

**Corollary 2.21.** Let $B$ be an animated ring, $M \in \mathrm{APerf}(B)_{\ge 0}$ and $B \oplus_\alpha M$ a square zero extension of $B$ by $M$ classified by a map $\alpha: L_B[-1] \to M$. Then the arrow $(B \oplus_\alpha M \to B)$ is complete almost finitely augmented.

*Proof.* Let $I$ be the kernel of $\pi_0(B \oplus_\alpha M) \to \pi_0(B)$. This is a finitely generated ideal, with generators arising from the generators of $\pi_0(M)$. By Proposition 2.18, it suffices to prove that $B \oplus_\alpha M \to B$ is almost of finite presentation and that $B \oplus_\alpha M$ is $I$-complete. The first condition follows from Lemma 2.20 and the second condition is immediate since $I^2 = 0$. $\square$

**Corollary 2.22.** Let $(A \to B)$ be an animated arrow which is complete almost finitely presented. For any almost perfect connective $A$-module $M$ and any square zero extension of $A$ by $M$, the arrow $(A \oplus_\alpha M \to B)$ is again complete almost finitely presented.
**Proof.** We apply Proposition 2.18. As the composite $A \oplus_{\alpha} M \to A \to B$ of two maps that are almost finitely presented, $A \oplus_{\alpha} M \to B$ is almost finitely presented. Let $I$ be the kernel of $\pi_0(A \oplus_{\alpha} M) \to \pi_0(B)$. This is generated by finitely many generators $x_i$ of $\pi_0(M)$, which square to zero, and finitely many elements $y_i$ that lift the generators of $\ker(\pi_0(A) \to \pi_0(B))$. Then $A \oplus_{\alpha} M$ is evidently $(x_i)$-complete, and it is $(y_i)$-complete since it is an extension of $A$ by $M$, both of which are $(y_i)$-complete. We conclude that $A \oplus_{\alpha} M$ is $I$-complete. $\square$

**2.4. Comonadicity of the cotangent fibre.** Throughout this section, we fix a base animated ring $R$. For each animated surjection $(A \to B)$ of $R$-algebras, the cotangent fibre $\cot(A \to B) = L_{B/A}[-1]$ is a connective $B$-module that comes equipped with a natural map

$$L_{B/R}[-1] \to \cot(A \to B).$$

The goal of this section will be to show that when $(A \to B)$ is complete almost finitely augmented, its structure can be completely encoded in terms of its cotangent fibre.

**Notation 2.23.** Let $B$ be an animated $R$-algebra. We will write

$$\mathrm{CAlg}_{R/B}^{\mathrm{an}, \wedge \mathrm{aft}} \longrightarrow \mathrm{CAlg}_{R/B}^{\mathrm{an}}$$

for the full subcategory of connective derived $R$-algebras $A$ endowed with a map $A \to B$ that exhibits $A$ as complete almost finitely augmented (Definition 2.15).

**Definition 2.24.** Let $B$ be a derived $R$-algebra. We define a *derivation* $(M, \alpha)$ on $B$ to be the datum of a $B$-module $M$, together with a map $\alpha: L_{B/R}[-1] \to M$. Write

$$\mathrm{Der}_{B/R} = (\mathrm{Mod}_B)_{L_{B/R}[-1]}/$$

for the $\infty$-category of derivations $(M, \alpha)$ on $B$. If $B$ is connective, let $\mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}} \subseteq \mathrm{Der}_{B/R}$ denote the full subcategory spanned by the derivations $(M, \alpha)$, where $M$ is an almost perfect connective.

Our goal will then be the following:

**Theorem 2.25.** Let $B$ be an animated $R$-algebra. Then the cotangent fibre defines the left adjoint in a comonadic adjunction

$$(1) \qquad \cot: \mathrm{CAlg}_{R/B}^{\mathrm{an}, \wedge \mathrm{aft}} \longrightarrow \mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}} : \mathrm{sqz}.$$

Let us first construct the desired adjoint pair. In fact, for later purposes we will construct a version of this adjunction that also encodes naturality in the animated $R$-algebra $B$.

**Construction 2.26.** Let Mod be the $\infty$-category of pairs $(B, M)$ consisting of a derived $R$-algebra and a $B$-module (cf. Construction A.17). We then have a commuting diagram as follows:

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

Here the two downwards pointing functors send a map of derived rings $A \to B$, resp. a derived ring with a module $(B, M)$, to $B$; both functors are cocartesian and cartesian fibrations. The pair
$(L, \text{triv})$ is the relative adjunction given by the cotangent complex and trivial square zero extension functors

$$L(A \to B) = (B, L_{A/R} \otimes_A B) \qquad \text{triv}(B, M) = (B \oplus M \to B).$$

We abuse notation and also write $L: \text{CAlg}_R^{\text{der}} \to \text{Mod}$ for the functor sending $A \mapsto (A, L_{A/R})$. We then define the $\infty$-category $\text{Der}_R$ of derivations to be the fibre product

$$ \begin{array}{ccc} \text{Der}_R & \to & \text{Fun}([1], \text{Mod}) \times_{\text{Fun}([1], \text{CAlg}^{\text{der}})} \text{CAlg}_R^{\text{der}} \\ \downarrow & & \downarrow^{\text{ev}_0} \\ \text{CAlg}_R^{\text{der}} & \xrightarrow{L[-1]} & \text{Mod} \times_{\text{CAlg}^{\text{der}}} \text{CAlg}_R^{\text{der}}. \end{array} $$

An object of $\text{Der}_R$ is given by a tuple $(B, M, \alpha)$ consisting of a derived ring $B$ and a derivation on $B$, i.e. a $B$-module $M$ and a map $\alpha: L_{B/R}[-1] \to M$. By adjunction, one can also identify this with a tuple $(B, M, \alpha)$ of a derived ring, a module and a map of derived pairs $\alpha: (B, B) \to (B \oplus M[1], B)$ which is the identity on the codomains.

**Remark 2.27.** The projection $\text{Der}_R \to \text{CAlg}_R^{\text{der}}$; $(B, M, \alpha) \mapsto B$ is a cartesian fibration by construction, classifying the functor

$$B \mapsto \text{Der}_{B/R}.$$

Each map $f: B \to B'$ induces a functor $f_*: \text{Der}_{B'/R} \to \text{Der}_{B/R}$ by restriction of scalars, sending a derivation $(N, \beta)$ to $f_*(N, \beta) = (f_*N, \beta')$ with

$$\beta': L_{B/R}[-1] \longrightarrow f_* L_{B'/R}[-1] \xrightarrow{f_*\beta} f_* N$$

This admits a left adjoint, sending a derivation $(M, \alpha)$ to the derivation computed as the pushout of the diagram of $B'$-modules

$$L_{B'/R}[-1] \leftarrow f^* L_{B/R}[-1] \xrightarrow{f^* \alpha} f^* M.$$

Consequently, the projection $\text{Der}_R \to \text{CAlg}_R^{\text{der}}$ is a cocartesian fibration as well.

**Remark 2.28.** Let $f: B \to B'$ be a map of derived rings with $L_{B'/B} \simeq 0$. Then the induced adjoint pair from Remark 2.27 has a simpler description: the left adjoint sends $(M, \alpha)$ to the derivation $(f^*M, f^*\alpha)$, with $f^*\alpha: L_{B'/R}[-1] \simeq f^*L_{B/R}[-1] \to f^*M$. In other words, the adjoint pair is simply given by the functors

$$f^*: \text{Der}_{B/R} \xrightarrow{\text{ev}_1} \text{Der}_{B'/R}: f_*$$

taking base change and direct image at the level of modules.

**Definition 2.29.** We will write

$$ (2) \qquad \begin{array}{ccc} \text{CAlg}_R^{\text{der}, \Delta^1} & \xrightarrow{\text{cot}} & \text{Der}_R \\ \xleftarrow{\text{sqz}} & & \\ \text{ev}_1 & & \\ \text{CAlg}_R^{\text{der}}. & & \end{array} $$

for the relative adjunction defined as follows:
(1) The right adjoint sqz sends a derivation $(B, M, \alpha)$ to the square zero extension $B \oplus_{\alpha} M \to B$ classified by $\alpha$. More precisely, this is given by the fibre product in the $\infty$-category of maps of derived $R$-algebras

$$ \begin{tikzcd} [B \oplus_{\alpha} M \to B] \arrow[r] \arrow[d] & [B \to B] \arrow[d] \\ [B \to B] \arrow[u] \arrow[r, "\alpha"] & [B \oplus M[1] \to B]. \end{tikzcd} $$

(2) The left adjoint cot is given by

$$ [A \to B] \longmapsto (B, L_{B/R}[-1] \to L_{B/A}[-1]). $$

*Proof of Theorem 2.25.* The proof is essentially the same as in [BM], although one cannot apply the axiomatic argument from loc. cit. (nor the modification in Section 6) stricto sensu. To start, notice that Corollary 2.21 implies that for each animated $R$-algebra $B$, the adjoint pair (2) restricts to an adjoint pair cot: $\mathrm{CAlg}_{R/B}^{\mathrm{an},\wedge\mathrm{aft}} \iff \mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}}$: sqz. We have to verify that this adjoint pair is comonadic. To see that cot is conservative, let $(A \to B) \to (A' \to B)$ be a map in $\mathrm{CAlg}_{R/B}^{\mathrm{an},\wedge\mathrm{aft}}$ that induces an equivalence on cotangent fibres. The induced map $\mathrm{adic}(A \to B) \to \mathrm{adic}(A' \to B)$ then induces an equivalence on the associated graded by Lemma 2.12. Since the adic filtrations are complete by definition, this implies that $A \to A'$ is an equivalence.

Next, let $A^\bullet$ be a cosimplicial diagram in $\mathrm{CAlg}_{R/B}^{\mathrm{an},\wedge\mathrm{aft}}$ such that $\cot(A^\bullet)$ admits a splitting in $\mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}}$. Let $\mathrm{Tot}(A^\bullet)$ be the totalisation of $A^\bullet$, computed in the $\infty$-category $\mathrm{CAlg}_{R/B}^{\mathrm{der}}$ of (not necessarily connective) derived $R$-algebras augmented over $B$. We need to show that $\mathrm{Tot}(A^\bullet)$ is complete almost finitely presented and that the natural map $\cot(\mathrm{Tot}(A^\bullet)) \to \mathrm{Tot}(\cot(A^\bullet))$ is an equivalence.

To see this, let $A_{\mathrm{ad}}^\bullet = \mathrm{adic}(A^\bullet \to B)$ denote the associated diagram of complete almost finitely presented derived filtered algebras. Recall that there are functors

$$ \mathrm{GrCAlg}_R^{\mathrm{der}} \leftarrow \mathrm{Gr} \quad \mathrm{FilCAlg}_R^{\mathrm{der}} \xrightarrow{\mathrm{aug}} \mathrm{CAlg}_R^{\mathrm{der}, \Delta^1} $$

preserving both limits (computed in unbounded $R$-modules) and colimits, and commuting with the functors taking cotangent fibres. In particular, $\mathrm{aug}(\mathrm{Tot}(A_{\mathrm{ad}}^\bullet)) \simeq \mathrm{Tot}(\mathrm{aug}(A_{\mathrm{ad}}^\bullet)) \simeq \mathrm{Tot}(A^\bullet)$ and there is a commuting square

$$ \begin{tikzcd} \mathrm{aug}(\cot(\mathrm{Tot}(A_{\mathrm{ad}}^{\bullet}))) \arrow[r] \arrow[d] & \mathrm{aug}(\mathrm{Tot}(\cot(A_{\mathrm{ad}}^{\bullet}))) \arrow[d] \\ \cot(\mathrm{Tot}(A^{\bullet})) \arrow[u] \arrow[r, "\sim"] & \mathrm{Tot}(\cot(A^{\bullet})). \end{tikzcd} $$

It therefore suffices to verify that:

(a) the derived filtered ring $\mathrm{Tot}(A_{\mathrm{ad}}^\bullet)$ is complete almost finitely augmented (hence connective).

(b) the filtered cotangent fibre preserves this totalisation (in unbounded filtered modules).

For (a), note that $\mathrm{Tot}(A_{\mathrm{ad}}^\bullet)$ is complete as a limit of complete algebras. Its associated graded is

$$ \mathrm{Gr}(\mathrm{Tot}(A_{\mathrm{ad}}^{\bullet})) \simeq \mathrm{Tot}(\mathrm{Gr}(A_{\mathrm{ad}}^{\bullet})) \simeq \mathrm{Tot}[\mathrm{LSym}_B(\cot(A^{\bullet} \to B))] \simeq \mathrm{LSym}_B(\mathrm{Tot}[\cot(A^{\bullet} \to B)]). $$

Here the second equivalence is Lemma 2.12 and the last equivalence uses that the augmented cosimplicial diagram $\mathrm{Tot}[\cot(A^{\bullet} \to B)] \to \cot(A^{\bullet} \to B)$ admits a splitting, so that $\mathrm{LSym}_B$ (or any
other functor) preserves this totalisation. Since Tot[cot(A• → B)] is almost perfect by assumption, we conclude that Gr(Tot(A•_ad)) is an almost finitely augmented graded algebra (in particular connective). It follows that the complete derived filtered algebra Tot(A•_ad) is connective as well, and Lemma 2.17 implies that it is complete almost finitely augmented.

For (b), note that there is a commuting square

$$
\begin{tikzcd}
\operatorname{Gr}(\cot(\operatorname{Tot}(A_{\operatorname{ad}}^{\bullet}))) \arrow[r] & \operatorname{Gr}(\operatorname{Tot}(\cot(A_{\operatorname{ad}}^{\bullet}))) \\
\arrow[r, "\sim"] & \arrow[uur, "\sim"]
\end{tikzcd}
\qquad
\begin{tikzcd}
\cot(\operatorname{Tot}[\operatorname{Gr}(A_{\operatorname{ad}}^{\bullet})]) \arrow[r] & \operatorname{Tot}[\cot(\operatorname{Gr}(A_{\operatorname{ad}}^{\bullet}))] \\
\arrow[uur, "\sim"] & \arrow[uur, "\sim"]
\end{tikzcd}
$$

since taking the associated graded commutes with totalisations (in the unbounded setting) and with cotangent complexes. Since the cosimplicial diagram Gr(A•_ad) ≃ LSym_B(cot(A• → B)) has a split totalisation (preserved by any functor), the bottom map is an equivalence. Consequently, the map

$$ (3) \qquad \cot(\mathrm{Tot}(A_{\mathrm{ad}}^{\bullet})) \to \mathrm{Tot}(\cot(A_{\mathrm{ad}}^{\bullet})) $$

induces an equivalence on the associated graded. The domain is an almost perfect connective filtered module: this follows from Proposition 2.14, since Tot(A•_ad) is complete almost finitely presented by part (a). In particular, cot(Tot(A•_ad)) is a complete filtered module. The codomain of the map (3) is a limit of complete (in fact, almost perfect connective) filtered B-modules and is therefore complete itself. We conclude that (3) is an equivalence, as desired. □

In particular, Theorem 2.25 shows that for every animated surjection (A → B) which is complete almost finitely augmented, there exists a cosimplicial diagram of square zero extensions of B whose totalisation is equivalent to A: this is simply the comonadic cobar resolution of A. For later purposes, we record the following variant of this construction:

**Corollary 2.30.** Let $A \to B \in \mathrm{Calg}_{R/B}^{\mathrm{an},\wedge,\mathrm{aft}}$ and let $M \in \mathrm{APerf}_{B,\ge 0}$. For any map $\alpha: A \to B \oplus M$ of animated R-algebras augmented over B, there exists a natural coaugmented cosimplicial diagram

$$ A \to A^{\bullet} \to B \oplus M $$

in $(\mathrm{Calg}_{R/B}^{\mathrm{an},\wedge,\mathrm{aft}})_{B \oplus M}$ with the following properties:

(1) For each $i$, the map $A^i \to B \oplus M$ is equivalent to $\mathrm{sqz}(M^i, \alpha^i) \to \mathrm{sqz}(M, 0)$ for a map $(M^i, \alpha^i) \to (M, 0)$ in $\mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}}$.

(2) The augmented cosimplicial diagram $\cot(A) \to \cot(A^{\bullet})$ in $\mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}}$ is split.

(3) $A \simeq \mathrm{Tot}(A^{\bullet})$.

*Proof.* Suppose that $U: \mathcal{D} \hookrightarrow \mathcal{C}: R$ is a comonadic adjunction. For every object $c \in \mathcal{C}$, this induces an adjoint pair $U: \mathcal{D}_{/R(c)} \hookrightarrow \mathcal{C}_c: C$ on over-categories, which is comonadic as well. It follows that every object $\alpha: d \to R(c)$ in $\mathcal{D}_{/R(c)}$ admits a comonadic cobar resolution of the form

$$ d \to (RU)^{\bullet}(d) \to R(c). $$

Here each $\alpha^n: (RU)^n(d) \to R(c)$ is the image under $R$ of the map $U(RU)^{n-1}(d) \to UR(c) \to c$, where the first map is the image of $\alpha^{n-1}$ under the forgetful functor $U$. This has the property that the augmented cosimplicial object $U(d) \to U(RU)^{\bullet}(d)$ in $\mathcal{C}_c$ is split and (hence) that $d \simeq \mathrm{Tot}((RU)^{\bullet}(d))$. We now apply this to the situation where $(U, R) = (\cot, \mathrm{sqz})$ is the comonadic adjunction (1), $c$ is the derivation $(M, 0) \in \mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}}$ and $d \to R(c)$ is the map $\alpha: A \to B \oplus M$. □
### 3. PARTITION LIE ALGEBROIDS

Recall that a Lie algebroid over a smooth scheme $Z$ over a field $k$ is a quasi-coherent sheaf $\mathfrak{g} \in \mathrm{QC}_Z^{\mathbb{Z}}$ equipped with an $\mathcal{O}_Z$-linear anchor map $\rho: \mathfrak{g} \to T_{Z/R}$ to the tangent sheaf and a $k$-linear Lie bracket $[-, -]: \mathfrak{g} \otimes_k \mathfrak{g} \to \mathfrak{g}$ satisfying the Leibniz rule. The goal of this section will be to introduce a derived (and shifted) variant of such Lie algebroids over a qcqs locally coherent derived $R$-scheme $X$, which we will refer to as partition Lie algebroids. More precisely, we will define a partition Lie algebroid to be given by a map $\rho: \mathfrak{g} \to T_{X/R}[1]$ in the $\infty$-category

$$ \mathrm{QC}_X^{\mathbb{V}} = \mathrm{Ind}(\mathrm{Coh}_X^{\mathrm{op}}) $$

of pro-coherent sheaves on $X$, together with the structure of an algebra over a certain monad

$$ \mathrm{Lie}_{\Delta, X/R}^{\pi}: (\mathrm{QC}_{X}^{\mathbb{V}})_{/T_{X/R}[1]} \longrightarrow (\mathrm{QC}_{X}^{\mathbb{V}})_{/T_{X/R}[1]} $$

Here $T_{X/R} = L_{X/R}^{\mathbb{V}}$ denotes the pro-coherent dual of the cotangent sheaf of $X$ (Example B.27). One can therefore think of $\rho: \mathfrak{g} \to T_{X/R}[1]$ as the anchor map underlying the partition Lie algebroid.

Our construction of the monad $\mathrm{Lie}_{\Delta, X/R}^{\pi}$ will be based on two ingredients:

(1) Let $\mathrm{Der}_{X/R, \ge 0}^{\mathrm{aperf}}$ denote the $\infty$-category of derivations $\alpha: L_{X/R}[-1] \to M$ with values in a connective, almost perfect quasi-coherent sheaf on $X$. We then consider the comonad

$$ \mathrm{coLie}_{\Delta, X/R}^{\pi}: \mathrm{Der}_{X/R, \ge 0}^{\mathrm{aperf}} \longrightarrow \mathrm{Der}_{X/R, \ge 0}^{\mathrm{aperf}}; \quad (M, \alpha) \longrightarrow \cot(\mathcal{O}_X \oplus_{\alpha} M) $$

sending each derivation $\alpha: L_{X/R}[-1] \to M$ to the cotangent fibre of the square zero extension $\mathcal{O}_X \oplus_{\alpha} M$.

(2) Taking pro-coherent duals determines a fully faithful functor (Lemma A.34)

$$ (-)^{\vee}: \mathrm{Der}_{X/R, \ge 0}^{\mathrm{aperf, op}} \longrightarrow (\mathrm{QC}_X^{\mathbb{V}})_{/T_{X/R}[1]} $$

Its essential image consists of maps $F \to T_{X/R}[1]$ where $F$ is dually almost perfect (Definition A.33) and of non-positive tor-amplitude (Definition A.35).

Our main aim will then be to prove the following:

**Theorem 3.1.** Let $X$ be a locally coherent qcqs derived $R$-scheme. There exists a unique extension

$$ \begin{array}{ccc} \mathrm{Der}_{X/R, \ge 0}^{\mathrm{aperf, op}} & \xrightarrow{(\mathrm{coLie}_{\Delta, X/R}^{\pi})^{\mathrm{op}}} & \mathrm{Der}_{X/R, \ge 0}^{\mathrm{aperf, op}} \\ (-)^{\vee} & \xrightarrow{\hspace{2.5cm}} & (-)^{\vee} \\ (\mathrm{QC}_X^{\mathbb{V}})_{/T_{X/R}[1]} & \xrightarrow{\mathrm{Lie}_{\Delta, X/R}^{\pi}} & (\mathrm{QC}_X^{\mathbb{V}})_{/T_{X/R}[1]} \end{array} $$

to a monad $\mathrm{Lie}_{\Delta, X/R}^{\pi}$ that preserves sifted colimits.

**Definition 3.2.** We will refer to the monad $\mathrm{Lie}_{\Delta, X/R}^{\pi}$ of Theorem 3.1 as the partition Lie algebroid monad and define the $\infty$-category of partition Lie algebroids

$$ \mathrm{LieAlgd}_{\Delta, X/R}^{\pi} = \mathrm{Alg}_{\mathrm{Lie}_{\Delta, X/R}^{\pi}} \left( (\mathrm{QC}_X^{\mathbb{V}})_{/T_{X/R}[1]} \right) $$

to be its $\infty$-category of algebras.
We will prove the affine case of Theorem 3.1 in Section 3.1. In Section 3.2 we will study the naturality of the $\infty$-category LieAlgd$^{\pi}_{\Delta,B/R}$ in the coherent animated $R$-algebra $B$. More precisely, we will prove the following in Proposition 3.25:

(a) For each map $f: B \to B'$ of that is almost finitely presented, there exists a functor

$$f^{\sharp}: \mathrm{LieAlgd}^{\pi}_{\Delta,B/R} \longrightarrow \mathrm{LieAlgd}^{\pi}_{\Delta,B'/R}.$$

(b) When $f$ is étale, the functor $f^{\sharp} = f^{*}$ is given by the usual inverse image at the level of the underlying pro-coherent module, and admits a right adjoint.

We will then use this in Section 3.3 to prove Theorem 3.1 by a descent argument.

**3.1. Partition Lie algebroids on affine schemes.** Let $B$ be an animated $R$-algebra and recall the adjoint pair from Definition 2.29

$$\mathrm{cot}: \mathrm{CAlg}_{R/B}^{\mathrm{der}} \xrightarrow{\hspace{2.5cm}} \mathrm{Der}_{B/R} : \mathrm{sqz}.$$

Here $\mathrm{Der}_{B/R}$ is the $\infty$-category of derivations $(M, \alpha) = (M, \alpha: L_{B/R}[-1] \to M)$ on $B$ (Definition 2.24).

**Notation 3.3.** Let us write $\mathrm{coLie}^{\pi}_{\Delta,B/R} := \mathrm{cot} \circ \mathrm{sqz}$ for the composite comonad

$$\mathrm{coLie}^{\pi}_{\Delta,B/R} = \mathrm{cot} \circ \mathrm{sqz}: \mathrm{Der}_{B/R} \longrightarrow \mathrm{Der}_{B/R}.$$

**Remark 3.4.** Given a derivation $(M, \alpha)$, one can endow $\mathrm{coLie}^{\pi}_{\Delta,B/R}(M, \alpha)$ with a natural decreasing filtration, as follows. Consider $L_{B/R}[-1]$ as a filtered module in weight 0 (i.e. $F^1 = 0$) and consider $M$ as a filtered module in weight 1, i.e., $F^2 M = 0$ and $F^1 M = M$. The derivation $\alpha$ then lifts to a derivation $\alpha: L_{B/R}[-1] \to F^*M$ and we obtain a filtration

$$F^* \mathrm{coLie}^{\pi}_{\Delta,B/R}(M, \alpha) = \mathrm{cot}(B \oplus_{\alpha} F^*M).$$

Note that $L_{B/R}[-1] \to F^*M$ is null on the associated graded, so that the associated graded of this filtration is equivalent to $\mathrm{coLie}^{\pi}_{\Delta,B/R}(M, 0)$.

We will use the following notation throughout this paper:

**Definition 3.5.** Let $\mathcal{C}$ be an $\infty$-category such that the inclusion $\tau_{\le n} \mathcal{C} \subseteq \mathcal{C}$ of its $n$-truncated objects admits a left adjoint $\tau_{\le n}$ for each $n \in \mathbb{N}$. We will say that a tower

$$\dots \to X_1 \to X$$

is almost eventually constant if for each $n \in \mathbb{N}$, there exists an $m$ such that

$$\tau_{\le n} X_{i+1} \to \tau_{\le n} X_i$$

is an equivalence for all $i \ge m$.

**Lemma 3.6.** The comonad $\mathrm{coLie}^{\pi}_{\Delta,B/R}: \mathrm{Der}_{B/R} \to \mathrm{Der}_{B/R}$ has the following properties:

(1) It preserves derivations $(M, \alpha)$ where $M$ is an almost perfect connective $B$-module.

(2) It preserves sifted colimits.

(3) For each finite cosimplicial diagram $(M^{\bullet}, \alpha^{\bullet})$ in $\mathrm{Der}_{B/R}$ such that $\mathrm{Tot}(M^{\bullet})$ and all $M^t$ are in $\mathrm{APerf}_{B, \ge 0}$, there is a natural equivalence

$$\mathrm{coLie}^{\pi}_{\Delta,B/R}(\mathrm{Tot}(M^{\bullet}, \alpha^{\bullet})) \simeq \mathrm{Tot}(\mathrm{coLie}^{\pi}_{\Delta,B/R}(M^{\bullet}, \alpha^{\bullet})).$$
(4) For any tower $\dots \to (M_2, \alpha_2) \to (M_1, \alpha_1)$ in $\mathrm{Der}_{B/R}$ such that $\dots \to M_2 \to M_1$ is eventually constant (i.e. the connectivity of $\mathrm{fib}(M_{n+1} \to M_n)$ tends to $\infty$), there is an equivalence

$$ \mathrm{coLie}^{\pi}_{\Delta, B/R}(\lim_n(M_n, \alpha_n)) \simeq \lim_n (\mathrm{coLie}^{\pi}_{\Delta, B/R}(M_n, \alpha_n)). $$

*Proof.* Property (1) follows from Lemma 2.20 and (2) follows from the fact that cot and sqz both preserve sifted colimits. For (3), we use the filtration from Remark 3.4. Since limits in $\mathrm{Der}_{B/R}$ are computed on the underlying $B$-modules, it then suffices to show that the natural map

$$ \mu: \cot(\mathrm{Tot}(B \oplus_{\alpha} F^*M^*)) \to \mathrm{Tot}(\cot(B \oplus_{\alpha} F^*M^*)) $$

is an equivalence of filtered modules. Proposition 2.14 and Lemma 2.20 imply that $\cot(B \oplus_{\alpha} F^*M)$ is an almost perfect filtered $B$-module whenever $M$ is almost perfect. In particular, the domain and codomain of $\mu$ are complete, and it remains to verify that the induced map on the associated graded is an equivalence. As the maps $\alpha^*$ are null on the associated graded, it will be enough to show that the functor

$$ F: \mathrm{Mod}_B \to \mathrm{GrMod}_{\mathbb{B}}^{\ge 1}; \quad M \to \cot(B \oplus M(1)) $$

preserves finite totalisations, where $\mathrm{GrMod}_{\mathbb{B}}^{\ge 1}$ is the $\infty$-category of strictly positively graded $B$-modules. Using the bar resolution, the functor $F$ can be obtained via colimits and compositions from the functors $\mathrm{LSym}_B^n: \mathrm{GrMod}_{\mathbb{B}}^{\ge 1} \to \mathrm{GrMod}_{\mathbb{B}}^{\ge 1}$. These are $n$-excisive and hence preserve finite totalisations [BM, Proposition 3.37]. Since the functors preserving finite totalisations are stable under composition and colimits, the result follows.

For (4), note that if $\mathrm{fib}(M_{n+1} \to M_n)$ is $i$-connective, then $B \oplus_{\alpha_{n+1}} M_{n+1} \to B \oplus_{\alpha_n} M_n$ has $i$-connective fibres. Consequently, its relative cotangent complex is $(i+1)$-connective [Lur16, Corollary 25.3.6.4]. Taking cotangent fibres, one finds that $\mathrm{coLie}_{\Delta}^{\pi}(M_{n+1}) \to \mathrm{coLie}_{\Delta}^{\pi}(M_n)$ has $i$-connective fibres. The tower therefore stabilises after $k$-truncation for any $k$, from which the result follows. $\square$

Let us now suppose that $B$ is a coherent animated $R$-algebra and consider the dual picture, using linear duality in the $\infty$-category $\mathrm{QC}_{\mathbb{B}}^{\vee} = \mathrm{Ind}(\mathrm{Coh}_{\mathbb{B}}^{\mathrm{op}})$ of pro-coherent $B$-modules. We refer to Appendix B for the theory of pro-coherent sheaves and just recall that there are left adjoint functors (cf. Observation A.30 and Definition A.31)

$$ \iota: \mathrm{Mod}_B \to \mathrm{QC}_{\mathbb{B}}^{\vee} $$

$$ (-)^{\vee}: \mathrm{Mod}_B \to \mathrm{QC}_{\mathbb{B}}^{\vee, \mathrm{op}}. $$

The first functor is fully faithful on eventually connective $B$-modules and the second is fully faithful on almost perfect $B$-modules. We will write $\mathrm{dPerf}_B \subseteq \mathrm{QC}_{\mathbb{B}}^{\vee}$ for the full subcategory of dually almost perfect pro-coherent $B$-modules (Definition A.33). Explicitly, one can identify $\mathrm{QC}_{\mathbb{B}}^{\vee}$ with the $\infty$-category of exact functors $F: \mathrm{Coh}_B \to \mathrm{Sp}$. The pro-coherent module $\iota(M)$ then sends $K \mapsto K \otimes_B M$ and the pro-coherent dual $M^{\vee}$ sends $K \mapsto \mathrm{Hom}_B(M, K)$.

**Notation 3.7.** Given a coherent animated $R$-algebra $B \in \mathrm{CAlg}_R^{\mathrm{an}}$, we will write $T_{B/R} = L_{B/R}^{\vee}$ for the pro-coherent dual of the cotangent complex and refer to it as the (pro-coherent) tangent complex.

By Lemma A.34, pro-coherent duality determines an equivalence

$$ (-)^{\vee}: \mathrm{Der}_{B/R}^{\mathrm{aperf,op}} = ((\mathrm{APerf}_B)_{L_{B/R}[-1]})^{\mathrm{op}} \xrightarrow{\sim} (\mathrm{dPerf}_B)_{T_{B/R}[1]} $$

Using this, we construct the partition Lie algebroid monad in the affine setting:
**Proposition 3.8.** Let $B$ be a coherent animated $R$-algebra. Then there exists a unique extension

$$
\begin{tikzcd}
\operatorname{Der}_{B/R, \ge 0}^{\text{aperf,op}} \arrow[r, "(\operatorname{coLie}_{\Delta,B/R}^{\pi})^{\text{op}}"'] & \operatorname{Der}_{B/R, \ge 0}^{\text{aperf,op}} \\
(-)^{\vee} \arrow[u,r] \arrow[d] & (\operatorname{QC}_{B}^{\vee})_{/T_{B/R}[1]} \arrow[r, "\operatorname{Lie}_{\Delta,B/R}^{\pi}", "\dots"] \arrow[d] & (\operatorname{QC}_{B}^{\vee})_{/T_{B/R}[1]} \arrow[u,r]
\end{tikzcd}
$$

to a monad $\operatorname{Lie}_{\Delta,B/R}^{\pi}$ that preserves sifted colimits.

**Definition 3.9.** Following Definition 3.2, we will write $\operatorname{LieAlgd}_{\Delta,B/R}^{\pi}$ for the $\infty$-category of partition $\operatorname{Lie}$ algebroids over $B$, that is, algebras over the monad $\operatorname{Lie}_{\Delta,B/R}^{\pi}$.

Before turning to the proof of Proposition 3.8, we record some first consequences of the above definition.

**Example 3.10.** If $B = R$ is a coherent animated ring, then the monad $\operatorname{Lie}_{\Delta,B/R}^{\pi}$ coincides with the partition $\operatorname{Lie}$ algebra monad from [BM, BCN], so that $\operatorname{LieAlgd}_{\Delta,B/R}^{\pi} \simeq \operatorname{LieAlg}_{\Delta,B}^{\pi}$ is equivalent to the $\infty$-category of partition $\operatorname{Lie}$ algebras over $B$. In particular, the free partition $\operatorname{Lie}$ algebra on a dually almost perfect $B$-module can be computed explicitly in terms of the partition complex, using the derived functor of $\Sigma_r$-invariants (see [BCN, Section 3.6])

$$
\operatorname{Lie}_{\Delta, B/R}^{\pi}(M) \simeq \sum_{r} (\tilde{C}^{*}(\Sigma|\Pi_{r}|^{\diamond}, B) \otimes_{B} M^{\otimes r})^{\Sigma_{r}}.
$$

**Remark 3.11.** For a coherent animated $R$-algebra $B$, the free partition $\operatorname{Lie}$ algebroid $\operatorname{Lie}_{\Delta,B/R}^{\pi}(M, \alpha)$ can be viewed as a deformation of the free $B$-linear partition $\operatorname{Lie}$ algebra on $M$. More precisely, $\operatorname{Lie}_{\Delta,B/R}^{\pi}(M, \alpha)$ admits a natural exhaustive increasing filtration whose associated graded is equivalent to the free partition $\operatorname{Lie}$ algebra $\operatorname{Lie}_{\Delta,B/R}^{\pi}(M, 0) \simeq \operatorname{Lie}_{\Delta,B/R}^{\pi}(M)$. Indeed, Remark 3.4 and Lemma A.22 provide such a filtration when $M$ is the pro-coherent dual of a connective almost perfect $B$-module; one then extends by sifted colimits. The resulting filtration is analogous to the filtration on free $\operatorname{Lie}$ algebroids constructed in [Kap07].

**Proposition 3.12.** Let $B$ be a coherent animated $R$-algebra. Then the following hold:

(1) There is an equivalence $(\operatorname{LieAlgd}_{\Delta,B/R}^{\pi})_{/0} \simeq \operatorname{LieAlg}_{\Delta,B}^{\pi}$.

(2) Taking the fibre of the anchor map defines a sifted colimit preserving right adjoint functor $\operatorname{LieAlgd}_{\Delta,B/R}^{\pi} \to \operatorname{LieAlg}_{\Delta,B}^{\pi}$.

It follows that one can identify $\operatorname{LieAlgd}_{\Delta,B/R}^{\pi}$ with the $\infty$-category of algebras over a monad on $\operatorname{LieAlg}_{\Delta,B}^{\pi}$ whose underlying endofunctor sends $g$ to $g \times T_{B/R}$.

*Proof.* For (1), note that the forgetful functor $(\operatorname{LieAlgd}_{\Delta,B/R}^{\pi})_{/0} \to \operatorname{QC}_{B}^{\vee}$ exhibits $(\operatorname{LieAlgd}_{\Delta,B/R}^{\pi})_{/0}$ as the $\infty$-category of algebras over a sifted colimit preserving monad on $\operatorname{QC}_{B}^{\vee}$. Unravelling the definitions, this monad coincides with $\operatorname{Lie}_{\Delta,B/R}^{\pi}$, so that the result follows from Example 3.10. Part (2) follows from part (1) and the fact that taking pullbacks along the map of partition $\operatorname{Lie}$ algebroids $0 \to T_{B/R}[1]$ preserves limits and sifted colimits. $\square$

Finally, we record the following reformulation of Theorem 2.25, using pro-coherent duality:

**Corollary 3.13.** Let $B$ be a coherent animated $R$-algebra. Then there is a fully faithful functor

$$
\mathfrak{D}: \mathrm{CAlg}_{R/B}^{\mathrm{an},\wedge\mathrm{aft}} \xrightarrow{\mathrm{cot}} \mathrm{Coalg}_{\mathrm{coLie}_{\Delta}^{\pi}}(\mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}}) \xrightarrow{(-)^{\vee}} (\mathrm{LieAlgd}_{\Delta,B/R}^{\pi})^{\mathrm{op}}
$$
whose essential image consists of the partition Lie algebroids whose underlying pro-coherent B-module is dually almost perfect of non-positive tor-amplitude.

Let us now turn to the proof of Proposition 3.8. We will show that the $\infty$-category $(QC_B^\vee)_{T_{B/R}[1]}$ is generated under sifted colimits by the full subcategory of $M \to T_{B/R}[1]$ where $M$ is dually almost perfect of nonpositive tor-amplitude (Definition A.35). To make this more precise, recall the process of adding colimits to $\infty$-categories from [Lur09, Section 5.3.6]:

**Definition 3.14.** Let $\mathcal{K}$ be a class of small $\infty$-categories. An $\infty$-category $\mathcal{V}$ is said to be $\mathcal{K}$-complete if it admits all colimits of diagrams indexed by $\infty$-categories in $\mathcal{K}$.

Suppose that $\mathcal{C}$ is a small $\infty$-category equipped with a set of colimiting cocones $\mathcal{R} = \{K_\alpha \to \mathcal{C}\}$ where each $K_\alpha \in \mathcal{K}$. We will say that a functor $j: \mathcal{C} \to \mathcal{V}$ exhibits $\mathcal{V}$ as the $\mathcal{K}$-completion of $\mathcal{C}$ relative to $\mathcal{R}$ if it is the initial $\mathcal{K}$-complete $\infty$-category with a functor from $\mathcal{C}$ that preserves the colimit diagrams in $\mathcal{R}$.

By [Lur09, Proposition 5.3.6.2], the $\mathcal{K}$-completion of $\mathcal{C}$ relative to $\mathcal{R}$ always exists and the functor $j: \mathcal{C} \to \mathcal{V}$ is fully faithful.

**Proposition 3.15.** Let $\mathcal{K}$ be a class of small $\infty$-categories, each of which is contractible. Suppose that $j: \mathcal{C} \hookrightarrow \mathcal{V}$ exhibits $\mathcal{V}$ as the $\mathcal{K}$-completion of $\mathcal{C}$ relative to a set of colimit diagrams $\mathcal{R} = \{K_\alpha \to \mathcal{C}\}$. For any object $X \in \mathcal{V}$, the induced fully faithful functor

$$j: \mathcal{C}_{/X} \dashv \mathcal{V}_{/X}$$

exhibits $\mathcal{V}_{/X}$ as the $\mathcal{K}$-completion of $\mathcal{C}_{/X}$ relative to the set $\mathcal{R}_{/X}$ of colimit diagrams $K_\alpha \to \mathcal{C}_{/X}$ whose image in $\mathcal{C}$ is contained in $\mathcal{R}$.

*Proof.* Let $\mathcal{P}_\mathcal{R}(\mathcal{C}) \subseteq \mathcal{P}(\mathcal{C})$ be the $\infty$-category of presheaves on $\mathcal{C}$ that send the colimit diagrams in $\mathcal{R}$ to limits. By the proof of [Lur09, Proposition 5.3.6.2], we can then identify $\mathcal{V} \subseteq \mathcal{P}_\mathcal{R}(\mathcal{C})$ with the smallest full subcategory that contains the representable presheaves and is closed under $\mathcal{K}$-indexed colimits. Using this, we can identify $X$ with a presheaf $X: \mathcal{C}^{\text{op}} \to \mathcal{S}$ and $\mathcal{C}_{/X}$ is the full subcategory of representable presheaves over $X$.

Now recall that the inclusion $\mathcal{C}_{/X} \hookrightarrow \mathcal{P}(\mathcal{C})_{/X}$ induces a natural equivalence $\mathcal{P}(\mathcal{C}_{/X}) \to \mathcal{P}(\mathcal{C})_{/X}$ [Lur09, Corollary 5.1.6.12] (the proof in loc. cit. also applies to non-representable $X$). This equivalence restricts to an equivalence $\mathcal{P}_\mathcal{R}(\mathcal{C})_{/X} \simeq \mathcal{P}_{\mathcal{R}_{/X}}(\mathcal{C}_{/X})$. Since the forgetful functor $\mathcal{P}_{\mathcal{R}_{/X}}(\mathcal{C}_{/X}) \to \mathcal{P}_\mathcal{R}(\mathcal{C})$ detects colimits of $\mathcal{K}$-indexed diagrams (which are all contractible), it follows that $\mathcal{V}_{/X} \subseteq \mathcal{P}_{\mathcal{R}_{/X}}(\mathcal{C}_{/X})$ is the smallest full subcategory that contains the representable presheaves and is closed under $\mathcal{K}$-indexed colimits. The result then follows from [Lur09, Proposition 5.3.6.2]. $\square$

*Proof of Proposition 3.8.* Let $\mathrm{dAPerf}_B^{\le 0} \subseteq \mathrm{QC}_B^\vee$ denote the full subcategory of dually almost perfect objects of non-positive tor-amplitude (Definition A.35). This $\infty$-category is equivalent to $\mathrm{APerf}_{B, \ge 0}^{\text{op}}$ by pro-coherent duality. It now follows from [BCN, Proposition 4.20] that $\mathrm{dAPerf}_B^{\le 0} \to \mathrm{QC}_B^\vee$ exhibits $\mathrm{QC}_B^\vee$ as the sifted colimit completion of $\mathrm{dAPerf}_B^{\le 0}$ relative to the set $\mathcal{R}_B$ of colimit diagrams in $\mathrm{dAPerf}_B^{\le 0}$ that are dual to the following diagrams in $\mathrm{APerf}_{B, \ge 0}$:

(a) Augmented cosimplicial diagrams $M^\bullet: \Delta_+ \to \mathrm{APerf}_{B, \ge 0}$ such that $M^{-1} \simeq \mathrm{Tot}(M^\bullet)$ (computed in $\mathrm{Mod}_B$).

(b) Limits $M_\infty \to \dots \to M_2 \to M_1$ of towers in $\mathrm{APerf}_{B, \ge 0}$ that are almost eventually constant (see Definition 3.5).

By Proposition 3.15, $(\mathrm{QC}_B^\vee)_{T_{B/R}[1]}$ is the sifted colimit completion of $(\mathrm{dAPerf}_B^{\le 0})_{T_{B/R}[1]}$ relative to $(\mathcal{R}_B)_{T_{B/R}[1]}$. This implies that restriction along $j$ determines an equivalence between:
(1) the (monoidal) $\infty$-category of sifted colimit preserving endofunctors of $(QC_B^\vee)_{T_{B/R}}[1]$ that preserve the full subcategory $(dA\text{Perf}_B^{\le 0})_{T_{B/R}}[1]$.

(2) the (monoidal) $\infty$-category of endofunctors of $(dA\text{Perf}_B^{\le 0})_{T_{B/R}}[1]$ that preserve all sifted colimits of the form (a) and (b) above.

Using that $(dA\text{Perf}_B^{\le 0})_{T_{B/R}}[1] \simeq \text{Der}_{B/R, \ge 0}^{\text{aperf,op}}$ by pro-coherent duality (Lemma A.34), the result then follows from Lemma 3.6. $\square$

**3.2. Base change.** We will now study the functoriality of the $\infty$-categories $\text{LieAlg}_\pi^{\Delta,B/R}$ of partition $\text{Lie algebroids}$ with respect to the animated $R$-algebra $B$. We will do this by directly construct the cartesian fibration classifying $B \to \text{LieAlg}_\pi^{\Delta,B/R}$. This is somewhat technical, and we refer to Proposition 3.25 for the final result.

We start by studying the functoriality of the $\infty$-category $(QC_B^\vee)_{T_{B/R}}[1]$ in the coherent animated $R$-algebra $B$. To do this, we will organise these $\infty$-categories into a cartesian fibration over the opposite of a certain subcategory of animated $R$-algebras.

**Notation 3.16.** Let $\text{CAlg}_R^{\text{an,coft}} \hookrightarrow \text{CAlg}_R^{\text{an}}$ denote the subcategory whose objects are coherent animated $R$-algebras and whose morphisms are maps $A \to B$ that are almost finitely presented. More generally, for any functor $M \to \text{CAlg}_R^{\text{an}}$ or $N \to \text{CAlg}_R^{\text{an,op}}$, we denote

$$ \mathcal{M}^{\text{coft}} = \mathcal{M} \times_{\text{CAlg}_R^{\text{an}}} \text{CAlg}_R^{\text{an,coft}} \qquad \mathcal{N}^{\text{coft}} = \mathcal{N} \times_{\text{CAlg}_R^{\text{an,op}}} \text{CAlg}_R^{\text{an,coft,op}}. $$

Recall the $\infty$-category $\text{Der}_{/R}$ of derivations $(A, M, \alpha) = (A, \alpha : L_{A/R}[-1] \to M)$ from Construction 2.26, and let $\text{Der}_{/R, \ge 0}^{\text{aperf}} \subseteq \text{Der}_{/R}$ denote the full subcategories of derivations $(A, M, \alpha)$ where $M$ is a connective almost perfect $A$-module.

**Lemma 3.17.** *The projection $\pi: \text{Der}_{/R, \ge 0}^{\text{aperf,coft}} \to \text{CAlg}_R^{\text{an,coft}}$ is a cocartesian fibration.*

*Proof.* Let $f: A \to B$ be a map of animated $R$-algebras. Recall from Remark 2.27 that a map $(A, M, \alpha) \to (B, N, \beta)$ covering $f$ is a cocartesian arrow in $\text{Der}_{/R}$ if the induced square of $B$-modules

$$ \begin{tikzcd} f^*L_{A/R}[-1] \arrow[r] \arrow[d] & L_{B/R}[-1] \arrow[d] \\ f^*(\alpha) \arrow[u] \arrow[r] & \beta \\ f^*M \arrow[u] \arrow[r] \arrow[d] & N \end{tikzcd} $$

is cocartesian. If $f$ is almost finitely presented, then the cofibres of the horizontal maps are almost perfect connective $B$-modules. Consequently, $N$ will be an almost perfect $B$-module if $M$ is an almost perfect connective $A$-module. We conclude that $\text{Der}_{/R, \ge 0}^{\text{aperf,coft}} \to \text{Der}_{/R}$ is stable under taking cocartesian lifts of maps in $\text{CAlg}_R^{\text{an,coft}}$, so that $\pi$ is indeed a cocartesian fibration. $\square$

**Construction 3.18.** Let $\text{Der}_{/R, \ge 0}^{\text{aperf,coft,op}} \to \text{CAlg}_R^{\text{an,coft,op}}$ be the opposite of the cocartesian fibration from Lemma 3.17. This classifies a functor $\text{CAlg}_R^{\text{an,coft}} \to \text{Cat}_\infty$ sending $A \mapsto \text{Der}_{A/R, \ge 0}^{\text{aperf,op}}$ and sending each map $f: A \to B$ to the functor

$$ (4) \qquad (L_{A/R}[-1] \to M) \longrightarrow (L_{B/R}[-1] \to L_{B/R}[-1] \oplus f^*L_{A/R}[-1] f^*M). $$

Write $\mathcal{R}_A$ for the collection of colimit diagrams in $\text{Der}_{A/R, \ge 0}^{\text{aperf,op}}$ opposite to pullback squares of derivations along $\pi_0$-surjections, as well as limits of towers of derivations that are almost eventually constant (cf. Definition 3.5).
(2) Let $(A, M, \alpha) \to (B, N, \beta)$ be a cocartesian arrow in $\text{Der}_{/R}$ such that $A, M, B, N$ are all connective and $L_{B/A} \simeq 0$. Then the induced square of animated rings

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

is cocartesian.

*Proof.* For property (1), note that the left commutative square of animated $R$-algebras gives rise to the right square of $B$-modules:

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

Since left square is a pushout of derived rings, the right vertical map $f^*L_{A/A'} \to L_{B/B'}$ is an equivalence. The left vertical map is an equivalence as well, since $f: A \to B$ was formally étale. It follows that the right square is a pushout square of $B$-modules, which means precisely that $\cot(A' \to A) \to \cot(B' \to B)$ is a cocartesian arrow in $\text{Der}_{/R}$ by Remark 2.28.

Property (2) follows from deformation theory. Indeed, consider the cartesian square of animated rings

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

and note that we need to show that the natural map $q_2^*(B \oplus_\beta N) \to B$ is an equivalence of connective $A$-modules. By [Lur16, Theorem 16.2.0.2], the above square induces an equivalence

$$ (q_1^*, q_2^*): \operatorname{Mod}_{A \oplus_{\alpha} M, \ge 0} \xrightarrow{\sim} \operatorname{Mod}_{A, \ge 0} \times_{\operatorname{Mod}_{A \oplus M[1], \ge 0}} \operatorname{Mod}_{A, \ge 0} $$

with inverse sending $(E_1, E_2, 0^*E_1 \simeq \alpha^*E_2)$ to the fibre product $q_{1*}(E_1) \times_{q_{0*}0^*(E_1)} q_{2*}(E_2)$. Since $L_{B/A} \simeq 0$, the cocartesian morphism $(A, M, \alpha) \to (B, N, \beta)$ induces an equivalence $N \simeq B \otimes_A M$ (Remark 2.28). Using this, one sees that the $A \oplus_\alpha M$-module $B \oplus_\beta N$ corresponds under the above equivalence to the triple $(B, B, 0^*(B) \simeq \alpha^*B)$, where the equivalence is induced by $\beta$. This implies that $q_2^*(B \oplus_\beta N) \to B$ is an equivalence. $\square$

**Lemma 3.21.** The comonad $\text{coLie}_\Delta^\pi$ restricts to a fibrewise comonad

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

where $\pi$ sends a tuple $(A, M, \alpha)$ to the underlying coherent animated $R$-algebra $A$. Furthermore, it preserves $\pi$-cocartesian maps covering étale maps.
*Proof.* The comonad restricts by Lemma 3.6 and preserves $\pi$-cocartesian morphisms covering étale maps by Lemma 3.20. □

We then have the following families version of Proposition 3.8:

**Proposition 3.22.** *There exists a unique extension*

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

to a monad $\text{Lie}_\Delta^\pi$ relative to $\text{CAlg}_R^{\text{an,coft,op}}$, that is, a monad in $\text{Cat}_{\infty/\text{CAlg}_R^{\text{an,coft,op}}}$, that preserves sifted colimits fibrewise.

**Remark 3.23.** The relative monad $\text{Lie}_\Delta^\pi$ restricts to a monad $(\text{QC}_B^\vee)_{T_{B/R}[1]} \to (\text{QC}_B^\vee)_{T_{B/R}[1]}$ on the fibre over a coherent animated $R$-algebra $B$. This monad preserves sifted colimits and extends the monad $(\text{coLie}_\Delta^\pi)_{B/R}^{\text{op}}$, so that it is naturally equivalent to the partition Lie algebroid monad $\text{Lie}_\Delta^\pi_{B/R}$ of Proposition 3.8. Consequently, Proposition 3.22 shows that the partition Lie algebroid monads $\text{Lie}_\Delta^\pi_{B/R}$ depend oplax naturally on the coherent animated $R$-algebra $B$.

As the proof of Proposition 3.22 is somewhat technical, we first describe the consequences of Proposition 3.22 at the level of algebras.

**Definition 3.24.** We define the $\infty$-category $\text{LieAlgd}_\Delta^{\pi}_{R}$ of partition Lie algebroids to be the $\infty$-category of algebras for the monad $\text{Lie}_\Delta^\pi: \text{QC}^\vee_{T[1]} \to \text{QC}^\vee_{T[1]}$.

Since $\text{Lie}_\Delta^\pi$ covers the identity monad on $\text{CAlg}_R^{\text{an,coft,op}}$, we obtain a natural projection

$$ \text{LieAlgd}_{\Delta, /R}^{\pi} \longrightarrow \text{Alg}_{\text{id}}(\text{CAlg}_{R}^{\text{an,coft,op}}) \simeq \text{CAlg}_{R}^{\text{an,coft,op}}. $$

For each coherent animated $R$-algebra $B$, the fibre over $B$ is equivalent to the $\infty$-category of partition Lie algebroids over $B$ from Definition 3.9.

**Proposition 3.25.** *The following assertions hold:*

(1) The forgetful functor defines a map of cartesian fibrations preserving cartesian arrows

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

In particular, every almost finitely presented map $f: A \to B$ of animated $R$-algebras induces a commuting square

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

where the bottom functor is given by $f^\sharp(\mathfrak{g}) = f^*(\mathfrak{g}) \times_{(f^*L_{A/R})^\vee[1]} T_{B/R}[1]$.
(2) If $f: A \to B$ is étale, then the square (7) is left adjointable, i.e. $f^\sharp$ intertwines the free partition Lie algebroid functors. Consequently, $f^\sharp$: LieAlg$_{\Delta,A/R}^\pi \to \text{LieAlg}_{\Delta,B/R}^\pi$ preserves colimits and is given by $g \mapsto f^*g$ on the underlying pro-coherent modules.

(3) The cartesian fibration $\text{LieAlg}_{\Delta,A/R}^{\pi} \to \text{CAlg}^{\text{an,coft,op}}$ is classified by a functor $A \mapsto \text{LieAlg}_{\Delta,A/R}^{\pi}$ satisfying étale descent.

**Remark 3.26.** Note that for a general finitely presented map $f: A \to B$, the functor $f^\sharp$ preserves sifted colimits, but it need not preserve limits and it typically does not preserve all colimits.

*Proof*. Assertion (1) is a formal property of categories of algebras over a fibrewise monad: if $p: \mathcal{Y} \to \mathcal{X}$ is a cartesian fibration and $T \in \text{End}_{/\mathcal{X}}(\mathcal{Y})$ is a fibrewise monad on $\mathcal{Y}$, then the forgetful functor $\text{Alg}_T(\mathcal{Y}) \to \mathcal{Y}$ is a map of cartesian fibrations over $\mathcal{X}$ that preserves cartesian edges. To see this, let $\mathcal{O}$ be the operad for algebras and modules and consider the map of $\mathcal{O}$-monoidal categories $(\text{End}_{/\mathcal{X}}(\mathcal{Y}), \mathcal{Y}) \to (\ast, \mathcal{X})$. Taking algebras, we then obtain

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

It follows from [Lur17, Corollary 3.2.2.3] that this is a map of cartesian fibrations over $\mathcal{X}$ preserving cartesian arrows. Taking the fibre over a fixed monad $T \in \text{Alg}_{\mathbb{E}_1}(\text{End}_{/\mathcal{X}}(\mathcal{Y}))$, one obtains that $\text{Alg}_T(\mathcal{Y}) \to \mathcal{X}$ is a cartesian fibration as well (with cartesian arrows detected in $\mathcal{Y}$).

Assertion (2) follows from Lemma 3.6 and (3) follows from (2) and étale descent for $A \mapsto \text{QC}^\vee(A)$. $\square$

We now turn to the proof of Proposition 3.22. We will need the following technical category-theoretic result about completing families of $\infty$-categories under colimits:

**Definition 3.27.** Let $\mathcal{C} \to \mathcal{B}$ be a cartesian fibration over a small $\infty$-category and let $\mathcal{K}$ be a class of small $\infty$-categories.

* Let us say that a colimit diagram $K^\Delta \to \mathcal{C}_b$ in a fibre is *conserved* if it is preserved by the base change functor $f^*: \mathcal{C}_b \to \mathcal{C}_{b'}$ for each $b' \to b$ in $\mathcal{B}$.

* We will say that $p$ is $\mathcal{K}$-complete if each fibre $\mathcal{E}_b$ is $\mathcal{K}$-complete and if all of these colimits are conserved.

* By a *collection of conserved colimit diagrams* we will mean a collection $\mathcal{R} = \bigcup_{b \in \mathcal{B}} \mathcal{R}_b$, where each $\mathcal{R}_b = \{f_\alpha: K_\alpha^\Delta \to \mathcal{C}_b\}$ is a set of conserved colimit diagrams in the fibre $\mathcal{C}_b$ and each base change functor sends $\mathcal{R}_b$ to $\mathcal{R}_{b'}$.

* Given another cartesian fibration $q: \mathcal{W} \to \mathcal{B}$, write $\text{Fun}_{\mathcal{R}/\mathcal{B}}(\mathcal{C}, \mathcal{W})$ for the full subcategory of $\text{Fun}_{\mathcal{R}/\mathcal{B}}(\mathcal{C}, \mathcal{W})$ on those functors $F: \mathcal{C} \to \mathcal{W}$ over $\mathcal{B}$ such that each map on fibres $F_b: \mathcal{C}_b \to \mathcal{W}_b$ preserves the colimits in $\mathcal{R}_b$. If $\mathcal{R}$ is the collection of all colimits indexed by diagrams from $\mathcal{K}$, we will simply write $\text{Fun}_{\mathcal{K}/\mathcal{B}}(\mathcal{C}, \mathcal{W})$.

**Proposition 3.28.** Let $\mathcal{B}$ be a small $\infty$-category and consider a fully faithful map of cartesian fibrations, preserving cartesian arrows

<figure><img src="image_18.png" /></figure>
Let $\mathcal{K}$ be a collection of small $\infty$-categories and let $\mathcal{R}$ be a set of conserved colimit diagrams, indexed by $\infty$-categories in $\mathcal{K}$. Suppose that the following condition holds:

(1) $\hat{p}: \mathcal{V} \to \mathcal{B}$ is $\mathcal{K}$-complete.

(2) Each $j_b: \mathcal{C}_b \to \mathcal{V}_b$ exhibits $\mathcal{V}_b$ as the $\mathcal{K}$-completion of $\mathcal{C}_b$ relative to $\mathcal{R}_b$.

Then for every $\mathcal{K}$-complete cartesian fibration $q: \mathcal{W} \to \mathcal{B}$, restriction along $j$ induces an equivalence of $\infty$-categories

$$ \mathrm{Fun}_{\mathcal{K}/\mathcal{B}}(\mathcal{V}, \mathcal{W}) \simeq \mathrm{Fun}_{\mathcal{R}/\mathcal{B}}(\mathcal{C}, \mathcal{W}). $$

*Proof.* We will need some details about the construction of the $\mathcal{K}$-completion $\mathcal{V}_b$ of $\mathcal{C}_b$ relative to $\mathcal{R}_b$ from the proof of [Lur09, Proposition 5.3.6.2]. Let $\mathcal{P}_{\mathcal{R}_b}(\mathcal{C}_b) \subseteq \mathcal{P}(\mathcal{C}_b)$ be the $\infty$-category of presheaves on $\mathcal{C}_b$ sending all colimits in $\mathcal{R}_b$ to limits and recall that we can identify $\mathcal{V}_b \subseteq \mathcal{P}_{\mathcal{R}_b}(\mathcal{C}_b)$ with the smallest subcategory that contains the representable presheaves and is closed under $\mathcal{K}$-indexed colimits. Let $L_b: \mathcal{P}(\mathcal{C}_b) \to \mathcal{P}_{\mathcal{R}_b}(\mathcal{C}_b)$ be the left adjoint to the inclusion and let $\mathcal{E}_b \subseteq \mathcal{P}(\mathcal{C}_b)$ be the inverse image $L_b^{-1}(\mathcal{V}_b)$; note that $\mathcal{E}_b$ is closed under $\mathcal{K}$-indexed colimits in $\mathcal{P}(\mathcal{C}_b)$. The functor $j_b: \mathcal{C}_b \to \mathcal{V}_b$ then factors as

$$ j_b: \mathcal{C}_b \xrightarrow{h_b} \mathcal{E}_b \xrightarrow{L_b} \mathcal{V}_b $$

where the first functor is the Yoneda embedding and the second is a localisation (at the $L_b$-local equivalences). All of this depends functorially on the pair $(\mathcal{C}_b, \mathcal{R}_b)$. Consequently, under unstraightening it gives rise to a diagram of cartesian fibrations and maps preserving cartesian arrows

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

where $\mathcal{E}$ and $\mathcal{V}$ are both $\mathcal{K}$-complete. By [Hin16], the map $L$ is a localisation, inverting the $L_b$-local equivalences in each fibre $\mathcal{E}_b$. Since each localisation $L_b$ preserves $\mathcal{K}$-indexed colimits and admits a fully faithful right adjoint, it follows that restriction along $L$ defines a fully faithful embedding

$$ \mathrm{Fun}_{\mathcal{K}/\mathcal{B}}(\mathcal{V}, \mathcal{W}) \hookrightarrow \mathrm{Fun}_{\mathcal{K}/\mathcal{B}}(\mathcal{E}, \mathcal{W}). $$

Write $\mathrm{Fun}'_{\mathcal{K}/\mathcal{B}}(\mathcal{E}, \mathcal{W})$ for the essential image; it consists of those functors $F: \mathcal{E} \to \mathcal{W}$ over $\mathcal{B}$ such that each $F_b: \mathcal{E}_b \to \mathcal{W}_b$ preserves $\mathcal{K}$-indexed colimits and sends the $L_b$-local equivalences to equivalences in $\mathcal{W}_b$. We now claim that the following two assertions hold:

(a) Let $F \in \mathrm{Fun}_{\mathcal{B}}(\mathcal{E}, \mathcal{W})$ such that its restriction $F|_{\mathcal{C}}$ is contained in $\mathrm{Fun}_{\mathcal{R}/\mathcal{B}}(\mathcal{C}, \mathcal{W})$. Then $F \in \mathrm{Fun}'_{\mathcal{K}/\mathcal{B}}(\mathcal{E}, \mathcal{W})$ is the $q$-left Kan extension of $F|_{\mathcal{C}}$ along $h$ if and only if $F$ sends $L$-local equivalences and preserves $\mathcal{K}$-indexed colimits fibrewise.

(b) If $F_0 \in \mathrm{Fun}_{\mathcal{R}/\mathcal{B}}(\mathcal{C}, \mathcal{W})$, then there exists a $q$-left Kan extension $F \in \mathrm{Fun}_{\mathcal{K}/\mathcal{B}}(\mathcal{E}, \mathcal{W})$ of $F_0$ along $h$.

Given (a) and (b), the result follows: restriction along the localisation $L$ yields an equivalence $\mathrm{Fun}_{\mathcal{K}/\mathcal{B}}(\mathcal{V}, \mathcal{W}) \simeq \mathrm{Fun}'_{\mathcal{K}/\mathcal{B}}(\mathcal{E}, \mathcal{W})$ and restriction along $h$ yields an equivalence $\mathrm{Fun}'_{\mathcal{R}/\mathcal{B}}(\mathcal{C}, \mathcal{W}) \simeq \mathrm{Fun}_{\mathcal{R}/\mathcal{B}}(\mathcal{C}, \mathcal{W})$ with inverse given by $q$-left Kan extension.

It remains to verify claims (a) and (b). In the case where $\mathcal{B} = *$, these assertions are proven in the proof of [Lur09, Proposition 5.3.6.2]; we will reduce to this case by a cofinality argument. To this end, let us fix a presheaf $X \in \mathcal{E}_b \subseteq \mathcal{P}(\mathcal{C}_b)$ in the fibre over $b \in \mathcal{B}$. Then the functor

$$ (8) \qquad (\mathcal{C}_b)_X := \mathcal{C}_b \times_{\mathcal{E}_b} (\mathcal{E}_b)_X \longrightarrow \mathcal{C} \times_{\mathcal{E}} \mathcal{E}_X := \mathcal{C}_X $$
is cofinal. To see this, we have to show that for each $g: c \to X$ in $C_X$ covering a map $f: b' \to b$, the $\infty$-category $((C_b)_X)_{g/}$ is contractible. Unravelling the definitions, the left fibration $((C_b)_X)_{g/} \to (C_b)_X$ classifies the functor

$$ (C_b)_X \longrightarrow \mathcal{S}; \qquad (c' \to X) \longrightarrow \mathrm{Map}_{\mathcal{E}_{b'}}(c, f^*c') \times_{\mathrm{Map}_{\mathcal{E}_{b'}}(c, f^*X)} \{g\}. $$

The colimit of this diagram is contractible: indeed, $f^*: \mathcal{E}_b \to \mathcal{E}_{b'}$ and $\mathrm{Map}_{\mathcal{E}_{b'}}(c, -): \mathcal{E}_{b'} \to \mathcal{S}$ both preserve colimits (recall that $\mathcal{E}'_b \subseteq \mathcal{P}(\mathcal{C}'_b)$ and that $f^*$ is the colimit-preserving extension of $f^*: \mathcal{C}_b \to \mathcal{C}'_b$ by definition).

Using [Lur09, Corollary 4.3.1.16] and the fact that (8) is cofinal, we find that:

(a') $F \in \mathrm{Fun}_{\mathcal{B}}(\mathcal{E}, \mathcal{W})$ is the $q$-left Kan of its restriction to $\mathcal{C}$ if and only if each $F_b: \mathcal{E}_b \to \mathcal{W}_b$ is the left Kan extension of its restriction to $\mathcal{C}_b$.

(b') $F_0 \in \mathrm{Fun}_{\mathcal{B}}(\mathcal{C}, \mathcal{W})$ admits a $q$-left Kan extension along $h$ if and only if each $F_{0b}: \mathcal{C}_b \to \mathcal{W}_b$ admits a left Kan extension along $h_b$.

Using this, it suffices to verify assertions (a) and (b) fibrewise, where they hold by the proof of [Lur09, Proposition 5.3.6.2]. $\square$

**Proof of Proposition 3.22.** This follows the same strategy as the proof of Proposition 3.8. By Proposition 3.28 and the proof of Proposition 3.8 (applied fibrewise), restriction defines an equivalence of monoidal $\infty$-categories between:

(1) endofunctors of $QC^\vee_{T[1]}$ relative to $\mathrm{Alg}_R^{\mathrm{an}, \mathrm{coft}, \mathrm{op}}$ that preserve sifted colimits fibrewise and that preserve the full subcategory $\mathrm{Der}^{\mathrm{aperf}, \mathrm{coft}, \mathrm{op}}_{R, \ge 0}$.

(2) endofunctors of $\mathrm{Der}^{\mathrm{aperf}, \mathrm{coft}, \mathrm{op}}_{R, \ge 0}$ relative to $\mathrm{Alg}_R^{\mathrm{an}, \mathrm{coft}, \mathrm{op}}$ that preserve the sifted colimit diagrams (a) and (b) from the proof of Proposition 3.8 fibrewise.

The result now follows from the fact that the monad $(\mathrm{coLie}^\pi_\Delta)^{\mathrm{op}}$ defines an algebra in the $\infty$-category (2), by Lemma 3.6. $\square$

### 3.3. Partition Lie algebroids on schemes.
We will now deduce Theorem 3.1 from Proposition 3.25 by a descent argument. To establish unicity of the partition Lie algebroid monad $\mathrm{Lie}^\pi_\Delta, X/R$, we will need the following observation:

**Lemma 3.29.** Let $X$ be a locally coherent $qcqs$ scheme, $T \in \mathrm{QC}^\vee_X$ and write $\mathcal{C}_T \subseteq (\mathrm{QC}^\vee_X)_{T/T}$ for the full subcategory of $M \to T$ where $M$ is dually almost perfect of non-positive tor-amplitude. If $\mathcal{D}$ be an $\infty$-category with sifted colimits and $F: (\mathrm{QC}^\vee_X)_{T/T} \to \mathcal{D}$ preserves sifted colimits, then $F$ is left Kan extended from its restriction to $\mathcal{C}_T \subseteq (\mathrm{QC}^\vee_X)_{T/T}$.

**Proof.** The proof of Lemma A.42 shows that the $\infty$-category $\mathrm{QC}^\vee_X \simeq \mathrm{Ind}(\mathrm{Coh}_X^{\mathrm{op}})$ is equivalent to the $\infty$-category of functors $(\mathrm{APerf}_{X, \ge 0})^{\mathrm{op}} \to \mathcal{S}$ that are reduced excisive and preserve limits of almost eventually constant towers (cf. Definition 3.5). Under this equivalence, the dually almost perfect objects of non-positive tor-amplitude simply correspond to the corepresentable functors. Consequently, for every $M \in \mathrm{QC}^\vee_X$ the canonical map $\mathrm{colim}_{N \in \mathcal{C}_M} N \to M$ is an equivalence.

Now let $M \in (\mathrm{QC}^\vee_X)_{T/M}$ and notice that there is an equivalence $(\mathcal{C}_T)_{M/M} \simeq \mathcal{C}_M$. To see that $F$ is left Kan extended from $\mathcal{C}_T$, we need to show that

$$ \mathrm{colim}_{N \in (\mathcal{C}_T)_{M/M}} F(N) \longrightarrow F(\mathrm{colim}_{N \in (\mathcal{C}_T)_{M/M}} N) \longrightarrow F(M) $$
are both equivalences. Now note that $(\mathcal{C}_T)_M \simeq \mathcal{C}_M$, so that the last map is an equivalence. Since $\mathcal{C}_M$ is sifted (it has finite coproducts) and $F$ preserves sifted colimits, the first map is an equivalence as well. □

*Proof of Theorem 3.1.* It follows directly from Lemma 3.29 that space of sifted colimit preserving extensions is either empty or contractible. It therefore remains to prove the existence of the monad $\mathrm{Lie}^\pi_{\Delta,X/R}$. Writing $\mathfrak{U}$ for the poset of affine opens of $X$, Proposition 3.25 provides a natural diagram

$$ \mathfrak{U}^{\mathrm{op}} \longrightarrow \mathrm{Fun}^{\mathrm{RAd}}(\Delta^1, \mathrm{Cat}); \quad U \longleftarrow \left( (\mathrm{QC}_U^\vee)_{T_{U/R}[1]} \xrightarrow{\mathrm{free}} \mathrm{LieAlgd}^\pi_{\Delta, \mathcal{O}(U)/R} \right) $$

to the $\infty$-category of left adjoints and right adjointable squares between them [Lur17, Definition 4.7.4.16]. The limit of this diagram then yields an adjoint pair [Lur17, Proposition 4.7.4.19]

$$ \mathrm{Lie}^\pi_\Delta: \lim_{U \in \mathfrak{U}^{\mathrm{op}}} (\mathrm{QC}_U^\vee)_{T_{U/R}[1]} \xrightarrow{\hspace{2cm}} \lim_{U \in \mathfrak{U}^{\mathrm{op}}} \mathrm{LieAlgd}^\pi_{\Delta, \mathcal{O}(U)/R}: \text{forget} $$

where the left (right) adjoint is given by the limit of the diagram of free (forgetful) functor for all affine open subspaces. Consequently, both adjoints preserve sifted colimits. Since pro-coherent sheaves satisfy descent (Corollary A.39), one obtains a sifted colimit-preserving monad

$$ \mathrm{Lie}^\pi_{\Delta, X/R}: (\mathrm{QC}_X^\vee)_{T_{X/R}[1]} \longrightarrow (\mathrm{QC}_X^\vee)_{T_{X/R}[1]}. $$

By construction, the restriction of this monad to the full subcategory $\mathrm{Der}^{\mathrm{aperf,op}}_{X/R, \ge 0}$ is the limit of the monads

$$ (\cot \circ \mathrm{triv}(-^\vee))^\vee: \mathrm{Der}^{\mathrm{aperf,op}}_{\mathcal{O}(U)/R, \ge 0} \longrightarrow \mathrm{Der}^{\mathrm{aperf,op}}_{\mathcal{O}(U)/R, \ge 0} $$

for all affine opens $U \subseteq X$. By descent for almost perfect complexes, this monad naturally equivalent to $(\cot \circ \mathrm{triv}(-^\vee))^\vee$, as desired. □

**Remark 3.30.** The proof of Theorem 3.1 shows that the $\infty$-category $\mathrm{LieAlgd}^\pi_{\Delta,X/R}$ from Definition 3.2 coincides with the definition as the limit of the categories $\mathrm{LieAlgd}^\pi_{\Delta,U/R}$ from Definition 3.9 for all affine open subschemes.
## 4. INFINITESIMAL DEFORMATIONS OF FAMILIES

In this section, we will show that the $\infty$-category of partition Lie algebroids over a coherent animated $R$-algebra $B$ is equivalent to the $\infty$-category of deformation functors, or formal moduli problems in the terminology of Lurie [Lur16, Chapter 12], defined on a certain $\infty$-category of Artinian extensions of $B$. In Section 4.4, we will use this to show that every partition Lie algebroid over $B$ has an underlying $R$-linear partition Lie algebra.

**4.1. Artinian extensions.** We begin by introducing the following variant of Artin local rings relative to $B$:

**Definition 4.1 (Artinian extensions).** Let $B$ be a coherent animated $R$-algebra. We will say that a map of animated $R$-algebras $A \to B$ is an Artinian extension if it satisfies the following three conditions:

(1) $A \to B$ is a nilpotent extension, that is, $\pi_0(A) \to \pi_0(B)$ is surjective with nilpotent kernel.

(2) $A \to B$ is almost of finite presentation.

(3) the fibre of $A \to B$ is eventually coconnective.

A map of animated $R$-algebras $A \to B$ is said to be an almost Artinian extension if it satisfies conditions (1) and (2). We will write $\text{Art}_{R/B} \subseteq A\text{Art}_{R/B} \subseteq C\text{Alg}_{R/B}^{\text{an}}$ for the full subcategory on the Artinian extensions of $B$.

**Remark 4.2.** Let $A$ be a discrete commutative ring and let $I \subseteq A$ be a finitely presented nilpotent ideal such that $A/I$ is a coherent ring. Let us recall the following facts:

(1) $A$ is coherent [Gla89, Theorem 4.1.1].

(2) For every $k \ge 0$, the ideal $I^k$ is finitely generated, and hence finitely presented over $A$.

(3) If $M$ is a finitely presented $A$-module, then $I^k M / I^{k+1} M$ is a finitely presented $A$-module and hence a finitely presented $A/I$-module [Gla89, Theorem 2.1.8, Theorem 2.2.1].

**Proposition 4.3.** Let $A' \to A$ be a map in $C\text{Alg}_{R/B}^{\text{an}}$ with $A \in \text{Art}_{R/B}$. Then the following assertions are equivalent:

(1) $A' \in \text{Art}_{R/B}$ and $A' \to A$ is a nilpotent extension.

(2) There exists a finite chain of maps $A' = A_n \to A_{n-1} \to \dots \to A_0 = A$ in $C\text{Alg}_{R/B}^{\text{an}}$ where each $A_i \to A_{i-1}$ is a square zero extension by a connective coherent $B$-module, viewed as an $A_{i-1}$-module via the canonical map $A_{i-1} \to B$.

*Proof.* If $A' \to A$ satisfies condition (2), it is clearly a nilpotent extension. We will prove by induction on $i$ that the map $A_i \to B$ exhibits $B$ as an almost perfect $A_i$-module. This is evident for $i=0$. Assume that $B \in \text{APerf}_{A_{i-1}}$, so that every almost perfect $B$-module is almost perfect as an $A_{i-1}$-module. It follows that $A_i \to A_{i-1}$ is a square zero extension by an almost perfect $A_{i-1}$-module. Lemma 2.20 now implies that every almost perfect $A_{i-1}$-module is almost perfect as an $A_i$-module, so that $B \in \text{APerf}_{A_i}$ as well. Proposition 2.14 now implies that $A_i \to B$ is almost finitely presented. Since $A_i \to B$ is clearly a nilpotent extension with eventually coconnective fibre, we conclude that $A_i \to B$ is an Artinian extension, so that (1) follows.

For the converse, we first suppose that $A' \to A$ is a nilpotent extension with $A' \in \text{Art}_{R/B}$ such that $A \in \text{APerf}_{A'}$. Write

$$I = \ker(\pi_0(A') \to \pi_0(B)) \quad \text{and} \quad J = \ker(\pi_0(A') \to \pi_0(A)).$$

Both $I$ and $J$ are finitely presented nilpotent ideals, since $A' \to A$ and $A' \to B$ are almost finitely presented. In particular, $\pi_0(A')$ is a coherent ring by Remark 4.2, and likewise for $\pi_0(A)$.
We now fix $m \ge 0$ such that $I^m = 0$ and $t \ge 1$ such that cofib($A' \to A$) is $t$-coconnective; such $t$ exists since $A' \to B$ and $A \to B$ both have eventually coconnective fibres. We will construct a sequence of animated $A'$-algebras $A' = A_{(t+1)} \to \dots \to A_{(3)} \to A_{(2)} \to A_{(1)} \to A_{(0)} = A$ with the following properties:

(a) For each $i$, cofib($A \to A_{(i)}$) is a $i$-connective and $t$-coconnective almost perfect $A'$-module.

(b) Each $A_{(i+1)} \to A_{(i)}$ decomposes into $m$ square zero extensions, each by the $i$-fold suspension of a finitely presented discrete $\pi_0(B)$-module.

Since finitely presented discrete $\pi_0(B)$-modules are coherent as $B$-modules (Definition A.41), this provides the desired decomposition.

To begin, we define $A_{(1)} = \pi_0(A') \times_{\pi_0(A)} A$ and note that $A_{(1)} \to A_{(0)} = A$ is the base change of the sequence of square zero extensions

$$ \pi_0(A') \to \pi_0(A')/I^{m-1}J \to \dots \to \pi_0(A')/IJ \to \pi_0(A')/J = \pi_0(A) $$

by the finitely presented discrete $\pi_0(B)$-modules $I^kJ/I^{k+1}J$ (by Remark 4.2). Note that $\pi_0(A') \cong \pi_0(A_{(1)})$ and that cofib($A' \to A_{(1)}$) is 1-connective and $t$-coconnective, as desired.

For $i \ge 1$, suppose that we have constructed $A_{(i)}$ and let $C = \text{cofib}(A' \to A_{(i)}) \in \text{APerf}_{A'}$. Since $C$ is $i$-connective and $\pi_0(A') \cong \pi_0(A_{(i)})$, $L_{A_{(i)}/A'}$ is $i$-connective and $\pi_i(L_{A_{(i)}/A'}) \cong \pi_i(C)$ is a finitely presented $\pi_0(A')$-module. We define $A_{(i+1)}$ to be the square zero extension of $A_{(i)}$ classified by the map $L_{A_{(i)}/A'} \to \pi_i(C)[i]$. Then cofib($A' \to A_{(i+1)}$) $\cong \tau_{\ge i+1}C$ is $(i+1)$-connective and $t$-coconnective. The finitely presented $\pi_0(A')$-module $\pi_i(C)$ fits into a sequence of extensions

$$ \pi_i(C) \to \pi_i(C)/I^{m-1} \to \dots \to \pi_i(C)/I^2 \to \pi_i(C) \otimes_{\pi_0(A')} \pi_0(B) $$

by finitely presented discrete $\pi_0(B)$-modules. This implies that $A_{(i+1)} \to A_{(i)}$ has property (b).

It now remains to verify by induction that $A_{(i+1)}$ is indeed an almost perfect $A'$-module for all $i \ge -1$. For $i = -1$, this holds by hypothesis and for $i \ge 0$ we constructed $A_{(i+1)}$ as an iterated extension of $A_{(i)}$ by coherent $B$-modules, all of which are in $\text{APerf}_A$ by inductive hypothesis.

Finally, let us show that if $A' \to A$ is a map in $\text{Art}_{R/B}$, then $A \in \text{APerf}_{A'}$. To this end, note that $A \to B$ exhibits $B$ as an almost perfect $A$-module because $A \to B$ is an Artinian extension. The previous part of the proof therefore shows that $A$ is an iterated square zero extension of $B$ by coherent $B$-modules. Because $A' \in \text{Art}_{R/B}$, these are all almost perfect as $A'$-modules, so that $A \in \text{APerf}_{A'}$ as well. $\square$

**Corollary 4.4.** The subcategory $\text{Art}_{R/B} \subset \text{CAlg}_{R/B}^{\text{an}}$ is the smallest full subcategory satisfying the following two conditions:

(1) It contains the terminal object $B \to B$.

(2) Given $M \in \text{Coh}_{B, \ge 0}$ and a pullback square exhibiting $A_\eta$ as the square zero extension of $A$ by $M$

$$ \begin{tikzcd} (9) \arrow[r, "A_\eta"] \arrow[d, "\eta"] & B \arrow[d] \\ & A \arrow[r, "\cong"] \arrow[d] & B \oplus M[1] \arrow[d] \end{tikzcd} $$

we have that $A_\eta \in \text{Art}_{R/B}$ as soon as $A \in \text{Art}_{R/B}$.

Furthermore, for each diagram $A_1 \to A_0 \leftarrow A_2$ in $\text{Art}_{R/B}$ where one arrow is a nilpotent extension, the pullback is again contained in $\text{Art}_{R/B}$.
**Proof.** The first assertion follows from Proposition 4.3 applied to $A = B$: this shows that $A' \to B$ is an Artinian extension if and only if it decomposes into a finite sequence of square zero extensions by connective coherent $B$-modules. The second assertion follows from Proposition 4.3 by decomposing the nilpotent extension into square zero extensions by coherent $B$-modules and using the first assertion.

□

**Corollary 4.5.** Let $B$ be a coherent animated ring and let $A \to B$ be an Artinian extension. Then $A$ is coherent as well.

**Proof.** By Proposition 4.3, it suffices to treat the case where $A$ is a square zero extension of $B$ by a coherent $B$-module. In this case, Lemma 2.20 implies that the truncation of an almost perfect $A$-module remains almost perfect.

□

**Corollary 4.6.** Let $B$ be a coherent animated ring and let $A \to A'$ be a map of Artinian extensions of $B$. Then $A'$ is almost finitely presented as an animated $A$-algebra.

**Proof.** By Proposition A.19, it suffices to verify that $A'$ is an almost perfect $A$-module. Since $A'$ arises from $B$ by square zero extensions by coherent $B$-modules, it suffices to verify that $B$ is almost perfect as an $A$-module, which holds by assumption (and Proposition A.19).

□

The $\infty$-category of almost Artinian extensions $A\mathrm{Art}_{R/B}$ is essentially obtained from the $\infty$-category of Artinian extensions by adding limits of relative Postnikov towers.

**Lemma 4.7.** Let $\mathrm{CAlg}_{R/B}^{\mathrm{an,sur}} \subseteq \mathrm{CAlg}_{R/B}^{\mathrm{an}}$ denote the full subcategory of animated surjections $A \to B$. For each $(A \to B) \in \mathrm{CAlg}_{R/B}^{\mathrm{an,sur}}$, there exists an initial map $(A \to B) \to (\tau_{\le n}(A/B) \to B)$ to an animated surjection with $n$-coconnective fibre.

We will refer to $\tau_{\le n}(A/B)$ as the *fibrewise $n$-truncation* of $A$ over $B$.

**Proof.** Consider the full subcategory $\mathcal{C} \subset \mathrm{Fun}(\Delta^1, \mathrm{CAlg}_R^{\mathrm{an}})$ spanned by the animated surjections, and recall that $\mathcal{C}$ has compact projective generators of the form $R[x_1, \dots, x_{n+m}] \to R[x_1, \dots, x_n]$. Using this, one sees that an object $(A \to B)$ of $\mathcal{C}$ is $n$-truncated if and only if $A$ and $\mathrm{fib}(A \to B)$ are $n$-truncated (which implies that $B$ is $(n+1)$-truncated). Furthermore, each animated surjection $(A \to B)$ admits an $n$-truncation $\tau_{\le n}(A \to B)$ whose domain and fibre are $\tau_{\le n}A$ and $\tau_{\le n}\mathrm{fib}(A \to B)$; write $\tau'_{\le n}B$ for the codomain, which need not be the $n$-truncation of $B$. Given an animated surjection $A \to \bar{B}$, we then define

$$ \tau_{\le n}(A/B) = \tau_{\le n}A \times_{\tau'_{\le n}B} B. $$

This determines a functor $\tau_{\le n}: \mathrm{CAlg}_{R/B}^{\mathrm{an,sur}} \to \mathrm{CAlg}_{R/B}^{\mathrm{an,sur}}$ equipped with a natural transformation $\mathrm{id} \to \tau_{\le n}$. By construction, the map $\mathrm{fib}(A \to B) \to \mathrm{fib}(\tau_{\le n}(A/B) \to B)$ exhibits its target as the $n$-truncation of the domain. In particular, this implies that $\tau_{\le n}$ is a localisation and that the $\tau_{\le n}$-local objects are the animated surjections with $n$-coconnective fibre.

□

**Lemma 4.8.** Let $A \to B$ be an animated surjection. Then the following are equivalent:

(1) $A \to B$ is an almost Artinian extension.

(2) For each $n \ge 0$, the fibrewise $n$-truncation $\tau_{\le n}(A/B) \to B$ is an Artinian extension.

Consequently, $A\mathrm{Art}_{B/R} \subseteq \mathrm{CAlg}_{R/B}^{\mathrm{an}}$ is closed under square zero extensions by almost perfect connective $B$-modules.
**Proof.** Note that $\pi_i(A) \cong \pi_i(\tau_{\le n}(A/B))$ for all $i \le n$. Assuming (2), it therefore follows that $A \to B$ is a nilpotent extension and that $A$ is a coherent animated ring. Since each $\pi_i(B)$ is a finitely presented discrete module over $\pi_0(A) \cong \pi_0(\tau_{\le 0}(A/B))$, we conclude that $B$ is an almost perfect $A$-module, so that $A \to B$ is almost finitely presented.

Assuming (1), the argument from Proposition 4.3 shows that there exists a tower $\cdots \to A_n \to A_{n-1} \to \cdots \to B$ of Artinian extensions such that $\mathrm{fib}(A \to A_n)$ becomes increasingly connective. In particular, choosing $n$ large enough we see that $\pi_0(A) \cong \pi_0(A_n)$ is a coherent ring and that each $\pi_i(A) \cong \pi_i(A_n)$ is a finitely presented $\pi_0(A)$-module. It follows that $A$ is coherent, so that each truncation $\tau_{\le n}$ $\mathrm{fib}(A \to B)$ is an almost perfect $A$-module. Using this and Proposition 2.14, one sees that each $A \to \tau_{\le n}(A/B)$ is almost finitely presented. It follows that $\tau_{\le n}(A/B) \to B$ is almost finitely presented, and it is clearly a nilpotent extension with $n$-coconnective fibre. $\square$

**Definition 4.9.** Let $\mathrm{Art}_R \hookrightarrow \mathrm{AArt}_R \hookrightarrow \mathrm{Fun}(\Delta^1, \mathrm{CAlg}_R^{\mathrm{an}})$ denote the subcategories whose:

(1) objects are (almost) Artinian extensions $A \to B$ where $B$ is coherent.

(2) morphisms $(\tilde{f}, f): (A \to B) \to (A' \to B')$ are commuting squares in which $f$ is étale.

$$ \begin{array}{ccc} A & \xrightarrow{\tilde{f}} & A' \\ \downarrow & & \downarrow \\ B & \xrightarrow{f} & B' \end{array} $$

**Proposition 4.10.** Let $\mathrm{CAlg}_R^{\mathrm{an},\mathrm{coh},\mathrm{et}} \hookrightarrow \mathrm{CAlg}_R^{\mathrm{an}}$ denote the subcategory of coherent animated $R$-algebras and étale maps between them. Then the codomain projections define cocartesian fibrations

$$ \mathrm{Art}_R \longrightarrow \mathrm{CAlg}_R^{\mathrm{an},\mathrm{coh},\mathrm{et}} \qquad \mathrm{AArt}_R \longrightarrow \mathrm{CAlg}_R^{\mathrm{an},\mathrm{coh},\mathrm{et}}. $$

Furthermore, for each étale map $f: B \to B'$, the induced functor $f^*: \mathrm{AArt}_{R/B} \to \mathrm{AArt}_{R/B'}$ preserves pullbacks along nilpotent extensions, as well as limits of fibrewise Postnikov towers.

**Proof.** A commuting square as in Definition 4.9 defines a locally cocartesian arrow in $\mathrm{AArt}_R$ if $A \to A'$ is étale. Indeed, any map of Artinian extensions $(A \to B) \to (A'' \to B')$ covering the map $B \to B'$ factors uniquely over $(A' \to B')$ because the étale map $A \to A'$ has the unique lifting property against the nilpotent extension $A'' \to B'$. It is clear that these types of locally cocartesian arrows are stable under composition.

It therefore remains to verify that for every Artinian extension $(A \to B)$ and every étale map $f: B \to B'$, there exists a lift $(\tilde{f}, f): (A \to B) \to (A' \to B')$ such that $\tilde{f}: A \to A'$ is étale. For every animated ring $B$, the functor $B' \mapsto \pi_0(B) \otimes_B B'$ defines an equivalence between the categories of étale $B$-algebras and étale $\pi_0(B)$-algebras. In turn, the functor $B'' \mapsto B'' \otimes_{\pi_0(B)} \pi_0(B)/I$ determines an equivalence between the (ordinary) categories of étale $\pi_0(B)$-algebras and $\pi_0(B)/I$-algebras whenever $I$ is a nilpotent ideal [Sta19, Tag 039R]. Using this, one sees that there is a unique lift of $f$ to an étale map $\tilde{f}: A \to A'$.

To see that the resulting map $A' \to B'$ is (almost) Artinian, note that it is almost of finite presentation since both $A'$ and $B'$ are almost of finite presentation over $A$. Furthermore, $\ker(\pi_0(A') \to \pi_0(B')) \cong \pi_0(A') \otimes_{\pi_0(A)} \ker(\pi_0(A) \to \pi_0(A'))$ is the base change of a nilpotent ideal along a flat map, and hence a nilpotent ideal itself. Likewise, $\mathrm{fib}(A' \to B') \simeq A' \otimes_A \mathrm{fib}(A \to B)$ is eventually coconnective if $\mathrm{fib}(A \to B)$ is eventually coconnective, as $A \to A'$ is flat.

Finally, consider the change of fibre functor $f^*: \mathrm{AArt}_{B/R} \to \mathrm{AArt}_{B'/R}$. For any map $A_1 \to A_2$ of (almost) Artinian extensions of $B$, we have that $A_i \to f^*(A_i)$ is the unique étale map covering
f. This implies that $f^*(A_2) \simeq f^*(A_1) \otimes_{A_1} A_2$. Using this, one readily verifies that $f^*$ preserves pullbacks along nilpotent extensions. Furthermore, the fact that $A \to f^*(A)$ is flat implies that $f^*(A) \otimes_A \tau_{\le n}(A/B) \simeq \tau_{\le n}(f^*(A)/B')$, so that $f^*$ preserves limits of fibrewise Postnikov towers as well. $\square$

**4.2. Formal moduli problems.** We recall the notion of a formal moduli problem [Lur16, Chapter 12]:

**Definition 4.11.** A functor $X: \mathrm{Art}_{R/B} \to S$ is a formal moduli problem if it satisfies the following two conditions:

(1) $X(B) \simeq *.$

(2) For any pullback square in $\mathrm{Art}_{R/B}$ of the form

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

where $A_0 \to A_{01}$ is a nilpotent extension, $X(A) \to X(A_0) \times_{X(A_{01})} X(A_1)$ is an equivalence. We will write $\mathrm{Moduli}_{B/R}$ for the $\infty$-category of formal moduli problems.

**Remark 4.12.** Using Proposition 4.3, one sees that $X: \mathrm{Art}_{R/B} \to S$ is a formal moduli problem if and only if $X(B) \simeq *$ and $X$ preserves each pullback square (9) describing a square zero extension of $A \in \mathrm{Art}_{R/B}$ by a coherent connective $B$-module.

**Example 4.13.** Let $B' \to B$ be a map of derived $R$-algebras. We will write $\mathrm{Spf}(B')$ for the corresponding corepresentable formal moduli problem

$$ \mathrm{Spf}(B') : \mathrm{Art}_{R/B} \longrightarrow S; \quad A \longleftarrow \mathrm{Map}_{\mathrm{CAlg}_{R/B}^{\mathrm{der}}}(B', A). $$

This defines the right adjoint in an adjoint pair

$$ \mathcal{O}: \mathrm{Moduli}_{B/R} \xrightarrow{\leftarrow\rightleftharpoons} \mathrm{CAlg}_{R/B}^{\mathrm{der,op}} : \mathrm{Spf} $$

where $\mathcal{O}(X)$ can be viewed as the global sections of the formal moduli problem $X$.

**Lemma 4.14.** *Restriction along $i$: $\mathrm{Art}_{R/B} \hookrightarrow \mathrm{AArt}_{R/B}$ defines an equivalence between $\mathrm{Moduli}_{R/B}$ and the $\infty$-category of functors $X: \mathrm{AArt}_{R/B} \to S$ satisfying the following conditions:*

(1) $X(B) \simeq *.$

(2) $X$ preserves pullbacks along nilpotent extensions in $\mathrm{AArt}_{R/B}$.

(3) $X$ preserves limits of fibrewise Postnikov towers.

*Proof.* Lemma 4.8 implies that for any functor $Y: \mathrm{Art}_{R/B} \to S$, its right Kan extension along $i$ is given by $i_*Y(A) = \lim_n Y(\tau_{\le n}(A/B))$. The result follows readily from this. $\square$

Recall that every formal moduli problem has a tangent complex:

**Construction 4.15.** Let $\mathrm{Der}_{B/R, \ge 0}^{\mathrm{coh}}$ denote the $\infty$-category of derivations $\alpha: L_{B/R}[-1] \to M$ where $M$ is a connective coherent $B$-module.

Given a formal moduli problem $X: \mathrm{Art}_{R/B} \to S$, write

$$ T_{B/X}[1]: \mathrm{Der}_{B/R, \ge 0}^{\mathrm{coh}} \longrightarrow S; \quad (M, \alpha) \longleftarrow X(B \oplus_{\alpha} M) $$
for the restriction of $X$ to the square zero extensions of $B$. Because $X$ is a formal moduli problem, this functor preserves pullbacks of $\pi_0$-surjections. Consequently, Proposition 3.15 and Lemma A.42 (or Example 4.25 below) imply that $T_{B/X}[1]$ is classified by an object of $\mathrm{QC}^\nabla_{B/T_{B/R}[1]}$ that we will refer to as the *tangent fibre* of the formal moduli problem $X$, and denote by $T_{B/X}[1]$ as well.

**Example 4.16.** Let $B' \to B$ be a map of derived $R$-algebras and let $\mathrm{Spf}(B')$ be the associated corepresentable formal moduli problem (Example 4.13). The tangent fibre of $\mathrm{Spf}(B')$ is then given by the functor $\mathrm{Der}^{\mathrm{coh}}_{B/R, \ge 0} \to S$ corepresented by the cotangent complex $L_{B/R}[-1] \to L_{B/B'}[-1]$. Consequently, $T_{B/\mathrm{Spf}(B')}$ is the pro-coherent dual of the cotangent fibre.

Finally, we record the functoriality of the $\infty$-category Moduli$_{B/R}$ with respect to étale maps:

**Construction 4.17.** Write $\mathcal{A} \to \mathcal{B}$ for the cocartesian fibration $\mathrm{Art}_R \to \mathrm{CAlg}^{\mathrm{an},\mathrm{coh},\mathrm{et}}_R$ from Proposition 4.10 and consider the relative functor $\infty$-category $\mathrm{Fun}_{\mathcal{B}}(\mathcal{A}, S) \to \mathcal{B}$. Recall that this is the $\infty$-category over $\mathcal{B}$ defined by the universal property that functors $\mathcal{D} \to \mathrm{Fun}_{\mathcal{B}}(\mathcal{A}, S)$ are naturally equivalent to pairs consisting of a functor $\mathcal{D} \to \mathcal{B}$ and a functor $\mathcal{A} \times_{\mathcal{B}} \mathcal{D} \to S$. Then projection $\mathrm{Fun}_{\mathcal{B}}(\mathcal{A}, S) \to \mathcal{B}$ is a cartesian and cocartesian fibration, classifying the functor sending $B \mapsto \mathrm{Fun}(\mathrm{Art}_{R/B}, S)$ and a map $f: B \to B'$ to restriction and left Kan extension along $\mathrm{Art}_{R/B} \to \mathrm{Art}_{R/B'}$ [GHN17, Proposition 7.3]. We write

$$ \mathrm{Moduli}_{R} \subseteq \mathrm{Fun}_{\mathcal{B}}(\mathcal{A}, S) $$

for the full subcategory spanned by the tuples $(B, X: \mathrm{Art}_{R/B} \to S)$ where $X$ is a formal moduli problem. Since each $\mathrm{Art}_{R/B} \to \mathrm{Art}_{R/B'}$ preserves pullbacks along nilpotent extensions, the projection Moduli$_R \to \mathrm{CAlg}^{\mathrm{an},\mathrm{coh},\mathrm{et}}_R$ is a cartesian and cocartesian fibration.

**4.3. Deformation theory.** Let $B$ be a coherent animated $R$-algebra. Our goal will be to show that taking the tangent complex of a formal moduli problem (Construction 4.15) refines to an equivalence

$$ T_{B/-}[1]: \mathrm{Moduli}_{B/R} \xrightarrow{\sim} \mathrm{LieAlgd}_{\Delta, B/R}^{\pi} $$

In fact, we will construct a version of this equivalence that is functorial in $B$ with respect to étale maps. To this end, we introduce the following notation:

**Notation 4.18.** Consider the functor $\mathrm{CAlg}^{\mathrm{an},\mathrm{coh},\mathrm{et}}_R \to \mathrm{Cat}_\infty$ sending $B \mapsto \mathrm{LieAlgd}^\pi_{\Delta, B/R}$ and sending each étale map $f: B \to B'$ to $f^*: \mathrm{LieAlgd}^\pi_{\Delta, B/R} \to \mathrm{LieAlgd}^\pi_{\Delta, B'/R}$. This classifies a cartesian fibration and a cocartesian fibration that we will denote by

$$ p: \mathrm{LieAlgd}^{\pi, \mathrm{et}}_{\Delta/R} \longrightarrow \mathrm{CAlg}^{\mathrm{an}, \mathrm{coh}, \mathrm{et}, \mathrm{op}}_{R} $$

$$ p^{\vee}: \mathrm{LieAlgd}^{\Delta/R}_{\pi} \longrightarrow \mathrm{CAlg}^{\mathrm{an}, \mathrm{coh}, \mathrm{et}}_{R}. $$

More precisely, $p$ is the restriction of the cartesian fibration from Proposition 3.25 to the subcategory of étale maps, and we define $p^\vee$ to be the cocartesian fibration classified by the same diagram of $\infty$-categories. Part (2) of Proposition 3.25 implies that $p^\vee$ is a cartesian fibration as well.

**Theorem 4.19.** There is an equivalence of $\infty$-categories

<figure><img src="image_21.png" /></figure>
Furthermore, the induced equivalence between fibres over a coherent animated R-algebra B fits into a commuting diagram

<figure><img src="image_22.png" /></figure>

We will first construct the functor $\mathrm{LieAlgd}^{\Delta/R}_{\pi,et} \to \mathrm{Moduli}_R$ and then use the language of deformation theories from [Lur16, Chapter 12] to prove that it is an equivalence.

**Construction 4.20.** If an animated surjection $A \to B$ is an (almost) Artinian extension, then it is in particular complete almost finitely augmented (Definition 2.15). Using Corollary 3.13, we thus obtain a composite functor

$$ \mathfrak{D}: \mathrm{Art}_R \xrightarrow{\mathrm{cot}} \mathrm{Coalg}_{\mathrm{coLie}_\Delta^\pi}(\mathrm{Der}^{\mathrm{aperf}}_{R,\ge 0}) \xrightarrow{(-)^\vee} (\mathrm{LieAlgd}^{\pi}_{\Delta,R})^{\mathrm{op}}. $$

sending each Artinian extension $A \to B$ to the partition $\mathrm{Lie algebroid} \mathfrak{D}(A) = \mathrm{cot}(A)^\vee$ over $B$. Since the maps in $\mathrm{Art}_R$ all cover étale maps $B \to B'$, this assembles into a functor

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

The functor $\mathfrak{D}$ preserves cocartesian arrows: indeed, for any cocartesian arrow $(A \to B) \to (A' \to B')$ in $\mathrm{Art}_R$ covering an étale map $f: B \to B'$, we have that $B' \simeq A' \otimes_A B$ so that $\mathfrak{D}(A') \simeq f^*\mathfrak{D}(A)$. The functor $\mathfrak{D}$ induces a fully faithful functor on fibres by Corollary 3.13 and is hence fully faithful itself, since it preserves cocartesian arrows.

**Proposition 4.21.** Let $B$ be a coherent animated ring and let $A \to B \oplus M[1]$ be a map in $\mathrm{Art}_{R/B}$ classifying a square zero extension $A_\eta \to A$ by a coherent connective $B$-module. Then induced map of partition $\mathrm{Lie algebroids}$ over $B$

$$ (10) \qquad \mathfrak{D}(A) \amalg_{\mathfrak{D}(B \oplus M[1])} \mathfrak{D}(B) \to \mathfrak{D}(A_\eta) $$

is an equivalence.

**Lemma 4.22.** Let $B$ be a coherent animated ring and let $F^*M \in \mathrm{FilMod}_B$ be an almost perfect filtered $B$-module. Then the diagram of pro-coherent duals

$$ (F^0 M/F^1 M)^\vee \to (F^0 M/F^2 M)^\vee \to \dots \to (F^0 M)^\vee $$

is a colimit diagram in $\mathrm{QC}^\vee_B$.

*Proof.* Recall that the pro-coherent dual of a $B$-module $M$ can be described by the left exact functor $\mathrm{Coh}_B \to \mathfrak{S}$ corepresented by $M$. It therefore suffices to verify that for any coherent $B$-module $N$, the map colim $\mathrm{Map}_B(F^0 M/F^n M, N) \to \mathrm{Map}_B(F^0 M, N)$ is an equivalence of spaces. Suppose that $N$ is $k$-coconnective. Because $F^*M$ is almost perfect, there exists an $m$ such that $F^n M$ is $k$-connective for all $n \ge m$. It follows that $\mathrm{Map}_B(F^0 M/F^n M, N) \to \mathrm{Map}_B(F^0 M, N)$ is an equivalence for all $n \ge m$. $\square$
Proof of Proposition 4.21. Let us say that an object $A \in \mathrm{Alg}_{R/B}^{\mathrm{der},\wedge,\mathrm{aft}}$ is adequate if for every $N \in \mathrm{APerf}_{B,\ge 0}$, the natural map

$$ \mathcal{D}(A) \sqcup \mathcal{D}(B \oplus N) \longrightarrow \mathcal{D}(A \oplus N) $$

is an equivalence, where $B \oplus N$ and $A \oplus N = A \times_B (B \oplus N)$ denote the trivial square zero extensions by $N$. To prove the proposition, it will suffice to verify the following assertions:

(1) Every square zero extension $B \oplus_{\alpha} M$ of $B$ by $M \in \mathrm{APerf}_{B,\ge 0}$ is adequate.

(2) Suppose that $A$ is adequate and consider a pullback square (9) classifying a square zero extension $A_{\eta} \to A$ by $M \in \mathrm{APerf}_{B,\ge 0}$. Then the following hold:

(2a) the map (10) is an equivalence.

(2b) $A_{\eta}$ is adequate.

Indeed, Corollary 4.4 then shows by induction that every $A \in \mathrm{Art}_{R/B}$ is adequate, so that $\mathcal{D}$ sends every pullback square (9) to a pushout of partition Lie algebroids.

Recall that for any square zero extension $B \oplus_{\alpha} M$ classified by $\alpha: L_{B/R}[-1] \to M$, the partition Lie algebroid $\mathcal{D}(B \oplus_{\alpha} M)$ is the free partition Lie algebroid on $\alpha^{\vee}: M^{\vee} \to T_{B/R}[1]$. Consequently, $\mathcal{D}((B \oplus_{\alpha} M) \oplus N)$ is the free partition Lie algebroid on $(\alpha^{\vee}, 0): M^{\vee} \oplus N \to T_{B/R}[1]$. Assertion (1) then follows from the fact that taking free Lie algebroids is a left adjoint.

We will simultaneously verify assertions (2a) and (2b). To this end, suppose that $A$ is adequate and let $A_{\eta} \to A$ be a square zero extension of $A$ classified by $\alpha: A \to B \oplus M[1]$. Let $A \to A_{\bullet} \to B \oplus M[1]$ be the cosimplicial resolution provided by Corollary 2.30. Given $N \in \mathrm{APerf}_{B,\ge 0}$, we then obtain a diagram of complete almost finitely augmented algebras over $B$

$$ \begin{tikzcd} A_{\eta} \oplus N \arrow[r] \arrow[d] & A_{\eta}^{\bullet} \oplus N \arrow[r] \arrow[d] & B \oplus N \arrow[d] \arrow[r] \arrow[d] & (id,0) \arrow[d] \\ A \arrow[r] \arrow[d] & A_{\bullet}^{\bullet} \arrow[r] \arrow[d] & B \oplus M[1]. \arrow[d] \arrow[r] \arrow[d] & \end{tikzcd} \quad (11) $$

where the top row is obtained from the bottom row by base change. This gives rise to a diagram of partition Lie algebroids of the form

$$ \begin{tikzcd} |\mathcal{D}(A^{\bullet}) \amalg_{\mathcal{D}(B \oplus M[1])} \mathcal{D}(B \oplus N)| \arrow[r] \arrow[d] & \mathcal{D}(A) \amalg_{\mathcal{D}(B \oplus M[1])} \mathcal{D}(B \oplus N) \arrow[d] \\ |\mathcal{D}(A_{\eta}^{\bullet} \oplus N)| \arrow[r] \arrow[d] & \mathcal{D}(A_{\eta} \oplus N). \arrow[d] \end{tikzcd} \quad (12) $$

It suffices to verify that the right vertical map is an equivalence: if this is the case, then taking $N = 0$ yields assertion (2a) and assertion (2b) then follows from the equivalences

$$ \begin{tikzcd} (\mathcal{D}(A) \amalg_{\mathcal{D}(B \oplus M[1])} \mathcal{D}(B)) \amalg_{\mathcal{D}(B \oplus N)} \mathcal{D}(B \oplus N) \arrow[r, "\sim"] & \mathcal{D}(A) \amalg_{\mathcal{D}(B \oplus M[1])} \mathcal{D}(B \oplus N) \arrow[d, "\sim"] \\ \mathcal{D}(A_{\eta}) \amalg_{\mathcal{D}(B \oplus N)} \mathcal{D}(B \oplus N) \arrow[r, "\sim"] & \mathcal{D}(A_{\eta} \oplus N). \arrow[d, "\sim"] \end{tikzcd} $$

To see that the right vertical map in (12) is an equivalence, we will show that each of the other three maps is an equivalence.

*Top horizontal map.* To see that the top horizontal map in (12) is an equivalence, it suffices to show that the map $|\mathcal{D}(A^{\bullet})| \to \mathcal{D}(A)$ is an equivalence. Since geometric realisations of partition
Lie algebroids are computed in $(\mathrm{QC}^\vee_B)_{/T_{B/R}[1]}$, it suffices to verify this at the level of pro-coherent modules. Now recall that the cosimplicial resolution $A \to A^\bullet$ from Corollary 2.30 has the property that the augmented cosimplicial diagram $\cot(A) \to \cot(A^\bullet)$ in $\mathrm{Der}^{\mathrm{aperf}}_{B/R, \ge 0}$ is split. Taking pro-coherent duals, one obtains that the augmented simplicial diagram $\mathfrak{D}(A^\bullet) \to \mathfrak{D}(A)$ is split in $(\mathrm{QC}^\vee_B)_{/T_{B/R}[1]}$, so that $|\mathfrak{D}(A^\bullet)| \simeq \mathfrak{D}(A)$.

**Left vertical map.** It suffices to verify that each square of partition Lie algebroids

$$ \begin{tikzcd} \mathfrak{D}(B \oplus M[1]) \arrow[r] \arrow[d] & \mathfrak{D}(B \oplus N) \arrow[d] \\ \mathfrak{D}(A^i) \arrow[r] & \mathfrak{D}(A_\eta^i \oplus N) \arrow[d] \end{tikzcd} $$

is a pushout. The cosimplicial resolution $A^\bullet$ from Corollary 2.30 has the property that $A^i \simeq B \oplus_\alpha K$ is a square zero extension of $B$ and that the map $A^i \simeq B \oplus_\alpha K \to B \oplus M[1]$ arises from a map $e: K \to M[1]$ in $\mathrm{Der}^{\mathrm{aperf}}_{B/R, \ge 0}$. It follows that the above square is the image of a pushout square in $(\mathrm{QC}^\vee_B)_{/T_{B/R}[1]}$ under the free partition Lie algebroid functor, and hence a pushout square itself.

**Bottom horizontal map.** It suffices to check that the augmented simplicial diagram $\mathfrak{D}(A_\eta^\bullet \oplus N) \to \mathfrak{D}(A_\eta \oplus N)$ is a colimit diagram in $\mathrm{QC}^\vee_B$. Recall that this is the pro-coherent dual of the augmented cosimplicial diagram of $B$-modules $\cot(A_\eta \oplus N) \to \cot(A_\eta^\bullet \oplus N)$. We will refine this to a diagram of filtered $B$-modules as follows. The diagram (11) of derived $R$-algebras over $B$ lifts to a diagram of filtered $R$-algebras over $B$: we endow $B \oplus N$, $A$ and $A^\bullet$ with the filtration where $F^i = 0$ for all $i \ge 1$, and $B \oplus M[1]$ with the filtration

$$ F^1(B \oplus M[1]) = M[1] \qquad F^i(B \oplus M[1]) = 0 \quad \text{for } i \ge 2. $$

Taking fibre products, we then obtain (finite) filtrations on $A_\eta \oplus N$ and $A_\eta^\bullet \oplus N$.

The maps of animated filtered algebras $F^*(A_\eta \oplus N) \to B$ and $F^*(A_\eta^i \oplus N) \to B$ are almost of finite presentation. Indeed, each $A^i \to B$ (concentrated in weight 0) is almost of finite presentation and $A_\eta^i \oplus N \to A^i$ is a square zero extension by an almost perfect filtered $A^i$-module, and thus almost of finite presentation by Lemma 2.20. By Proposition 2.14, we therefore obtain an augmented cosimplicial diagram of almost perfect filtered $B$-modules

$$ \cot(F^*(A_\eta \oplus N)) \to \cot(F^*(A_\eta^\bullet \oplus N)). $$

At the level of $F^0$, this is given by $\cot(A_\eta \oplus N) \to \cot(A_\eta^\bullet \oplus N)$ and at the level of the associated graded, it is given by

$$ \cot(A \oplus M(1) \oplus N) \to \cot(A^\bullet \oplus M(1) \oplus N). $$

Here $A \oplus M(1) \oplus N$ denotes the derived graded $R$-algebra given by the trivial square zero extension of $A$ by $N$ (in weight 0) and $M$ (in weight 1).

Using Lemma 4.22, we therefore obtain a certain augmented simplicial diagram of pro-coherent $B$-modules equipped with an *increasing* filtration

$$ (13) \qquad F_*\mathfrak{D}(A_\eta^\bullet \oplus N) \to F_*\mathfrak{D}(A_\eta \oplus N) $$

At the level of the underlying objects (i.e. taking the colimit as $\star \to \infty$), this yields the augmented simplicial diagram $\mathfrak{D}(A_\eta^\bullet \oplus N) \to \mathfrak{D}(A_\eta \oplus N)$. To see that this is a colimit diagram, it now suffices
to show that (13) is a colimit diagram at the level of the associated graded. The associated graded of (13) is equivalent to the augmented simplicial diagram (with a certain weight grading that will not play a role)

$$ \mathcal{D}(A^{\bullet} \oplus M \oplus N) \to \mathcal{D}(A \oplus M \oplus N). $$

Since $A$ and all $A^i$ (which are of the form $B \oplus_{\alpha} I$) are adequate, this is equivalent to the map

$$ \mathcal{D}(A^{\bullet}) \amalg \mathcal{D}(B \oplus M \oplus N) \to \mathcal{D}(A) \amalg \mathcal{D}(B \oplus M \oplus N). $$

This is an equivalence because (as we already saw), $|\mathcal{D}(A^{\bullet})| \simeq \mathcal{D}(A)$. $\square$

<figure><img src="image_24.png" /></figure>

By Proposition 4.21, any partition Lie algebroid $\mathfrak{g}$ over $B$ determines a formal moduli problem

$$ (14) \qquad \Psi_{\mathfrak{g}}: \mathrm{Art}_{R/B} \to \mathcal{S}; \quad \Psi_{\mathfrak{g}}(A) = \mathrm{Map}_{\mathrm{LieAlgd}_{\Delta,B/R}^{\pi}}(\mathcal{D}(A), \mathfrak{g}). $$

Using this, we will now construct the putative equivalence of Theorem 4.19.

**Construction 4.23.** Consider the functor

$$ \mathrm{Art}_R \times_{\mathrm{CAlg}_R^{\mathrm{an},\mathrm{coh},\mathrm{et}}} \mathrm{LieAlgd}_{\pi,\mathrm{et}}^{\Delta/R} \xrightarrow{(\mathcal{D},\mathrm{id})} (\mathrm{LieAlgd}_{\Delta/R}^{\pi,\mathrm{et}})^{\mathrm{op}} \times_{\mathrm{CAlg}_R^{\mathrm{an},\mathrm{coh},\mathrm{et}}} \mathrm{LieAlgd}_{\pi,\mathrm{et}}^{\Delta/R} \xrightarrow{\mathrm{Map}} \mathcal{S} $$

where the last functor is the fibrewise mapping space functor. Using Construction 4.17, this is adjoint to a functor $\mathrm{LieAlgd}_{\pi,\mathrm{et}}^{\Delta/R} \to \mathrm{Fun}_{\mathrm{CAlg}_R^{\mathrm{an},\mathrm{coh},\mathrm{et}}} (\mathrm{Art}_R, \mathcal{S})$. Unravelling the definitions, this sends each $\mathfrak{g} \in \mathrm{LieAlgd}_{\Delta,B/R}^{\pi}$ to the formal moduli problem $\Psi_{\mathfrak{g}}$ from (14). We thus obtain a functor

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

Because $\mathcal{D}$ preserved cocartesian fibrations, the functor $\Psi$ is a map of cartesian fibrations that preserves cartesian arrows.

It now remains to verify that $\Psi$ restricts to an equivalence on fibres. For this we will use Lurie's formalism of deformation theories, slightly reformulated as follows:

**Theorem 4.24** (cf. [Lur16, Theorem 12.3.3.5]). Let $\mathcal{B}$ be a presentable $\infty$-category equipped with a set of adjunctions $L_{\alpha}: \mathrm{Sp} \simeq \mathcal{B} : R_{\alpha}$ such that each $R_{\alpha}$ preserves sifted colimits and the $R_{\alpha}$ are jointly conservative. Let $\mathcal{B}_0 \subseteq \mathcal{B}$ be the smallest full subcategory of $\mathcal{B}$ satisfying the following conditions:

(1) It contains the initial object $0$.

(2) If $K \in \mathcal{B}_0$, the for each pushout square in $\mathcal{B}$ of the form

$$ (15) \qquad \begin{tikzcd} L_{\alpha}(\mathcal{S}[n]) \arrow[r] & K \arrow[d] \\ 0 \arrow[u] \arrow[r] & K' \end{tikzcd} $$

for some $\alpha$ and $n < 0$, the object $K'$ is contained in $\mathcal{B}_0$ as well.

Then the restricted Yoneda embedding $h: \mathcal{B} \to \mathcal{P}(\mathcal{B}_0)$; $K \mapsto \mathrm{Map}_{\mathcal{B}}(-, K)$ is fully faithful, with essential image consisting of those presheaves $X$ on $\mathcal{B}_0$ such that $X(0) \simeq *$ and such that (15) is sent to a pullback diagram of spaces.
Proof. Consider the adjoint pair $\mathfrak{D}^*: \mathfrak{B} \hookrightarrow \mathcal{P}(\mathcal{B}_0)^{\text{op}}: \mathfrak{D}_*$ where $\mathfrak{D}^*(K) = \text{Map}_{\mathfrak{B}}(K, -)$. The conditions imply that this is a deformation theory in the sense of [Lur16, Definition 12.3.3.2], and the result follows from [Lur16, Theorem 12.3.3.5]. $\square$

**Example 4.25.** Let $B$ be a coherent animated $R$-algebra. For each $M \in \text{Coh}_{B, \ge 0}$, consider

$$R_M: \mathrm{QC}^\vee_{B/T_{B/R}[1]} \longrightarrow \mathrm{Sp}; \quad (\rho: N \to T_{B/R}[1]) \longmapsto \mathrm{Hom}_{\mathrm{QC}^\vee(B)}(M^\vee, \mathrm{fib}(\rho)).$$

This is a right adjoint functor preserving sifted colimits since $M^\vee \in \mathrm{QC}^\vee_B$ is compact. Since the collection of $M^\vee$ with $M \in \text{Coh}_{B, \ge 0}$ generates $\mathrm{QC}^\vee_B$, this satisfies the conditions of Theorem 4.24. The full subcategory $\mathcal{B}^{\text{lin}}_0 \subseteq \mathrm{QC}^\vee_{B/T_{B/R}[1]}$ appearing in Theorem 4.24 can then be identified as follows. Note that

$$L_M(\mathfrak{S}) \simeq (0: M^\vee \to T_{B/R}[1]).$$

For any $\rho: N^\vee \to T_{B/R}[1]$ with $N \in \text{Coh}_{B, \ge 0}$, we then have that:

(a) For any $n < 0$ and $M^\vee[n] \to N^\vee$, the cofibre $\mathrm{fib}(N \to M[-n])^\vee$ is dual to a coherent connective $B$-module.

(b) it can be obtained as a pushout of a diagram $0 \leftarrow L_N(\mathfrak{S}[-1]) \to 0$ in $\mathrm{QC}^\vee_{B/T_{B/R}[1]}$.

It follows that $\mathcal{B}^{\text{lin}}_0$ is the full subcategory of $\rho: N^\vee \to T_{B/R}[1]$ where $N^\vee$ is the pro-coherent dual of a coherent connective $B$-module. Using Lemma 2.16, we can identify $\mathcal{B}^{\text{lin,op}}_0 \simeq \text{Der}^{\text{coh}}_{B/R, \ge 0}$ with the full subcategory of derivations $\alpha: L_{B/R}[-1] \to M$ for which $M \in \text{Coh}_{B, \ge 0}$. Consequently, there is a fully faithful functor

$$\mathrm{QC}^\vee_{B/T_{B/R}} \longrightarrow \mathrm{Fun}(\mathrm{Der}^{\mathrm{coh}}_{B/R, \ge 0}, \mathfrak{S})$$

whose essential image consists of functors $X$ such that $X(0) \simeq *$ and preserving pullbacks along all maps $0 \to (M[1], \alpha)$ in $\text{Der}^{\text{coh}}_{B/R, \ge 0}$ with $M \in \text{Coh}_{B, \ge 0}$.

**Proof of Theorem 4.19.** We will mimic the strategy from Example 4.25. For each $M \in \text{Coh}_{B, \ge 0}$, consider the functor

$$R_M: \text{LieAlgd}^\pi_{\Delta, B/R} \longrightarrow \text{Sp}; \quad (\rho: \mathfrak{g} \to T_{B/R}[1]) \longmapsto \text{Hom}_{\mathrm{QC}^\vee_B}(M^\vee, \text{fib}(\rho)).$$

Proposition 3.8 implies that the forgetful functor $\text{LieAlgd}^\pi_{\Delta, B/R} \to \mathrm{QC}^\vee_{B/T_{B/R}[1]}$ is a right adjoint that preserves sifted colimits and detects equivalences. Using this and Example 4.25, it follows that the set of functors $R_M$ satisfies the conditions of Theorem 4.24.

To identify the full subcategory $\mathcal{B}_0 \subseteq \text{LieAlgd}^\pi_{\Delta, B/R}$, note that $L_M(\mathfrak{S}) \simeq \text{Lie}^\pi_{\Delta, B/R}(0: M^\vee \to T_{B/R})$ is the free partition $\text{Lie algebroid}$ generated by the zero map from $M^\vee$. By Proposition 3.8, this means that there are natural equivalences

$$L_M(\mathfrak{S}) \simeq \text{Lie}^\pi_{\Delta, B/R}(0: M^\vee \to T_{B/R}) \simeq \mathfrak{D}(B \oplus M)$$

for all $M \in \text{Coh}_{B, \ge 0}$. It follows that $\mathcal{B}_0 \subseteq \text{LieAlgd}^\pi_{\Delta, B/R}$ is the smallest full subcategory containing the initial partition $\text{Lie algebroid} \ 0 \simeq \mathfrak{D}$ which is closed under pushouts along the maps $\mathfrak{D}(B \oplus M[1]) \to \mathfrak{D}(B)$ for all $M \in \text{Coh}_{B, \ge 0}$. Corollary 4.4 and Proposition 4.21 now imply that there is a
commuting diagram

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

where equivalence $\mathfrak{D}$ sends each pullback square (9) to the opposite of a pushout square of the form (15). Using this, Theorem 4.24 then implies that $\Psi: \mathrm{LieAlgd}_{\pi,et}^{\Delta/R} \to \mathrm{Moduli}_R$ restricts to an equivalence $\mathrm{LieAlgd}_{\Delta,B/R}^{\pi} \simeq \mathrm{Moduli}_{B/R}$ on fibres with the desired properties. $\square$

**Remark 4.26.** Combining the equivalence of Theorem 4.19 with the adjoint pair from Example 4.13, we obtain an adjoint pair

$$ C^*: \mathrm{LieAlgd}_{\Delta,B/R}^{\pi} \xrightarrow{\leftarrow} \mathrm{CAlg}_{R/B}^{\mathrm{der},\mathrm{op}} : \mathfrak{D} $$

where $\mathfrak{D}(B') = T_{B/\mathrm{Spf}(B')}$ with the partition Lie algebroid structure arising from Theorem 4.19. Example 4.16 implies that the underlying object of $\mathfrak{D}(B')$ in $\mathrm{QC}^{\vee}_{B/T_{B/R}[1]}$ is naturally equivalent to $L^{\vee}_{B/B'}[1] \to L^{\vee}_{B/R}[1] = T_{B/R}[1]$.

The right adjoint $C^*$ can be thought of as a version of the *Chevalley–Eilenberg complex* for partition Lie algebroids. Explicitly, $C^*(\mathfrak{g})$ is the derived algebra of functions on the formal moduli problem classified by $\mathfrak{g}$. One can now check that the adjoint pair $(C^*, \mathfrak{D})$ also defines a deformation theory in the sense of [Lur16, Definition 12.3.3.2].

**Remark 4.27.** Suppose that $X$ is a locally coherent $\mathrm{qcqs}$ derived scheme. By Remark 3.30, the $\infty$-category of partition Lie algebroids on $X$ can be identified with the $\infty$-category of dotted sections

<figure><img src="image_27.png" /></figure>

sending each map to a cocartesian arrow. Using Theorem 4.19 and unravelling Construction 4.17, one sees that the $\infty$-category of partition Lie algebroids on $X$ is equivalent to the $\infty$-category of functors $F: \mathrm{Art}_R \times_{\mathrm{CAlg}_R^{\mathrm{an},\mathrm{coh},\mathrm{et}}} \mathrm{Open}^{\mathrm{aff}}(X)^{\mathrm{op}} \to S$ satisfying the following two properties:

(1) For each affine open $U \subseteq X$, $F$ restricts to a formal moduli problem $F_U: \mathrm{Art}_{R/\mathcal{O}(U)} \to S$.

(2) For each inclusion of affine opens $V \subseteq U \subseteq X$, $F_V$ is the initial formal moduli problem equipped with a map from $F_U$ to its restriction along $\mathrm{Art}_{R/\mathcal{O}(U)} \to \mathrm{Art}_{R/\mathcal{O}(V)}$.

**4.4. Underlying partition Lie algebra.** Suppose that $R$ is coherent and that $B$ is a coherent animated $R$-algebra. We will use Theorem 4.19 to show that the underlying pro-coherent $R$-module of a partition Lie algebroid $\mathfrak{g}$ over $B$ can be endowed with a $(R$-linear) partition Lie algebra structure. To this end, we begin by recalling the following properties of the $\infty$-categories $\mathrm{CAlg}_A^{\mathrm{an}}$ of animated $A$-algebras.
**Proposition 4.28.** Given a cartesian square of animated rings on the left in which $f: A_0 \to A_{01}$ induces a surjection on $\pi_0$, the right square of $\infty$-categories is cartesian as well:

<figure><img src="image_28.png" /></figure>

*Proof.* Write $h: A \to A_{01}$ for the composite map and consider the following diagram of $\infty$-categories:

<figure><img src="image_29.png" /></figure>

Here the rows are both adjoint pairs: the left adjoint $(g'^*, f'^*)$ sends an animated $A$-algebra (resp. $A$-module) $B$ to the matching triple of objects $(A_0 \otimes_A B, A_{01} \otimes_A B, A_1 \otimes_A B)$ and the right adjoint sends a matching triple of animated algebras (resp. modules) $(B_0, B_{01}, B_1)$ to the fibre product $g'_*B_0 \times_{h_*B_{01}} f'_*B_1$ of $A$-algebras (resp. $A$-modules). In particular, the vertical forgetful functors commute both with the left adjoints and with the right adjoints. Since the forgetful functor detects equivalences, the unit and counit of the top adjunction are an equivalence if the unit and counit of the bottom adjunction are equivalences. This follows from [Lur16, Theorem 16.2.0.2], so that the top adjunction is an equivalence. $\square$

**Definition 4.29.** Suppose that $R$ is coherent and let $B \in \text{Alg}_R^{\text{an}}$ be a $R$-algebra. We will write $\text{Def}_{B/R}$ for the full subcategory of $\text{Fun}(\Delta^1, \text{Alg}_R^{\text{an}})/_{R \to B}$ spanned by those commuting squares

<figure><img src="image_30.png" /></figure>

that are cocartesian and in which $A \to R$ is an Artinian extension. We will typically write $A \to B_A$ for an object in $\text{Def}_{B/R}$, omitting the data of the cocartesian square.

There is a canonical functor $\text{Def}_{B/R} \to \text{Art}_{R/R}$ sending each such square to $A \to R$. We then have the following corollary of Proposition 4.28:

**Lemma 4.30.** Suppose that $R$ is coherent and let $B$ be an animated $R$-algebra. Then the functor $\text{Def}_{B/R} \to \text{Art}_{R/R}$ is a left fibration, classified by a formal moduli problem $\text{def}_B: \text{Art}_{R/R} \to S$.

*Proof.* The functor $\text{Def}_{B/R} \to \text{Art}_{R/R}$ is a cocartesian fibration, classified by $\text{def}_B: \text{Art}_{R/R} \to \text{Cat}_\infty$ sending $A \mapsto \text{CAlg}_A^{\text{an}} \times_{\text{CAlg}_R^{\text{an}}} \{B\}$. Proposition 4.28 implies that $\text{def}_B$ is a formal moduli problem. In particular, $\text{def}_B(R \oplus M) \simeq \Omega\text{def}_B(R \oplus M[1])$ is a space for every coherent $R$-module $M$. By Artinian induction, this implies that $\text{def}_B(A)$ is a space for all $A$, so that the result follows. $\square$

**Notation 4.31.** Write $\Gamma(T_{B/R}[1]) \in \text{LieAlg}_{\Delta,R}^\pi$ for the partition Lie algebra classified by the formal moduli problem $\text{def}_B$.

One can identify the complex underlying $\Gamma(T_{B/R}[1])$ as follows.
The last equivalence follows from the fact that, under unstraightening, both $\infty$-categories are equivalent to the $\infty$ left fibrations over $\int X$. Unravelling the definitions, one sees that the essential image of the above inclusion consists of those functors $\int X \to S$ that preserve the terminal object, as well as every pullback square in $\int X$ whose image in $\mathrm{Art}_{R/R}$ is as in Definition 4.11.

**Construction 4.35.** Let $B$ be a coherent $R$-algebra. Given an object $A \to B_A$ in $\mathrm{Def}_{B/R}$, the map $B_A \to B$ is an almost Artinian extension. Indeed, it is the base change of the almost finitely presented map $A \to R$, hence almost finitely presented, and the kernel of $\pi_0(B_A) \to \pi_0(B)$ is generated by the images of the (nilpotent) generators of the kernel of $\pi_0(A) \to \pi_0(R)$. We thus obtain a functor $t: \mathrm{Def}_{B/R} \to \mathrm{AArt}_{R/B}$ given by $t(A \to B_A) = B_A$. Now suppose that $A_0 \to A_{01}$ is a nilpotent extension and consider a pullback square in $\mathrm{Def}_{B/R}$

$$ \begin{tikzcd} (A \to B_A) \arrow[r] \arrow[d] & (A_0 \to B_{A_0}) \arrow[d] \\ (A_1 \to B_{A_1}) \arrow[r] \arrow[d] & (A_{01} \to B_{A_{01}}). \arrow[d] \end{tikzcd} $$

Then $B_{A_0} \to B_{A_{01}}$ is a nilpotent extension as well, and the proof of Proposition 4.28 shows that $B_A \simeq B_{A_0} \times_{B_{A_{01}}} B_{A_1}$. It follows that restriction along $t$ preserves formal moduli problems. Using Remark 4.34, we therefore obtain a functor

$$ \Gamma: \mathrm{LieAlg}^{\pi}_{\Delta, B/R} \xrightarrow{\sim} \mathrm{Moduli}_{B/R} \xrightarrow{t^*} (\mathrm{Moduli}_{R/R})_{/\mathrm{def}_B} \xrightarrow{\sim} (\mathrm{LieAlg}^{\pi}_{\Delta, R})_{/\Gamma(T_{B/R}[1])}. $$

**Proposition 4.36.** Let $f: R \to B$ be a map of coherent animated rings. Then there is a commuting diagram of right adjoint functors

$$ \begin{tikzcd} (17) \arrow[d, "\mathrm{forget}"] \arrow[r, "\Gamma"] & \mathrm{LieAlg}^{\pi}_{\Delta, B/R} \arrow[r, "\Gamma"] \arrow[d, "\mathrm{forget}"] & (\mathrm{LieAlg}^{\pi}_{\Delta, R})_{/\Gamma(T_{B/R}[1])} \arrow[d] \\ (\mathrm{QC}^{\vee}_{B})_{/T_{B/R}[1]} \arrow[r, "f_*"] & (\mathrm{QC}^{\vee}_{R})_{/\Gamma(T_{B/R}[1])}. \arrow[d] & \end{tikzcd} $$

In particular, $\Gamma(T_{B/R}[1]) \simeq f_*(T_{B/R}[1])$ as pro-coherent $R$-modules.

*Proof.* Consider the commuting square of $\infty$-categories

$$ \begin{tikzcd} \mathrm{Def}_{B/R}^{\mathrm{sqz}} \arrow[r, "t'"] & \mathrm{Der}_{B/R, \ge 0}^{\mathrm{aperf}} \arrow[d] \\ \mathrm{Def}_{B/R} \arrow[r, "t"] & \mathrm{AArt}_{R/B}. \arrow[d] \end{tikzcd} $$

where the top functor $t'$ sends $(M, \alpha: L_{B/R}[-1] \to f^*M)$ to $(f^*M, \alpha)$. By restriction, this induces a commuting square of $\infty$-categories of formal moduli problems and right adjoint functors between them. Using the equivalences from Example 4.25, Theorem 4.19 and Remark 4.34, this square of $\infty$-categories of formal moduli problems is equivalent to a commuting square of the form (17). Indeed, the functor $\Gamma$ is equivalent to restriction along $t$ by definition, and one readily verifies that restriction along the functor $t'$ sending $(M, \alpha)$ to $(f^*M, \alpha)$ corresponds to $f_*$ at the level of pro-coherent modules. $\square$
## 5. FORMALLY INTEGRATING PARTITION LIE ALGEBROIDS

Let $X$ be a locally coherent $qcqs$ derived scheme over a coherent ground ring $R$. In this section, we will formally integrate partition Lie algebroids

$$ \mathfrak{g} \to T_{X/R}[1] $$

on locally coherent $qcqs$ derived schemes $X$ over animated rings $R$. The result of our integration procedure will be *formal moduli stacks*

$$ X \to Y $$

under $X$, where $Y$ can be thought of as a formal leaf space. More precisely, we define:

**Definition 5.1** (Formal moduli stacks). A *formal moduli stack* under $X$ is a map of $R$-prestacks

$$ X \to Y $$

satisfying the following conditions:

(1) $Y$ has deformation theory, which means that it preserves limits of Postnikov towers and pullbacks along nilpotent extensions of $R$-algebras, see Definition B.6;

(2) $X \to Y$ is locally almost of finite presentation ('laft'), see Definition B.3;

(3) $X \to Y$ is a nil-isomorphism. Equivalently, it restricts to an equivalence on all reduced affine schemes, see Remark B.13.

Let $\mathrm{ModuliStk}_{X/R} \subset (\mathrm{PrStk}_R)_X$ be the full subcategory spanned by all formal moduli stacks under $X$.

**Remark 5.2.** Note that Definition 5.1 also makes sense for $X$ a general prestack.

In Section 5.1 below, we will link formal moduli stacks under *affine* schemes to formal moduli problems in the sense of Definition 4.11, and hence to partition Lie algebroids. In Section 5.2, we will then prove our main integration equivalence in Theorem 5.26.

**5.1. Formal moduli problems and formal moduli stacks.** Throughout this section, let $X = \mathrm{Spec}(B)$ be a coherent affine derived scheme over $R$. We will show:

**Theorem 5.3.** Let $X = \mathrm{Spec}(B)$ be a coherent affine derived $R$-scheme. Then there is an equivalence between the $\infty$-category $\mathrm{ModuliStk}_{X/R}$ of formal moduli stacks under $X$ and the $\infty$-category of formal moduli problems

$$ F: \mathrm{Art}_{B/R} \to S $$

in the sense of Definition 4.11. If $X \to Y$ is a formal moduli stack, then the corresponding formal moduli problem $\mathrm{Art}_{B/R} \to S$ sends $A \to B$ to the space of diagonal lifts

<figure><img src="image_31.png" /></figure>

The proof closely follows (a simple version of) the argument in [?] and proceeds in two steps.

**Notation 5.4.** Throughout, we write

$$ X_{\mathrm{prored}} = \mathrm{Spec}(B_{\mathrm{prored}}), \qquad X_{\mathrm{inf}} := (X/R)_{\mathrm{inf}} $$

for the pro-reduction (Definition B.8) and infinitesimal stack of the affine derived scheme $X$. We will write $\sqcup$ for the pushout of (pro-)affine derived schemes (i.e. the fibre product of (ind-)rings) and consider the following categories:
(1) Write $\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$ for the $\infty$-category of affine $R$-schemes equipped with a nilpotent embedding $X \to S$ that exhibits $X$ as an almost finitely presented $S$-scheme, such that $\mathcal{O}(S) \to \mathcal{O}(X) = B$ has eventually coconnective fibre. Equivalently, we simply have

$$ \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+} = \mathrm{Art}_{B/R}^{\mathrm{op}}. $$

In particular, each $S \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$ is a coherent affine derived scheme as well (Corollary 4.5).

(2) Let $\mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ be the $\infty$-category of affine $R$-schemes equipped with a nilpotent embedding $X \to S$ such that $\mathcal{O}(S) \to \mathcal{O}(X)$ has eventually coconnective fibre.

(3) Let $\mathrm{Aff}_{X/\mathrm{inf}}$ be the $\infty$-category of affine schemes over $X_{\mathrm{inf}}$, i.e. $S \in \mathrm{Aff}_R$ together with a map $S \leftarrow S_{\mathrm{prored}} \to X$.

Our goal will be to first identify formal moduli problems with certain functors on the $\infty$-category $\mathrm{Aff}_{X/R}^{\mathrm{nil},+}$. Next, we will relate functors on $\mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ with prestacks $X \to Y \to X_{\mathrm{inf}}$ in between $X$ and its infinitesimal prestack. The $\infty$-category of formal moduli stacks can then be identified with a natural subcategory of this $\infty$-category.

**Definition 5.5.** We will say that a functor $F: (\mathrm{Aff}_{X/R}^{\mathrm{nil},+})^{\mathrm{op}} \to S$ has deformation theory if:

(1) its value on the initial object is contractible,

(2) it sends pushouts along nilpotent embeddings to pullbacks of spaces.

Let $u: \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+} \hookrightarrow \mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ be the evident fully faithful embedding and consider the induced adjoint pair between $\infty$-categories of presheaves:

$$ u_!: \mathcal{P}(\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}) \xrightarrow{\hspace{2em}} \mathcal{P}(\mathrm{Aff}_{X/R}^{\mathrm{nil},+}) : u^* . $$

The main technical step in the proof of Theorem 5.3 is the following:

**Proposition 5.6.** Let $F: (\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+})^{\mathrm{op}} \to S$ be a formal moduli problem. Then the left Kan extension $u_!F: (\mathrm{Aff}_{X/R}^{\mathrm{nil},+})^{\mathrm{op}} \to S$ has deformation theory.

The proof will require some preliminaries:

**Lemma 5.7.** Let $F: (\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+})^{\mathrm{op}} \to S$ be a formal moduli problem and consider a pushout diagram in $\mathrm{Aff}_{X/R}^{\mathrm{nil},+}$

$$ \begin{tikzcd} S_0 \arrow[r, "f"] & S'_0 \arrow[d] \\ \arrow[d] & S \arrow[r, "S'"] \arrow[d] \arrow[Rightarrow,

such that $S_0$ and $S'_0$ are contained in $\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$ and $S_0 \to S$ is a square zero extension. Then the induced map

$$ u_!F(S') \longrightarrow u_!F(S) \times_{u_!F(S_0)} u_!F(S'_0) $$

is an equivalence.
**Proof.** We will compare the fibres over a fixed $x: S'_0 \to u_!F$. Suppose $S_0 \to S$ is a square zero extension by an (necessarily eventually coconnective) ideal $I$ and let $S_{0,I[1]}$ be the trivial square zero extension of $S_0$ by its suspension. Since $S_0 \in \text{Aff}_{X/R}^{\text{laft-nil,+}}$ is a coherent affine derived scheme (Corollary 4.5), we can choose a filtered system $I_*: \mathcal{K} \to \text{Coh}(S_0)_{\ge 0}$ with $\text{colim}_\alpha I_\alpha = I$.
Having fixed this, let $\mathcal{C}'$ be the $\infty$-category of factorizations

<figure><img src="image_32.png" /></figure>

with $T' \in \text{Aff}_{X/R}^{\text{laft-nil,+}}$. A map is a natural transformation that restricts to the identity on the solid part of the diagram. The geometric realization of this $\infty$-category computes the fibre $u_!F(S')_x$. Likewise, let $\mathcal{C}$ be the $\infty$-category of factorizations

<figure><img src="image_33.png" /></figure>

which computes the fibre $u_!F(S)_x$. The restriction $u_!F(S')_x \to u_!F(S)_x$ then arises as the realization of the functor $\pi: \mathcal{C}' \to \mathcal{C}$ that restricts a factorization along $S \to S'$.

We therefore have to prove that $\pi$ is a weak equivalence. To do this, we will replace $\mathcal{C}'$ and $\mathcal{C}$ be weakly equivalent categories $\mathcal{D}'$ and $\mathcal{D}$. First, let us consider the functor $\mathcal{K} \to \text{Cat}$ sending each $\alpha \in \mathcal{K}$ to the $\infty$-category $\mathcal{D}(\alpha)$ of diagrams of the form

<figure><img src="image_34.png" /></figure>

where $T \in \text{Aff}_{X/R}^{\text{laft-nil,+}}$ and a morphism is a natural transformation which is the identity on the solid part of the diagram. This depends functorially on $\alpha$ by restriction along the map of trivial square zero extensions $S_{0,I[1]} \to S_{0,I_\beta[1]} \to S_{0,I_\alpha[1]}$ for each map of ideals $I_\alpha \to I_\beta$. We let $\mathcal{D} = \int_{\alpha \in \mathcal{K}} \mathcal{D}(\alpha)$ be the (cocartesian) unstraightening.

There is a natural functor $q: \mathcal{D} \to \mathcal{C}$ restricting to the right face of the cube in (18). This map is a cocartesian fibration (by postcomposition with $T \to T'$) whose fibre over a factorization $S \to T \to u_!F$ is given as follows. Note that the map $S \to T$ exhibits $S$ as a square zero extension of $S_0$ by $I$ relative to $T$, i.e. it is classified by a map $L_{S_0/T} \to I[1]$. One can then identify

$$q^{-1}(T) = \int_{\alpha \in \mathcal{K}} \text{Map}_{\text{QC}(S_0)/I[1]} (L_{S_0/T}, I_\alpha[1])$$
with the unstraightening of the functor sending each $\alpha$ to the space of lifts of the map $L_{S_0/T} \to I[1]$ to $I_\alpha[1]$. Since $S_0$ and $T$ are in $\mathrm{Aff}^{\mathrm{laft}-\mathrm{nil},+}_{X/R}$, the map $S_0 \to T$ is almost finitely presented (Corollary 4.6). Using that $L_{S_0/T}$ is an almost perfect $S_0$-module, we find that

$$|q^{-1}(T)| \simeq \underset{\alpha \in K}{\mathrm{colim}} \mathrm{Map}_{\mathrm{QC}(S_0)/I[1]} (L_{S_0/T}, I_\alpha[1]) \simeq \mathrm{Map}_{\mathrm{QC}(S_0)/I[1]} (L_{S_0/T}, \underset{\alpha}{\mathrm{colim}} I_\alpha[1]) = *.$$

We conclude that $q: \mathcal{D} \to \mathcal{C}$ is a weak equivalence.

Next, we define $\mathcal{D}'$ to the $\infty$-category of diagrams of the form

<figure><img src="image_35.png" /></figure>

where again, $T' \in \mathrm{Aff}^{\mathrm{laft}-\mathrm{nil},+}_{X/R}$ and a morphism is a natural transformation which is the identity on the solid part of the diagram. Restriction to the right face determines a cocartesian fibration $q': \mathcal{D}' \to \mathcal{C}'$ whose fibre over $T'$ satisfies

$$|q'^{-1}(T')| \simeq \underset{\alpha \in K}{\mathrm{colim}} \mathrm{Map}_{\mathrm{QC}(S_0)/I[1]} (L_{S_0/T'}, I_\alpha[1]) \simeq \mathrm{Map}_{\mathrm{QC}(S_0)/I[1]} (L_{S_0/T'}, \underset{\alpha}{\mathrm{colim}} I_\alpha[1]) = *.$$

We thus obtain a commuting square

<figure><img src="image_36.png" /></figure>

where the horizontal functors are weak equivalences and $\pi'$ forgets the copies of $S'_0$ and $S'$ in Diagram (19). It therefore remains to verify that $\pi'$ is a weak equivalence.

Now let $\mathcal{E} \hookrightarrow \mathcal{D}$ and $\mathcal{E}' \hookrightarrow \mathcal{D}'$ be the full subcategories of diagrams of the form (18), resp. (19), whose front face is a pushout in $\mathrm{Aff}^{\mathrm{laft}-\mathrm{nil},+}_{X/R}$. These inclusions admit left adjoints $L, L'$ that replace the object $T$ by the pushout of the front face; in particular, $\mathcal{E}$ and $\mathcal{E}'$ are weakly equivalent to $\mathcal{D}$ and $\mathcal{D}'$. Furthermore, there is a natural functor $\phi: \mathcal{E} \to \mathcal{E}'$ sending each $T$ as in (18) to $T' = S'_0 \sqcup_{S_0} T$, which fits naturally into a diagram of the form (19).

Now note that for any $T \in \mathcal{E} \subseteq \mathcal{D}$, there is a natural transformation $T \to \pi' \circ \phi(T) = S'_0 \sqcup_{S_0} T$. In particular, the map $\pi': |\mathcal{D}'| \to |\mathcal{D}|$ admits a right inverse (presented by $\phi$). Conversely, for every object $T' \in \mathcal{D}'$ as in (19), there is a natural map

$$\phi \circ L \circ \pi'(T) = S'_0 \sqcup_{S_0, I_\alpha[1]} S_0 \to T$$

which shows that $\pi': |\mathcal{D}'| \to |\mathcal{D}|$ admits a left inverse (presented by $\phi \circ L$). We conclude that $\pi'$, and therefore the map $u_!F(S')_x \to u_!F(S)_{x|S_0}$ is an equivalence. $\square$
**Lemma 5.8.** Let $F: (\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+})^{\mathrm{op}} \to S$ be a formal moduli problem and consider a pushout square

$$ \begin{array}{ccc} S_0 & \xrightarrow{f} & S'_0 \\ \downarrow & & \downarrow \\ S & \longrightarrow & S' \end{array} $$

in $\mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ such that $S'_0 \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$ and $S_0 \to S$ is a square zero extension. Then the induced map

$$ u_!F(S') \longrightarrow u_!F(S) \times_{u_!F(S_0)} u_!F(S'_0) $$

is an equivalence.

*Proof.* We fix $x: S \to u_!F$ with $x_0 = x|_{S_0}$. Consider the $\infty$-category $\mathcal{D}$ of diagrams of the form

$$ (20) \qquad \begin{array}{ccc} S_0 & \longrightarrow & S'_0 \\ \downarrow & & \downarrow \\ S & \longrightarrow & S' \end{array} \xrightarrow[x]{x'} u_!F $$

where the square is cocartesian and $S'_0 \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$. A map is a natural transformation that is constant on the solid part of the diagram. Likewise, let $\mathcal{C}_0$ be the $\infty$-category of factorizations

$$ S_0 \underset{x_0}{\longrightarrow} S'_0 \xrightarrow{x'_0} u_!F $$

where $S'_0 \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$. There is an evident right fibration $\pi: \mathcal{D} \to \mathcal{C}_0$ sending $x' \mapsto x'|_{S'_0}$. The lemma asserts that the fibres of $\pi$ are contractible.

By Lemma 5.7, the functor $\pi$ is also a left fibration. Furthermore, the $\infty$-category $\mathcal{C}_0$ is contractible by definition of $u_!F$ as a left Kan extension. Consequently, it will suffice to prove that the $\infty$-category $\mathcal{D}$ itself is contractible.

To see this, let $\mathcal{D}'$ be the $\infty$-category of diagrams of the form (20) where the square is not necessarily cocartesian, but where $S'$ is contained in $\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$. There is an evident functor $\pi: \mathcal{D}' \to \mathcal{D}$ sending a factorization $x': S' \to u_!F$ with $S' \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$ to $S \sqcup_{S_0} S'_0 \to S' \to u_!F$. To prove the lemma it will thus suffice to show that:

(a) $\mathcal{D}'$ is weakly contractible.

(b) $\pi: \mathcal{D}' \to \mathcal{D}$ is a weak equivalence.

Let us start with (a): notice that restriction of the diagram (20) to the bottom row defines a functor $\phi: \mathcal{D}' \to \mathcal{C}'$ to the $\infty$-category of factorizations

$$ S \underset{x}{\longrightarrow} S' \xrightarrow{x'} u_!F $$
with $S' \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$. Note that $\mathcal{C}'$ is contractible by definition of $u_!F$ as a left Kan extension and that $\phi$ admits a right adjoint sending each factorization $S \to S' \to u_!F$ to

<figure><img src="image_37.png" /></figure>

This implies that $\mathcal{D}'$ is weakly contractible.

For (b), pick a factorization $x': S' = S \sqcup_{S_0} S'_0 \to u_!F$ in $\mathcal{D}$ and consider the comma $\infty$-category $x'/\mathcal{D}'$. Unraveling the definitions, one sees that $x'/\mathcal{D}'$ is just the $\infty$-category $\mathcal{D}'$ considered in our proof of (a), but with $S_0 \to S$ replaced $S'_0 \to S'$. The proof of (a) then shows that $x'/\mathcal{D}'$ is contractible, so that $\pi$ is a weak equivalence. $\square$

**Corollary 5.9.** Let $F: (\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+})^{\mathrm{op}} \to S$ be a formal moduli problem and consider a pushout square in $\mathrm{Aff}_{X/R}^{\mathrm{nil},+}$

<figure><img src="image_38.png" /></figure>

such that $S_0 \to S$ is a square zero extension. Then the induced map

$$u_!F(S') \to u_!F(S) \times_{u_!F(S_0)} u_!F(S'_0)$$

is an equivalence.

*Proof.* Let us fix a map $x: S'_0 \to u_!F$. It suffices to show that the induced map on fibres $u_!F(S')_x \to u_!F(S)_{x|S_0}$ is an equivalence. Now $x$ factors as $S'_0 \to T_0 \xrightarrow{t_0} u_!F$ with $T_0 \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$. Letting $T = T_0 \sqcup_{S_0} S$, we then have maps

$$u_!F(T)_{t_0} \to u_!F(S')_{t_0|S'_0} \to u_!F(S)_{t_0|S_0}.$$

The left map and the composite are equivalences by Lemma 5.8, so that the right map is an equivalence as well. $\square$

*Proof of Proposition 5.6.* Every nilpotent embedding of eventually coconnective affine derived schemes can be decomposed into a finite sequence of square zero extensions, so this follows from Corollary 5.9. $\square$

**Definition 5.10.** Let $f: G \to G'$ be a natural transformation between functors $(\mathrm{Aff}_{X/R}^{\mathrm{nil},+})^{\mathrm{op}} \to S$. We will say that $f$ is locally almost finitely presented if for any $n \in \mathbb{N}$ and any cofiltered diagram $S_* : \mathcal{K} \to \mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ with limit $S$ such that all $\mathcal{O}(S_\alpha) \to \mathcal{O}(X) = B$ have $(n-1)$-coconnective fibre, the map

$$\mathrm{colim}_{\alpha \in \mathcal{K}^{\mathrm{op}}} G(S_\alpha) \to G(S) \times_{G'(S)} (\mathrm{colim}_{\alpha \in \mathcal{K}^{\mathrm{op}}} G'(S_\alpha))$$

is an equivalence.
**Remark 5.11.** Suppose that $S_* : \mathcal{K} \to \mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ is a cofiltered diagram of affine schemes under $X$ such that all $\mathcal{O}(S_\alpha) \to \mathcal{O}(X)$ have $(n-1)$-coconnective fibre for some fixed $n$. Then

$$S_{\alpha} \simeq S_{\alpha}^{\le n} \sqcup_{X \le n} X$$

where we write $X^{\le n} = \mathrm{Spec}(\tau_{\le n} B)$ for the corresponding truncated affine derived scheme.

**Corollary 5.12.** The adjoint pair $(u_!, u^*)$ restricts to an equivalence between the full subcategories of:

(1) formal moduli problems $F: (\mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+})^{\mathrm{op}} \to S$.

(2) functors $G: (\mathrm{Aff}_{X/R}^{\mathrm{nil},+})^{\mathrm{op}} \to S$ with deformation theory such that the canonical map $X \to G$ is locally almost finitely presented.

*Proof.* It will suffice to show that $X \to u_!(F)$ is indeed locally almost finitely presented. Assuming this, one has that $u_!$ is fully faithful because it takes the left Kan extension along the fully faithful functor $u$. A map $G \to G'$ between functors with deformation theory is an equivalence if and only if it for every square zero extension $X \to X_I$ with $I \in \mathrm{QC}(X)_{\ge 0}^+$, the induced map

$$G(X_I) \simeq G(X) \times_{G(X_I[1])} G(X) \longrightarrow G'(X) \times_{G'(X_I[1])} G'(X) = G'(X_I)$$

is an equivalence. If $X \to G$ and $X \to G'$ are both locally almost finitely presented, it suffices to verify this when $I \in \mathrm{Coh}(X)$, in which case $X_I \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$. It follows that $u^*$ detects equivalences such functors, so that $(u_!, u^*)$ restricts to the desired equivalence.

To see that $X \to u_!(F)$ is indeed locally almost finitely presented, let $S_* : \mathcal{K} \to \mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ be a cofiltered diagram as in Definition 5.10, with limit $S$. By Remark 5.11, we can assume that $S_* = S_*^{\le n} \sqcup_{X \le n} X$ arises as the pushout of the cofiltered diagram of $n$-coconnective schemes $S_\alpha^{\le n}$ under $X^{\le n}$. Consequently, we have natural equivalences $S_\alpha \simeq S_\alpha^{\le n} \sqcup_{S^{\le n}} S$. We will need to prove that the square

$$\begin{tikzcd}
\mathrm{colim}_\alpha X(S_\alpha) \arrow[r] \arrow[d] & X(S) \arrow[d] \\
\mathrm{colim}_\alpha u_!F(S_\alpha) \arrow[r] & u_!F(S)
\end{tikzcd}$$

is cartesian. To this end, note that $u_!F(S)$ arises as the classifying space of the $\infty$-category $\mathcal{C}$ of factorizations $S \to T \to j_!F$ with $T \in \mathrm{Aff}_{X/R}^{\mathrm{laft}-\mathrm{nil},+}$. Likewise, for each $\alpha$ let $\mathcal{C}(\alpha)$ be the $\infty$-category of factorizations $S_\alpha \to T \to u_!F$. This determines a diagram of categories $\mathcal{K} \to \mathrm{Cat}$ whose unstraightening

$$\mathcal{C}' = \int_{\alpha \in \mathcal{K}} \mathcal{C}(\alpha)$$

has the property that $|\mathcal{C}'| = \mathrm{colim}_\alpha u_!F(S_\alpha)$.

We now consider the functor $\pi: \mathcal{C}' \to \mathcal{C}$ sending each $S_\alpha \to T \to u_!F$ to the composite $S \to S_\alpha \to T \to u_!F$. This is a cocartesian fibration (by postcomposition with $T \to T'$) whose fibre over $x = (S \to T \to u_!F) \in \mathcal{C}$ is given by the $\infty$-category

$$\pi^{-1}(x) = \int_{\alpha \in \mathcal{K}} \mathrm{Map}_{S/} (S_\alpha, T) \simeq \int_{\alpha \in \mathcal{K}} \mathrm{Map}_{S^{\le n}/} (S_\alpha^{\le n}, T).$$
The classifying space of this $\infty$-category can be identified with $|\pi^{-1}(x)| = \text{colim}_\alpha \text{Map}_{S\leq n/}(S\leq n, T)$. For every map in $\mathcal{C}$ of the form

$$S \to T \xrightarrow{f} T' \to u_!F$$

the induced change-of-fibres map $f_*: \pi^{-1}(x) \to \pi^{-1}(x')$ is then given at the level of classifying spaces by

$$\text{colim}_\alpha \text{Map}_{S\leq n/}(S\leq n, T) \xrightarrow{f_0-} \text{colim}_\alpha \text{Map}_{S\leq n/}(S\leq n, T').$$

This map is a weak equivalence since every $T \to T'$ in $\text{Aff}_{X/R}^{\text{laft-nil,+}}$ is locally almost finitely presented (Corollary 4.6). Consequently, given a point $x \in |\mathcal{C}| = u_!F(S)$, corresponding to a factorization $x: S \to T \to u_!F$, there is an equivalence

$$(\text{colim}_\alpha u_!F(S_\alpha)) \times_{u_!F(S)} \{x\} \simeq |\pi^{-1}(x)|.$$

Applying this to each point $x: S \to X \to u_!F$, one sees that

$$(\text{colim}_\alpha u_!F(S_\alpha)) \times_{u_!F(S)} \{x\} \simeq \text{colim}_\alpha \text{Map}_S(S_\alpha, X) \simeq (\text{colim}_\alpha X(S_\alpha)) \times_{X(S)} \{x\}$$

which yields the result. $\square$

Next, let $\text{PrStk}_{X_{\text{inf}}}^{\text{conv}}$ be the $\infty$-category of convergent prestacks over $X_{\text{inf}}$. Every $S \in \text{Aff}_{X/R}^{\text{nil,+}}$ in gives rise to a convergent prestack $S \to X_{\text{inf}}$ over $X_{\text{inf}}$, where the map is adjoint to the canonical map $S_{\text{prored}} = X_{\text{prored}} \to X$. Let us now consider the adjoint pairs

$$v_!: \mathcal{P}(\text{Aff}_{X/R}^{\text{nil,+}}) \xrightarrow{\leftarrow} (\text{PrStk}_{X_{\text{inf}}}^{\text{conv}}) : v^* \qquad \tilde{v}_!: \mathcal{P}(\text{Aff}_{X/R}^{\text{nil,+}}) \xrightarrow{\leftarrow} (\text{PrStk}_{X_{\text{inf}}}^{\text{conv}})_X : \tilde{v}^*$$

where $v_!$ and $\tilde{v}_!$ are the unique colimit-preserving functors sending each nilpotent embedding $S \in \text{Aff}_{X/R}^{\text{nil,+}}$ to the diagrams of convergent prestacks $S \to X_{\text{inf}}$ and $X \to S \to X_{\text{inf}}$, respectively. At the level of the right adjoints, given a diagram of convergent $R$-prestacks $X \to Y \to X_{\text{inf}}$, the presheaf $\tilde{v}^*Y$ sends each nilpotent embedding $X \hookrightarrow S$ in $\text{Aff}_{X/R}^{\text{nil,+}}$ to the space of dotted lifts

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    X \arrow[r, "Y"] & \arrow[d, "\gamma"] \\
    S \arrow[r, "\gamma"] & X_{\text{inf}}
  \end{tikzcd} \qquad (21)$$

in $\text{PrStk}_R^{\text{conv}}$. Let us point out that, since $X \hookrightarrow S$ is a nilpotent embedding, this space is equivalent to the space of dotted lifts where we forget compatibility with the projection to $X_{\text{inf}}$.

**Remark 5.13.** The functor $v_!$ can be computed explicitly as follows: if $x: S \to X_{\text{inf}}$ is a map from an *eventually coconnective* affine derived scheme, then

$$\text{Map}_{X_{\text{inf}}}(S, v_!F) \simeq F(X \sqcup_{S_{\text{prored}}} S) = \text{colim}_\alpha F(S \amalg_{S_\alpha} X)$$

with the colimit taken over the filtered $\infty$-category of nilpotent ideals in $\pi_0(\mathcal{O}(S))$. Indeed, both of these functors preserve colimits, as colimits of convergent prestacks are computed pointwise on eventually coconnective affine derived schemes. Furthermore, they are naturally equivalent for
functors $F$ represented by $T \in \mathrm{Aff}_{X/R}^{\mathrm{nil},+}$, because $\mathrm{Map}_{X_{\mathrm{inf}}}(S, v_!T)$ is naturally equivalent to the
diagram of dotted extensions in $\mathrm{Aff}_R$

<figure><img src="image_39.png" /></figure>

**Lemma 5.14.** *The adjoint pair $(\tilde{v}_1, \tilde{v}^*)$ restricts to an equivalence between the full subcategories of:*

(1) functors $F: (\mathrm{Aff}_{X/R}^{\mathrm{nil},+})^{\mathrm{OP}} \to \mathcal{S}$ with deformation theory.

(2) diagrams of nil-isomorphic convergent prestacks $X \to Y \to X_{\mathrm{inf}}$ where $Y$ has deformation theory.

*Proof.* Let us first prove that $\tilde{v}_1$ and $\tilde{v}^*$ preserve these two subcategories. First, given $X \to Y \to X_{\mathrm{inf}}$ where $Y$ has deformation theory, the diagram (21) directly shows that $\tilde{v}^*(Y)$ sends pushouts in $\mathrm{Aff}_{X/R}^{\mathrm{nil},+}$ along nilpotent embeddings to pullbacks of spaces and sends $X$ to a contractible space.

Next, we consider the functor

$$ \mathcal{P}(\mathrm{Aff}_{X/R}^{\mathrm{nil},+}) \xrightarrow{\tilde{v}_1} (\mathrm{PrStk}_{X_{\mathrm{inf}}}^{\mathrm{conv}})_{X/} \xrightarrow{\mathrm{forget}} \mathrm{PrStk}_{X_{\mathrm{inf}}}^{\mathrm{conv}} $$

and the unique colimit-preserving functor

$$ v_!: \mathcal{P}(\mathrm{Aff}_{X/R}^{\mathrm{nil},+}) \longrightarrow \mathrm{PrStk}_{X_{\mathrm{inf}}}^{\mathrm{conv}} $$

that sends $X \hookrightarrow S$ to the map $S \to X_{\mathrm{inf}}$ adjoint to $S_{\mathrm{prored}} = X_{\mathrm{prored}} \to X$. Both functors preserve colimits indexed by contractible diagrams and are naturally equivalent on representables. Consequently, they are naturally equivalent on all presheaves arising as colimits of contractible diagrams of representables. Since every functor $F: \mathrm{Aff}_{X/R}^{\mathrm{nil},+} \to \mathcal{S}$ with deformation theory has $F(X) \simeq \ast$, it is such a colimit of a contractible diagram, i.e. we can identify $\tilde{v}_1(F) \to X_{\mathrm{inf}}$ simply with $v_1(F) \to X_{\mathrm{inf}}$. Using the formula from Remark 5.13 and Lemma B.7, one now readily verifies that if $F$ has deformation theory, then the convergent prestack $\tilde{v}_1(F) \simeq v_1(F)$ has deformation theory and is nil-isomorphic to $X_{\mathrm{inf}}$ (or equivalently, to $X$).

The adjoint pair $(\tilde{v}_1, \tilde{v}^*)$ therefore restricts to an adjoint pair between functors $F$ with deformation theory and nil-isomorphisms $X \to Y \to X_{\mathrm{inf}}$ where $Y$ has deformation theory. When $F$ has deformation theory, the unit $F \to \tilde{v}^*\tilde{v}_1F$ is given at each nilpotent embedding $X \hookrightarrow S$ as follows. Write $x_0: X \to F$ for the canonical map and suppose that $\mathcal{O}(S) \to \mathcal{O}(X)$ has $(n-1)$-coconnective fibre for some $n$, so that $S = S^{\le n} \sqcup_{X^{\le n}} X$ is the pushout of the corresponding $n$-coconnective affine derived schemes. Then the unit map can be identified with

$$ (22) \qquad F(S) \simeq F(S^{\le n} \sqcup_{X^{\le n}} X) \longrightarrow F(S^{\le n} \coprod_{S_{\mathrm{prored}}} X) \times_{F(X^{\le n} \coprod_{X_{\mathrm{prored}}} X)} \{\nabla^* x_0\} $$

where $f: X^{\le n} \sqcup_{X_{\mathrm{prored}}} X \to X$ arises from the codiagonal map. Diagrammatically, this sends the
dotted map $x: S \to F$ in the following diagram to its restrictions:

<figure><img src="image_40.png" /></figure>
where $X \stackrel{\leq n}{\Pi}_{X_{\text{prored}}} X$ is a pro-object in $\text{Aff}_{X/R}^{\text{nil},+}$ using the inclusion of $X$ as the right component. Using that $X_{\text{prored}} \to S_{\text{prored}}$ is an equivalence and that $S \simeq S \stackrel{\leq n}{\sqcup}_{X \leq n} X$, one sees that the above square is a pushout square of pro-objects in $\text{Aff}_{X/R}^{\text{nil},+}$. Consequently, the unit map (22) is an equivalence, since $F$ had deformation theory.

On the other hand, for $X \xrightarrow{y_0} Y \to X_{\text{inf}}$, the counit $\tilde{v}_1\tilde{v}^*Y \to Y$ is given at each $x: S \to X_{\text{inf}}$, where $S$ is an eventually coconnective affine derived scheme, by the map

$$ (23) \qquad \text{Map}_{/X_{\text{inf}}} (S \Pi_{S_{\text{prored}}} X, Y) \times_{\text{Map}_{/X_{\text{inf}}} (X, Y)} \{y_0\} \longrightarrow \text{Map}_{/X_{\text{inf}}} (S, Y). $$

In terms of diagrams, the domain can be identified with the space of dotted extensions in the following diagram

<figure><img src="image_41.png" /></figure>

The counit sends such $y$ to its restriction to $S$. Notice that the pushout $S \sqcup_{S_{\text{prored}}} X$ arises as a pro-system of pushout squares $S \sqcup_{S_{\alpha}} X$, where each $S_{\alpha} \to S$ is a nilpotent embedding. Since $Y$ has deformation theory, it follows that the map

$$ Y(X \sqcup_{S_{\text{prored}}} S) \longrightarrow Y(X) \times_{Y(S_{\text{prored}})} Y(S) $$

is an equivalence. Consequently, the counit map (23) can be identified with the map

$$ Y(S) \times_{(Y(S_{\text{prored}}) \times_{X(S_{\text{prored}})} \{x\})} \{y_0|_{S_{\text{prored}}}\} \longrightarrow Y(S) $$

projecting onto the first factor. The fact that $Y \to X_{\text{inf}}$ is a nil-isomorphism is equivalent to the fact that $Y(S_{\text{prored}}) \simeq X(S_{\text{prored}})$, so that $Y(S_{\text{prored}}) \times_{X(S_{\text{prored}})} \{x\}$ is contractible and the above map is indeed an equivalence as desired. $\square$

Finally, we note:

**Lemma 5.15.** The functor oblv: $(\text{PrStk}_{/X_{\text{inf}}})_{X/} \to \text{PrStk}_{X/R}$ sending each $X \to Y \to X_{\text{inf}}$ to $X \to Y$ restricts to an equivalence between the full subcategories of:

(1) Diagrams $X \to Y \to X_{\text{inf}}$ where $Y$ has deformation theory and $X \to Y$ is a laft nil-isomorphism.

(2) Formal moduli stacks $X \to Y$ in the sense of Definition 5.1.

*Proof.* Note that if $f: X \to Y$ is locally almost finitely presented, then $f$ is a nil-isomorphism if and only if it restricts to an equivalence on reduced schemes, by Remark B.13. The forgetful functor then restricts to an equivalence because for any nil-isomorphism $X \to Y$, space $\text{Map}_{X/}(Y, X_{\text{inf}})$ is contractible, as all three functors are naturally equivalent on pro-reduced affine schemes $S_{\text{prored}}$. $\square$

*Proof of Theorem 5.3.* Proposition 5.6 and Lemma 5.14 imply that the composite adjunction

$$ w_!: \mathcal{P}(\text{Aff}_{X/R}^{\text{laft}-\text{nil},+}) \xrightarrow{u_!} \mathcal{P}(\text{Aff}_{X/R}^{\text{nil},+}) \xrightarrow{\tilde{v}_!} (\text{PrStk}_{X_{\text{inf}}}^{\text{conv}})_{X/}: w^* $$

restricts to an adjunction $(w_!^{\text{def}}, w_{\text{def}}^*)$ between formal moduli problems on $\text{Aff}_{X/R}^{\text{laft}-\text{nil},+} \simeq \text{Art}_{B/R}^{\text{op}}$ and nil-isomorphisms $X \to Y \to X_{\text{inf}}$ where $Y$ has deformation theory.
The resulting adjoint pair $(w_i^{\text{def}}, w_{\text{def}}^*)$ has a fully faithful left adjoint: $u_!$ is fully faithful by Corollary 5.12 and $\tilde{v}_!$ was fully faithful on functors with deformation theory by Lemma 5.14. It therefore remains to show that the essential image of $w_i^{\text{def}}$ consists precisely of nil-isomorphisms $X \to Y \to X_{\text{inf}}$ which exhibit $X$ as a laft $Y$-prestack. Lemma 5.15 then implies that $w_{\text{def}}^*$ induces an equivalence between formal moduli stacks $X \to Y$ and formal moduli problems. Unravelling the definitions, one sees that $w_{\text{def}}^*$ is precisely the functor from Theorem 5.3.

Let us first verify that $X \to w_i^{\text{def}}(F)$ is indeed laft, i.e. for each map $T \to w_i^{\text{def}}(F)$ from an eventually coconnective affine derived $R$-scheme $T$, the pullback $X \times_{w_i^{\text{def}}(F)} T$ is a laft $T$-prestack. Note that $w_i^{\text{def}}(F)$ is the colimit of a contractible diagram of nilpotent extensions $S \in \text{Aff}_{X/R}^{\text{laft}-\text{nil},+}$, and that such colimits in $\text{PrStk}_R^{\text{conv}}$ are computed pointwise on eventually coconnective affine derived schemes. Hence it suffices to treat the case where $T \to w_i^{\text{def}}(F)$ fits into a diagram

<figure><img src="image_42.png" /></figure>

where $X \to S$ is contained in $\text{Aff}_{X/R}^{\text{laft}-\text{nil},+}$ and $x \in F(S)$. It therefore suffices to treat the case where $T = S$. In that case,

$$X \times_{w_i^{\text{def}}(F)} S = w_i^{\text{def}}(X \times_F S)$$

arises as the image of a formal moduli problem over the base $S$, i.e. a formal moduli problem $(\text{Aff}_{X/S}^{\text{laft}-\text{nil},+})^{\text{op}} \to S$. This uses that $u_!$ preserves pullbacks of formal moduli problems (by Corollary 5.12) and that $v_!$ preserves pullbacks by the formula from Remark 5.13.

Since $X \to S$ is a nilpotent embedding which is almost finitely presented, each $S' \in \text{Aff}_{X/S}^{\text{laft}-\text{nil},+}$ is almost finitely presented over $S$ (Corollary 4.6). Consequently, the prestack $X \times_{w_i^{\text{def}}(F)} T$ is a colimit of almost finitely presented affine derived $T$-schemes; this implies that it is a laft $T$-prestack, as required.

To conclude, it now suffices to verify that $w_{\text{def}}^*$ detects equivalences between laft nil-isomorphisms $X \to Y$ where $Y$ has deformation theory. This follows from the fact that $\tilde{v}^*$ detects weak equivalences by Lemma 5.14 and that $u^*$ detects weak equivalences by Corollary 5.12. $\square$

**5.2. Formal integration of partition Lie algebroids.** Let us again fix a coherent animated base ring $R$ and a locally coherent qcqs derived $R$-scheme $X$. To prove our main theorem Theorem 5.26, we will proceed in two steps: first, we will shat the shifted tangent complex functor

$$ \begin{aligned} \text{ModuliStk}_{X/R} & \longrightarrow (\text{QC}_X^\nu)_{T_{X/R}[1]} \\ (X \to Y) & \longleftarrow (T_{X/Y}[1] \to T_{X/R}[1]) \end{aligned} $$

is monadic; next, we will identify the resulting monad with the partition Lie algebroid monad constructed in Theorem 3.1. In our proof, we will need to relate formal moduli stacks on different schemes.

**Construction 5.16** (Functorial moduli stacks and their tangent fibres). Let $X' \to X$ be a laft map of prestacks with deformation theory. Given a formal moduli stack $X \to Y$, consider the composite map

$$X' \to X \to Y$$
and let $Y_{X'}^{\wedge}$ be the formal completion of $Y$ along $X' \to Y$ as defined in Construction B.9. Via the natural map $X' \to Y_{X'}^{\wedge}$, the prestack $Y_{X'}^{\wedge}$ becomes a formal moduli stack under $X'$ by Lemma B.14. The construction $(X \to Y) \mapsto (X' \to Y_{X'}^{\wedge})$ defines a functor

$$ (-)^{\wedge}_{X'} : \mathrm{ModuliStk}_{X/R} \to \mathrm{ModuliStk}_{X'/R}. $$

More formally, let us write $\mathrm{PrStk}_R^{\mathrm{def,laft-map}}$ for the $\infty$-category of $R$-prestacks with deformation theory and laft maps between them. Consider the full subcategory

$$ \mathrm{ModuliStk}_R \subset \mathrm{Fun}(\Delta^1, \mathrm{PrStk}_R^{\mathrm{def,laft-map}}) $$

whose objects are maps of $R$-prestacks $X \to Y$ with $Y$ a formal moduli stack under $X$ (Definition 5.1) and whose morphisms are squares

$$ \begin{array}{ccc} X & \to & Y \\ \downarrow & & \downarrow \\ X' & \to & Y' \end{array} $$

for which the vertical maps are laft and have deformation theory. Evaluation at 0 defines a cartesian fibration

$$ \mathrm{ModuliStk}_R \to \mathrm{PrStk}_R^{\mathrm{def,laft-map}}, $$

which we straighten to obtain a functor $(\mathrm{PrStk}_R^{\mathrm{def,laft-map}})^{\mathrm{op}} \to \mathrm{Cat}_{\infty}$. It sends an object $X$ to $\mathrm{ModuliStk}_{X/R}$ and a morphism $X' \to X$ to the induced

$$ (-)^{\wedge}_{X'} : \mathrm{ModuliStk}_{X/R} \longrightarrow \mathrm{ModuliStk}_{X'/R} ; \quad (X \to Y) \longmapsto (X' \to Y_{X'}^{\wedge}). $$

Construction B.23 and Proposition B.32 then define (after suspending) the functorial shifted relative tangent complex

$$ \begin{array}{ccc} T[1] : \mathrm{ModuliStk}_R & \longrightarrow & \mathrm{QC}^{\vee}_{T_{-R}[1]} \\ (X \to Y) & \longleftarrow & (T_{X/Y}[1] \to T_{X/R}[1]) \end{array} $$

This defines a map of cartesian fibrations by Proposition B.32 and Proposition B.28(4). In other words, we obtain functors $T_{X/-}[1]: \mathrm{ModuliStk}_{X/R} \to (\mathrm{QC}^{\vee}_{X})_{T_{X/R}[1]}$ that are natural in $X$ with respect to laft maps, i.e., each laft map $f: X' \to X$ gives rise to a commutative square

$$ \begin{array}{ccc} \mathrm{ModuliStk}_{X/R} & \xrightarrow{T_{X/-}[1]} & (\mathrm{QC}^{\vee}_{X})_{T_{X/R}[1]} \\ (-)^{\wedge}_{X} \downarrow & & \downarrow^{f^{\sharp}} \\ \mathrm{ModuliStk}_{X'/R} & \xrightarrow{T_{X'/-}[1]} & (\mathrm{QC}^{\vee}_{X'})_{T_{X'/R}[1]} \end{array} $$

where $f^{\sharp}(\mathcal{F} \to T_{X/R}[1]) \simeq f^{*}(\mathcal{F}) \times_{(f^{*}T_{X/R})^{\mathrm{laft}}[1]} T_{X'/R}[1]$. If $f$ is formally étale, then this simplifies to $f^{\sharp}(\mathcal{F} \to T_{X/R}[1]) \simeq f^{*}\mathcal{F}$ by Proposition B.28(3).

**Proposition 5.17 (Conservativity).** Let $X$ be a prestack with deformation theory and $Y_1 \to Y_2$ a map of formal moduli stacks under $X$. If $T_{X/Y_1} \to T_{X/Y_2}$ is an equivalence, then so is $Y_1 \to Y_2$.
Proof. Since $Y_1$ and $Y_2$ are both convergent, it suffices to verify that $Y_1(A) \to Y_2(A)$ is an equivalence for every eventually coconnective animated ring $A$. To see this, consider the ind-ring

$$A_{\text{prored}} = \text{``colim''} \pi_0(A)/I$$

indexed by the filtered poset of nilpotent ideals of $\pi_0(A)$. We then have a commuting diagram

$$\begin{tikzcd}
Y_1(A) \arrow[r] \arrow[d] & Y_2(A) \arrow[d] \\
X(A_{\text{prored}}) \arrow[r, "\sim"] & Y_1(A_{\text{prored}}) \arrow[r, "\sim"] & Y_2(A_{\text{prored}}).
\end{tikzcd}$$

The bottom maps are equivalences since $X \to Y_1$ and $X \to Y_2$ are nil-isomorphisms and laft (Remark B.13). We therefore have to show that for each point $x \in X(A_{\text{prored}}) = \text{colim}_I X(\pi_0(A)/I)$, the map on fibres $Y_1(A)_x \to Y_2(A)_x$ is an equivalence. To see this, it will suffice to show that for any nilpotent ideal $I \subseteq \pi_0(A)$ and $x \in X(\pi_0(A)/I)$, the map

$$(24) \qquad Y_1(A) \times_{Y_1(\pi_0(A)/I)} \{x\} \to Y_2(A) \times_{Y_2(\pi_0(A)/I)} \{x\}$$

is an equivalence. Indeed, the map $Y_1(A)_x \to Y_2(A)_x$ is a filtered colimit of maps of the above form.

Now recall that $A \to \pi_0(A)/I$ decomposes as a finite sequence $A = A_n \to A_{n-1} \to \dots \to A_0 = \pi_0(A)/I$ where each $A_k \to A_{k-1}$ is a square zero extension by an $A_0$-module $J_k$ (concentrated in a single degree, although we will not need this). To prove that (24) is an equivalence, we will proceed by induction on this tower, the case of $A_0$ being evident. For the inductive step, note that there is a natural map of fibre sequences of spaces

$$\begin{tikzcd}
Y_1(A_k) \times_{Y_1(A_0)} \{x\} \arrow[r] \arrow[d] & Y_1(A_{k-1}) \times_{Y_1(A_0)} \{x\} \arrow[r] \arrow[d] & Y_1(A_0 \oplus J_k[1]) \times_{Y_1(A_0)} \{x\} \arrow[d] \\
Y_2(A_k) \times_{Y_2(A_0)} \{x\} \arrow[r] \arrow[d] & Y_2(A_{k-1}) \times_{Y_2(A_0)} \{x\} \arrow[r] \arrow[d] & Y_2(A_0 \oplus J_k[1]) \times_{Y_2(A_0)} \{x\}
\end{tikzcd}$$

since $Y_1$ and $Y_2$ have deformation theory. By inductive hypothesis the middle vertical map is an equivalence, so it suffices to verify that the right vertical map is an equivalence. Unravelling Definition B.17 and using that the maps $X \to Y_i$ are laft, one sees that the right vertical map fits into a pullback square

$$\begin{tikzcd}
Y_1(A_0 \oplus J_k[1]) \times_{Y_1(A_0)} \{x\} \arrow[r] \arrow[d] & T_{X/Y_1,x}(J_k[2]) \arrow[d] \\
Y_2(A_0 \oplus J_k[1]) \times_{Y_2(A_0)} \{x\} \arrow[r] \arrow[d] & T_{X/Y_2,x}(J_k[2]).
\end{tikzcd}$$

The right vertical map is an equivalence by the assumption that $T_{X/Y_1} \to T_{X/Y_2}$ was an equivalence, so the result follows. $\square$

**Definition 5.18** (Zariski descent). We say that a prestack $X \in \text{PrStk}$ satisfies Zariski descent if for all animated rings $A \in \text{CAlg}^{\text{an}}$ and all families of animated rings

$$\{A \to A_i\}$$

for which the maps $\text{Spec}(A_i) \to \text{Spec}(A)$ are jointly surjective open immersions, we have

$$X(A) \xrightarrow{\sim} \lim X(S),$$

where the limit is indexed by all $S \in \text{CAlg}^{\text{an}}$ such that $A \to S$ factors through some $A_i$.
Theorem 7.3.5.2 in [Lur09] implies the following ‘finite’ characterisation of Zariski descent:

**Proposition 5.19.** A prestack $X \in \mathrm{PrStk}$ satisfies Zariski descent if and only if the following conditions hold:

(1) $X(0) \simeq *$ is contractible;

(2) Let us say that a square of animated rings

<figure><img src="image_43.png" /></figure>

is a Zariski square if it is cocartesian ($A_{12} \simeq A_1 \otimes_A A_2$) and the maps $\mathrm{Spec}(A_i) \to \mathrm{Spec}(A)$ are jointly surjective open immersions. Then $X$ sends each Zariski square to a pullback square.

**Remark 5.20.** For any Zariski square, the induced square of $n$-truncations is a Zariski square as well, since $A \to A_1$ and $A \to A_2$ are flat. Consequently, each Zariski square is the limit of the Zariski squares of its $n$-truncations. A convergent prestack therefore satisfies Zariski descent if and only if it sends every Zariski square of eventually coconnective animated rings to a pullback.

**Proposition 5.21.** Let $X$ be a prestack with deformation theory that satisfies Zariski descent, and let $X \to Y$ be a formal moduli stack under $X$. Then the prestack $Y$ satisfies Zariski descent as well.

*Proof.* By Remark 5.20, it suffices to verify that $Y(A) \simeq Y(A_1) \times_{Y(A_{12})} Y(A_2)$ for any Zariski square of eventually coconnective animated rings. Let us fix such a Zariski square, as well as a filtered system of nilpotent ideals $I \subseteq \pi_0(A)$ such that $\mathrm{colim}_I \pi_0(A)/I \cong A_{\mathrm{red}}$. For each such ideal, let us furthermore write $I_i = \pi_0(A_i) \otimes_{\pi_0(A)} I_i$. For $i = \emptyset, 1, 2, 12$, we then have that $\mathrm{colim}_I \pi_0(A_i)/I_i \cong A_i \otimes_A A_{\mathrm{red}} \cong (A_i)_{\mathrm{red}}$. Since $X \to Y$ is laft, we have a cartesian square

<figure><img src="image_44.png" /></figure>

for each $i = \emptyset, 1, 2, 12$. The bottom map is an equivalence because $X \to Y$ is a nil-isomorphism, so that the top map is an equivalence as well. We therefore obtain a diagram of the form

<figure><img src="image_45.png" /></figure>

in which the bottom map is an equivalence because $X$ satisfies Zariski descent. Given a point $x \in \mathrm{colim}_I X(\pi_0(A)/I)$, we therefore need to check that the induced map on fibres $Y(A)_x \to Y(A_1)_{x_1} \times_{Y(A_{12})_{x_{12}}} Y(A_2)_{x_2}$ is an equivalence, where $x_i$ denotes the restriction of $x$ to the corresponding Zariski open. For this it suffices to show that for any nilpotent ideal $I$ and any point
$x \in X(\pi_0(A)/I)$, the square

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  & Y(A) \times_{Y(\pi_0 A/I)} \{x\} \arrow[r] \arrow[d] & Y(A_1) \times_{Y(\pi_0 A_1/I_1)} \{x_1\} \arrow[d] \\
  Y(A_2) \times_{Y(\pi_0 A_2/I_2)} \{x_2\} & &
\end{tikzcd}
\qquad (25)
$$

is cartesian. Indeed, one then finds that $Y(A)_x \simeq Y(A_1)_{x_1} \times_{Y(A_{12})_{x_{12}}} Y(A_2)_{x_2}$ by taking the filtered colimit over all $I$.

We now proceed as in the proof of Proposition 5.17: the map $A \to \pi_0(A)/I$ decomposes as a finite sequence $A = A(n) \to A(n-1) \to \dots \to A(0) = \pi_0 A/I$ where each $A(k) \to A(k-1)$ is a square zero extension by an $A(0)$-module $J(k)$. Localising, we similarly have that $A_i \to \pi_0(A_i)/I_i$ decomposes as a finite sequence $A_i = A(n)_i \to \dots \to A(0)_i = \pi_0(A_i)/I_i$ of square zero extensions by the $A(0)_i$-modules $J(k)_i = J(k) \otimes_{A(0)} A(0)_i$. We will prove by induction on this tower that the square (25) is cartesian; when $k=0$, all spaces are contractible and the assertion is evident.

For the inductive step, note that for each $i = \emptyset, 1, 2, 12$, there is a natural fibre sequence of spaces

$$
Y(A(k)_i) \times_{Y(A(0)_i)} \{x_i\} \to Y(A(k-1)_i) \times_{Y(A(0)_i)} \{x_i\} \to Y(A(0)_i \oplus J(k)_i[1]) \times_{Y(A(0)_i)} \{x_i\}
$$

because $Y$ has deformation theory. By inductive hypothesis, the middle terms for $i = \emptyset, 1, 2, 12$ fit into a cartesian square. It therefore suffices to show that the right terms also form a cartesian square. Since $Y$ has deformation theory, there is a natural equivalence

$$
Y(A(0) \oplus J(k)_i[1]) \times_{Y(A(0))} \{x\} \simeq Y(A(0)_i \oplus J(k)_i[1]) \times_{Y(A(0)_i)} \{x_i\}.
$$

and it suffices to verify that the square

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  & Y(A(0) \oplus J(k)[1]) \times_{Y(A(0))} \{x\} \arrow[r] \arrow[d] & Y(A(0) \oplus J(k)_1[1]) \times_{Y(A(0))} \{x\} \arrow[d] \\
  Y(A(0) \oplus J(k)_2[1]) \times_{Y(A(0))} \{x\} & & Y(A(0) \oplus J(k)_{12}[1]) \times_{Y(A(0))} \{x\}
\end{tikzcd}
$$

is cartesian. The functor $Y(A(0) \oplus -) \times_{Y(A(0))} \{x\} : \mathrm{Mod}_{A, \ge 0} \to \mathcal{S}$ preserves all pullback diagrams preserved by the inclusion into $\mathrm{Mod}_A$, again because $Y$ has deformation theory. We are therefore left with showing that $J(k)[1] \simeq J(k)_1[1] \times_{J(k)_{12}[1]} J(k)_2[1]$. Unravelling the definitions, this is nothing but Zariski descent for (the quasi-coherent sheaf associated to) the $A(0)$-module $J(k)$. $\square$

**Theorem 5.22** (Zariski descent for formal moduli stacks). *Given a derived scheme X, write $\mathcal{U}$ for the poset of affine opens of X. Then there is canonical equivalence*

$$
\operatorname{ModuliStk}_{X/R} \xrightarrow{\sim} \lim_{U \in \mathcal{U}^{\text{op}}} \operatorname{ModuliStk}_{U/R}.
$$

*Proof.* Let us consider the cartesian fibration

$$
\mathrm{ModuliStk}_{\mathcal{U}, R} \to \mathcal{U}
$$

obtained by restricting the cartesian fibration

$$
\mathrm{ModuliStk}_R \to \mathrm{PrStk}_R^{\mathrm{def}, \mathrm{laft-map}}
$$

from Construction 5.16 along the evident functor $\mathcal{U} \to \mathrm{PrStk}_R^{\mathrm{def,laft-map}}$. By [Lur09, Proposition 3.3.3.1], there is an equivalence

$$
\lim_{U \in \mathcal{U}} \mathrm{ModuliStk}_{U/R} \simeq \mathrm{Fun}_{\mathcal{U}}^{\mathrm{cart}} (\mathcal{U}, \mathrm{ModuliStk}_{\mathcal{U}, R}),
$$