where the right hand side denotes the $\infty$-category of cartesian sections of ModuliStk$_{U,R} \to U$.

We have a canonical functor

$$ \Psi: \mathrm{ModuliStk}_{X/R} \to \mathrm{Fun}_{U}^{\mathrm{cart}}(U, \mathrm{ModuliStk}_{U,R}) $$

sending $(X \to Y)$ to

$$ \{U \to Y_U^\wedge\}_{U \in U} $$

Here the formal completion $Y_U^\wedge \simeq Y \times_{Y^\sharp} U^\sharp$ is the prestack defined in Definition B.8, where we have used the notation $F^\sharp(-) := F(-_{\mathrm{red}})$. Note that each map $U \to Y$ is laft by [Lur04, Proposition 5.3.10], so $Y_U^\wedge$ agrees with $(Y/U)_{\mathrm{inf}}$.

The above functor is well-defined as each $U \to Y_U^\wedge$ is a formal moduli stack under $U$. Indeed, the fact that $Y_U^\wedge \simeq (Y/U)_{\mathrm{inf}}$ has deformation theory follows immediately from the definitions. Since $U \to Y$ is laft, $U \to Y_U^\wedge$ is laft by Lemma B.14 and clearly an equivalence on reduced objects.

To construct an inverse to $\Psi$, we will need some notation. Write $\mathrm{PrStk}_{R, <\infty}$ for the $\infty$-category of prestacks defined only on coconnective objects. Let $\mathrm{Stk}_R^{\mathrm{Zar}} \subset \mathrm{PrStk}_R$ ($\mathrm{Stk}_{R, <\infty}^{\mathrm{Zar}} \subset \mathrm{PrStk}_{R, <\infty}$) denote the full subcategories spanned by all prestacks $Y$ satisfying Zariski descent, i.e. satisfying $Y(0) \simeq *$ and sending Zariski squares (of eventually coconnective rings) to pullbacks. By Remark 5.20, the functor

$$ \mathrm{conv}: \mathrm{PrStk}_{R, <\infty} \to \mathrm{PrStk}_R; \quad X \mapsto \lim_n X(\tau_{\le n} -) $$

sends sheaves to sheaves.

We then define

$$ \Phi: \mathrm{Fun}_{U}^{\mathrm{cart}}(U, \mathrm{ModuliStk}_{U,R}) \to \mathrm{ModuliStk}_{X/R} $$

as the composite

$$ \mathrm{Fun}_{U}^{\mathrm{cart}}(U, \mathrm{ModuliStk}_{U,R}) \subset \mathrm{Fun}(U, \mathrm{Fun}(\Delta^1, \mathrm{Stk}_{R, <\infty}^{\mathrm{Zar}})) \xrightarrow{\mathrm{colim}} \mathrm{Fun}(\Delta^1, \mathrm{Stk}_{R, <\infty}^{\mathrm{Zar}}) \xrightarrow{\mathrm{conv}} \mathrm{Fun}(\Delta^1, \mathrm{Stk}_R^{\mathrm{Zar}}). $$

Note that $\mathrm{colim}$ is computed by sheafifying the pointwise colimit. We have used Proposition 5.21 to embed formal moduli stacks into Zariski sheaves.

We now want to verify that $\Phi$ takes values in formal moduli stacks under $X$. To this end, let us fix a family

$$ \{U \to Y_U\}_{U \in U} \in \mathrm{Fun}_{U}^{\mathrm{cart}}(U, \mathrm{ModuliStk}_{U,R}) $$

indexed by the poset $U$ of affine opens $U$ of $X$ and let $(X \to Y) \in \mathrm{Fun}(\Delta^1, \mathrm{Stk}_R^{\mathrm{Zar}})$ denote its image under $\Phi$. In a first step, we will prove that the natural map

$$ \{U \to Y_U\}_{U \in U} \to \{U \to Y_U^\wedge\}_{U \in U} $$

is an equivalence, i.e. that for each $U \in U$, the following square in $\mathrm{Stk}_{R, <\infty}^{\mathrm{Zar}}$ is cartesian:

<figure><img src="image_46.png" /></figure>

Note that the functor $(-)^{\sharp}: \mathrm{PrStk}_{R, <\infty} \to \mathrm{PrStk}_{R, <\infty}$ preserves colimits and Zariski sheaves, and commutes with Zariski sheafification. For the last assertion, it suffices to observe that for any covering sieve $S \to \mathrm{Spec}(A)$ for the Zariski topology and a map $f: \mathrm{Spec}(B) \to \mathrm{Spec}(A)^{\sharp}$, corresponding to $f_{\mathrm{red}}: \mathrm{Spec}(B_{\mathrm{red}}) \to \mathrm{Spec}(A_{\mathrm{red}})$, the base change $f^*S^{\sharp} \to \mathrm{Spec}(B)$ is a covering sieve as well: it is
the unique covering sieve whose restriction to $\text{Spec}(B_{\text{red}})$ coincides with $f_{\text{red}}^*S$. Consequently, the natural map of Zariski sheaves $\text{colim}_{U \in \mathcal{U}} Y_U^\sharp \to (\text{colim}_{U \in \mathcal{U}} Y_U)^\sharp$ is an equivalence.

To see that the above square is cartesian, it then suffices to consider the natural transformation $\alpha$ of $\mathcal{U}$-indexed diagrams in $\text{Stk}_{R, <\infty}^{\text{Zar}}$ given by

$$Y_U \to (Y_U)^\sharp$$

Write $\bar{\alpha}$ for the induced transformation between the $\mathcal{U}^\lambda$-indexed colimit cones. The transformation $\alpha$ is cartesian as for each $U \to V$, we have $Y_U \simeq (Y_V)^\wedge \simeq Y_V \times_{V^\sharp} U^\sharp$, which implies by [Lur09, Theorem 6.1.0.6] that the above square is also cartesian.

We will use this to show that $X \to Y$ is a formal moduli problem. The prestack $Y$ is convergent by construction and the map $X \to Y$ is laft by (the derived analogue of) [Lur16, 17.4.3.3]. To verify it has deformation theory, it suffices to check that for any pullback square of eventually coconnective animated rings

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

with $B \to A$ a nilpotent extension and any $f \in Y(B)$ with image $f_A \in Y(A)$, the map

$$Y(\mathcal{O}_{B'}) \times_{Y(\mathcal{O}_B)} \{f\} \to Y(\mathcal{O}_B \otimes_{B'} A') \times_{Y(\mathcal{O}_B \otimes_{B'} A)} \{f_A\}$$

of sheaves on the topological space $|\text{Spec}(B)| \cong |\text{Spec}(B')|$ is an equivalence on global sections. Taking global sections, this implies that the fibres $Y(B') \times_{Y(B)} \{f\} \to Y(A') \times_{Y(A)} \{f_A\}$ are equivalent, so that $Y(B') \simeq Y(B) \times_{Y(A)} Y(A')$.

Since $Y$ is a Zariski sheaf, it suffices to check this after restricting to a Zariski cover of $|\text{Spec}(B)|$. Let us pick a cover of $\text{Spec}(B)$ by affine opens $V_i \subset \text{Spec}(B)$ corresponding to affine opens $V_{i, \text{red}} \subset \text{Spec}(B_{\text{red}})$ such that each composite

$$f_{i, \text{red}}: V_{i, \text{red}} \subset \text{Spec}(B_{\text{red}}) \to Y_{\text{red}} \simeq X_{\text{red}}$$

factors through some affine open $U_i \subset X$. This implies that each restriction $f_i: V_i \to Y$ of the map $f$ factors through some $Y_{U_i}^\wedge$. The claim follows as $Y_{U_i}^\wedge \simeq Y_{U_i}$ by the above argument, and $Y_{U_i}$ has deformation theory.

Finally, to see that $X \to Y$ is an equivalence on reduced affine schemes, we note that precomposing with $A \mapsto A_{\text{red}}$ gives rise to a limit-preserving functor $\text{Fun}(\text{CR}^{\text{red}}, S) \to \text{Fun}(\text{CAlg}^{\text{an}}, S)$, $F \mapsto F(-_{\text{red}})$ which sends sheaves to sheaves. Here $\text{CR}^{\text{red}} \subset \text{CAlg}^{\text{an}}$ denotes the full subcategory of reduced discrete rings. Passing to left adjoints, we see that restriction to $\text{CR}^{\text{red}}$ commutes with sheafification. As each object $U \to Y_U$ induces an equivalence on reduced affine schemes, we deduce that $X = \text{colim}_{U \in \mathcal{U}} U \to \text{colim}_{U \in \mathcal{U}} Y_U = Y$ does so as well.

We may therefore conclude that the functor $\Phi$ lands in formal moduli stacks, and the above argument also shows that $\Psi \circ \Phi$ is equivalent to the identity.

To see that $\Phi \circ \Psi$ is equivalent to the identity, let us fix a formal moduli stack $X \to Y$ under $X$. It suffices to verify that the canonical map

$$\text{colim}_{U \in \mathcal{U}} Y_U^\wedge \to Y$$

is an equivalence of sheaves on eventually coconnective objects. Since colimits in the $\infty$-topos $\text{Stk}_{R, <\infty}^{\text{Zar}}$ are universal in the sense of [Lur09, Definition 6.1.1.2], it suffices to check that for all maps
$\mathrm{Spec}(A) \to Y$, the induced map

$$ \operatorname{colim}_{U \in \mathcal{U}} (Y_U^{\wedge} \times_Y \mathrm{Spec}(A)) \to \mathrm{Spec}(A) $$

is an equivalence. But

$$ Y_U^{\wedge} \times_Y \mathrm{Spec}(A) \simeq \mathrm{Spec}(A) \times_{Y^{\sharp}} U^{\sharp} \simeq \mathrm{Spec}(A) \times_{X^{\sharp}} U^{\sharp} $$

and hence the above map is an equivalence since $\operatorname{colim}_{U \in \mathcal{U}} U^{\sharp} \to X^{\sharp}$ is an equivalence. $\square$

**Proposition 5.23** (Monadicity of the tangent fibre). Let $X$ be a locally coherent $qcqs$ derived scheme over an animated ring $R$. Then the functor

$$ \begin{array}{ccc} T_{X/-}[1] : \operatorname{ModuliStk}_{X/R} & \longrightarrow & (\mathrm{QC}_X^{\vee})_{/T_{X/R}[1]} \\ & & \\ (X \to Y) & \longleftarrow & (T_{X/Y}[1] \to T_{X/R}[1]) \end{array} $$

in Construction 5.16 is a monadic right adjoint.

The proof uses the following $\infty$-categorical result:

**Lemma 5.24.** Let $F: \mathcal{C} \hookrightarrow \mathcal{D}: U$ be a monadic adjunction and let $G: \mathcal{D}' \to \mathcal{D}$ be a functor from an $\infty$-category $\mathcal{D}'$ with geometric realisations. If $UG$ admits a left adjoint, then $G$ admits a left adjoint.

*Proof.* As in [Lur17, Lemma 4.7.3.13], let $\mathcal{X} \subseteq \mathcal{D}$ be the full subcategory of objects $D \in \mathcal{D}$ such that $\mathrm{Map}_{\mathcal{D}}(D, G(-))$ is corepresentable and let $L: \mathcal{X} \to \mathcal{D}'$ be the corresponding partial left adjoint. For any simplicial diagram $D_\bullet$ in $\mathcal{X}$, the colimit $|L(D_\bullet)|$ corepresents $\mathrm{Map}_{\mathcal{D}}(|D_\bullet|, G(-))$, so that $\mathcal{X}$ is closed under geometric realisations. Since $UG$ admits a left adjoint, $\mathcal{X}$ contains all the free objects $F(C)$ with $C \in \mathcal{C}$. As every object in $\mathcal{D}$ is the geometric realisation of a (U-split) simplicial diagram of free objects, $\mathcal{X} = \mathcal{D}$. $\square$

*Proof of Proposition 5.23.* When $X = \mathrm{Spec}(B)$ is affine, the tangent functor $T_{X/-}[1]$ from Construction 5.16 can be identified with the composite

$$ T_{X/-}[1] : \operatorname{ModuliStk}_{X/R} \xrightarrow{\sim} \operatorname{Moduli}_{B/R} \xrightarrow{T_B/[1]} (\mathrm{QC}_B^{\vee})_{/T_{B/R}[1]} $$

where the first functor is the equivalence from Theorem 5.3, restricting formal moduli stacks to Artinian extensions of $B$, and the second functor takes the tangent complex of the corresponding formal moduli problem. It then follows from Proposition 3.12 and Theorem 4.19 that $T_{X/-}[1]$ is a monadic right adjoint preserving sifted colimits.

For non-affine $X$, let us first prove that $\operatorname{ModuliStk}_{X/R}$ has sifted colimits and that $T_{X/-}[1]$ preserves these. To this end, let us again write $\mathcal{U}$ for the category of Zariski open affine subsets of $X$. By Proposition B.32, have a commutative square

$$ \begin{array}{ccc} \operatorname{ModuliStk}_{X/R} & \longrightarrow & (\mathrm{QC}_X^{\vee})_{/T_{X/R}[1]} \\ & & \\ \xrightarrow{\approx} \downarrow & \downarrow \xrightarrow{\approx} \\ \lim_{U \in \mathcal{U}} \operatorname{ModuliStk}_{U/R} & \longrightarrow & \lim_{U \in \mathcal{U}} (\mathrm{QC}_U^{\vee})_{/T_{U/R}[1]} \end{array} $$

where the vertical functors are equivalences by Theorem 5.22 and Corollary A.39, respectively. Each functor $\operatorname{ModuliStk}_{U/R} \to (\mathrm{QC}_U^{\vee})_{/T_{U/R}[1]}$ preserves sifted colimits, and this also holds true for the transition maps associated to inclusions $j: U \hookrightarrow V$ of Zariski open subsets of $X$

$$ \operatorname{ModuliStk}_{V/R} \xrightarrow{(-)_{U}^{\wedge}} \operatorname{ModuliStk}_{U/R} \xrightarrow{(\mathrm{QC}_V^{\vee})_{/T_{V/R}[1]}} \xrightarrow{j^{*}} (\mathrm{QC}_U^{\vee})_{/T_{U/R}[1]} $$
For the functor $(-)^{\wedge}_{U}$, we use that the functors $T_{U/}[1]:$ ModuliStk$_{U/R} \to (QC_{U}^{\vee})_{T_{U/R}[1]}$ are conservative and hence detect sifted colimits. It then follows from [Lur09, Corollary 5.3.6.10] that $T_{X/-}[1]$ is also a sifted colimit preserving functor between $\infty$-categories with sifted colimits.

Next, we will show that $T_{X/-}[1]$ admits a left adjoint. By Lemma 5.24, it suffices to verify that

$$ \mathrm{ModuliStk}_{X/R} \xrightarrow{T_{X/-}[1]} (\mathrm{QC}^{\vee}_{X})_{T_{X/R}[1]} \xrightarrow{\mathrm{fib}} \mathrm{QC}^{\vee}_{X} $$

admits a left adjoint, since taking the fibre is a monadic right adjoint. This follows immediately from Proposition B.33, using that the left adjoint $F \mapsto X_F$ constructed there takes values in the full subcategory of formal moduli stacks since $X \to X_F$ is a nil-isomorphism.

The claim now follows from the Barr-Beck-Lurie theorem [Lur17, Theorem 4.7.3.5] as the tangent fibre is conservative by Proposition 5.17. $\square$

**Proposition 5.25 (Identification of the monad).** The right adjoint in Proposition 5.23 induces the partition Lie algebroid monad from Definition 3.2 the $\infty$-category $(QC_{X}^{\vee})_{T_{X/R}[1]}$.

*Proof.* Consider the commutative diagram

$$ \begin{tikzcd} \mathrm{ModuliStk}_{X/R} \arrow[r, "\sim"] \arrow[d, "T_{X/-}[1]"'] & \lim_{U \in \mathcal{U}} \mathrm{ModuliStk}_{U/R} \arrow[r, "\sim"] \arrow[d, "T_{U/-}[1]"'] & \lim_{U \in \mathcal{U}} \mathrm{Moduli}_{\mathcal{O}(U)/R} \arrow[r, "\sim"] \arrow[d, "T_{\mathcal{O}(U)/R}[1]"'] & \lim_{U \in \mathcal{U}} \mathrm{LieAlgd}_{\Delta, \mathcal{O}(U)/R} \\ (QC_{X}^{\vee})_{T_{X/R}[1]} \arrow[r, "\sim"'] & \lim_{U \in \mathcal{U}} (QC_{U}^{\vee})_{T_{U/R}[1]} \arrow[d, "\text{oblv}"'] & & \end{tikzcd} $$

where the left square is defined as in Proposition 5.23 and the two triangles arise from Theorem 5.3 and Theorem 4.19. The claim then follows from the construction of the monad $\mathrm{Lie}^{\pi}_{\Delta, X/R}$ in the proof of Theorem 3.1. $\square$

We deduce:

**Theorem 5.26 (Main theorem).** Let $X$ be a locally coherent $\mathrm{qcqs}$ derived $R$-scheme over an animated ring $R$.

(1) The shifted tangent fibre functor

$$ \begin{aligned} \mathrm{ModuliStk}_{X/R} & \longrightarrow (\mathrm{QC}^{\vee}_{X})_{T_{X/R}[1]} \\ (X \to Y) & \longleftarrow (T_{X/Y}[1] \to T_{X/R}[1]) \end{aligned} $$

lifts to an equivalence of $\infty$-categories

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

Hence every partition Lie algebroid on $X$ integrates uniquely to a formal moduli stack under $X$.

(2) This equivalence is functorial with respect to almost finitely presented maps. More precisely, if $X \to Y$ is a locally almost finitely presented map, there is a commuting diagram

$$ \begin{tikzcd} \mathrm{ModuliStk}_{Y/} \arrow[r, "T_{Y/-}[1]"'] & \mathrm{Lie}^{\pi}_{\Delta, Y/R} \arrow[d, "f^{\sharp}"] \\ (-)^{\wedge}_{X} \arrow[d, "f^{\sharp}"] & \mathrm{ModuliStk}_{X/-} \arrow[r, "T_{X/-}[1]"'] & \mathrm{Lie}^{\pi}_{\Delta, X/R} \arrow[d, "f^{\sharp}"] \arrow[Rightarrow, from=2-2, to=3-3, above=1pt, left=1pt, right=1pt] \end{tikzcd} $$

where $f^{\sharp}(\mathfrak{g} \to T_{Y/R}[1]) = f^{*}\mathfrak{g} \times_{(f^{*}L_{Y/R})^{\vee}[1]} T_{X/R}[1]$ as pro-coherent sheaves.
APPENDIX A. DERIVED ALGEBRA RECOLLECTIONS

The goal of this section is to introduce some elements of (homological) algebra that will be used throughout the text. We will start in Section A.1 by reviewing the theory of derived commutative rings, the non-connective analogues of animated (or simplicial) rings. We will work in a somewhat general setting that also includes the derived filtered and graded rings studied in Section 2. Section A.2 discusses the theory of pro-coherent modules over an animated ring.

A.1. **Derived commutative rings.** To set the stage, we will briefly recall the method of right-left extending monads from [BM, Section 3.2] and [BCN, Section 2.2] in the generality needed for us, see also [Rak20, Section 4.2] for the case of the commutative algebra monad.

**Definition A.1.** Let $\mathcal{C}$ be a presentable stable $\infty$-category with a left complete $t$-structure $(\mathcal{C}_{\ge 0}, \mathcal{C}_{\le 0})$. We call $\mathcal{C}$ is stably projectively generated if there exists a full additive subcategory $\mathrm{Vect}^t(\mathcal{C}) \subseteq \mathcal{C}_{\ge 0}$ whose objects are compact in $\mathcal{C}$ and projective in $\mathcal{C}_{\ge 0}$, generating $\mathcal{C}_{\ge 0}$ under sifted colimits.

In this situation, there is an equivalence

$$ \mathcal{C} \simeq \mathrm{Fun}^{\oplus}(\mathrm{Vect}^t(\mathcal{C})^{\mathrm{op}}, \mathrm{Sp}) $$

identifying $t$-structures [Rak20, Remark 4.2.2]. We define the subcategory of flat objects in $\mathcal{C}$ as the ind-completion of $\mathrm{Vect}^t(\mathcal{C})$:

$$ \mathrm{Flat}(\mathcal{C}) := \mathrm{Ind}(\mathrm{Vect}^t(\mathcal{C})) \subseteq \mathcal{C}. $$

**Example A.2.** For $A \in \mathrm{Alg}(\mathrm{Sp})$ a connective algebra object in spectra, the stable $\infty$-category $\mathrm{Mod}_A$ of (left) $A$-module spectra with its usual $t$-structure is stably projectively generated. In this case, we will take $\mathrm{Vect}_A^t$ to be the $\infty$-category consisting of finite direct sums of copies of $A$. Then $\mathrm{Flat}_A$ is the $\infty$-category of flat $A$-modules in the sense of [Lur17, Definition 7.2.2.10], by Lurie's version of Lazard's theorem (cf. Theorem 7.2.2.15 of [op.cit]).

**Definition A.3.** Let $\mathcal{C}$ be a projectively generated stable $\infty$-category with $t$-structure $(\mathcal{C}_{\ge 0}, \mathcal{C}_{\le 0})$ and $\mathcal{D}$ a presentable $\infty$-category. We will say that a functor $F: \mathcal{C} \to \mathcal{D}$ is right-left extended if it preserves sifted colimits and totalisations of finite cosimplicial diagrams with values in $\mathrm{Vect}^t(\mathcal{C}) \subseteq \mathcal{C}$.

The following follows from [BCN, Remark 2.45] and the definition of $\mathrm{Flat}(\mathcal{C})$ as an ind-completion:

**Proposition A.4.** *The restriction functor*

$$ \mathrm{Fun}_{\mathrm{RL}}(\mathcal{C}, \mathcal{D}) \longrightarrow \mathrm{Fun}^{\omega}(\mathrm{Flat}(\mathcal{C}), \mathcal{D}) \xrightarrow{\sim} \mathrm{Fun}(\mathrm{Vect}^{t}(\mathcal{C}), \mathcal{D}) $$

is fully faithful. Here $\mathrm{Fun}_{\mathrm{RL}}(\mathcal{C}, \mathcal{D}) \subseteq \mathrm{Fun}(\mathcal{C}, \mathcal{D})$ denotes the full subcategory of right-left extended functors and $\mathrm{Fun}^{\omega}(\mathrm{Flat}(\mathcal{C}), \mathcal{D}) \subset \mathrm{Fun}(\mathrm{Flat}(\mathcal{C}), \mathcal{D})$ is the full subcategory of functors preserving filtered colimits.

**Definition A.5.** Let us say that a functor $F: \mathrm{Vect}^t(\mathcal{C}) \to \mathcal{D}$, or equivalently, a filtered-colimit preserving functor $F: \mathrm{Flat}(\mathcal{C}) \to \mathcal{D}$, admits a right-left extension or is right-left extendable, if it is contained in the essential image. In that case, we will refer to its (unique) inverse image $\mathrm{LF}: \mathcal{C} \to \mathcal{D}$ as the derived functor, or right-left extension, of $F$. Note that this coincides with the non abelian derived functor in the sense of Dold-Puppe on the connective part $\mathcal{C}_{\ge 0} \simeq \mathcal{P}_{\Sigma}(\mathrm{Vect}^t(\mathcal{C}))$.

**Example A.6.** When $\mathcal{D}$ is stable, every functor $F: \mathrm{Vect}^t(\mathcal{C}) \to \mathcal{D}$ which is a filtered colimit of finite degree functors is right-left extendable. See [BCN, Corollary 2.49] for details.
**Example A.7.** Let $\mathrm{End}_{\mathrm{RL}}^{\mathrm{Flat}(\mathcal{C})}(\mathcal{C}) \subset \mathrm{End}_{\mathrm{RL}}(\mathcal{C})$ be the full subcategory of endofunctors $\mathcal{C} \to \mathcal{C}$ which preserve the full subcategory $\mathrm{Flat}(\mathcal{C}) \subset \mathcal{C}$. Restriction $\mathrm{End}_{\mathrm{RL}}^{\mathrm{Flat}(\mathcal{C})}(\mathcal{C}) \to \mathrm{End}^{\omega}(\mathrm{Flat}(\mathcal{C}))$ is a monoidal equivalence onto the full subcategory of $\mathrm{End}^{\omega}(\mathrm{Flat}(\mathcal{C}))$ spanned by right-left extendable functors. Consequently, any right-left extendable and filtered-colimit preserving monad $T: \mathrm{Flat}(\mathcal{C}) \to \mathrm{Flat}(\mathcal{C})$ induces a *derived monad* $\mathcal{L}T: \mathcal{C} \to \mathcal{C}$.

As an example of this kind of derived monad, consider the derived symmetric algebra monad. To define this, assume that $\mathcal{C}$ is a derived algebraic context in the sense of [Rak20, Section 4.2]:

**Definition A.8.** We will say that a stably projectively generated $\infty$-category $\mathcal{C}$ is a *derived algebraic context* if it comes equipped with a closed symmetric monoidal structure such that:

(1) $\mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$ is stable under tensor products and contains the monoidal unit.

(2) $\mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$ is an ordinary category.

(3) For each $M \in \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$, the object $\mathrm{Sym}^n(M) = \tau_{\le 0}((M_{h\Sigma_n}^{\otimes n}))$ is contained in $\mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$.

Condition (1) implies that $\mathcal{C}_{\ge 0}$ is stable under the tensor product. We include condition (2) for simplicity; it implies that $\mathrm{Flat}(\mathcal{C})$ is an ordinary category and that $\mathcal{C}^{\heartsuit}$ is a symmetric monoidal abelian category with enough projectives. Extending the functors $\mathrm{Sym}^n: \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C}) \to \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$ by filtered colimits and taking the sum over all $n \ge 0$, one then obtains a monad $\mathrm{Sym}: \mathrm{Flat}(\mathcal{C}) \to \mathrm{Flat}(\mathcal{C})$ preserving filtered colimits. Since each $\mathrm{Sym}^n: \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C}) \to \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$ is of degree $\le n$, this admits a derived monad $\mathcal{LSym}: \mathcal{C} \to \mathcal{C}$.

**Definition A.9.** If $\mathcal{C}$ is a derived algebraic context, we will refer to $\mathcal{LSym}$-algebras as *derived commutative algebras* and to connective $\mathcal{LSym}$-algebras as * animated commutative algebras* in $\mathcal{C}$, and write $\mathrm{CAlg}^{\mathrm{an}}(\mathcal{C}_{\ge 0}) \subseteq \mathrm{CAlg}^{\mathrm{der}}(\mathcal{C})$ for the corresponding $\infty$-categories.

The (ordinary) category $\mathrm{CAlg}^{\mathrm{der}}(\mathcal{C})^{\heartsuit}$ of discrete derived algebras is equivalent to the category of commutative algebras in $\mathcal{C}^{\heartsuit}$.

**Example A.10.** Let $\mathcal{C} = \mathrm{Mod}_Z$ with the usual $t$-structure, so that $\mathrm{Vect}^{\mathrm{ft}}(\mathcal{C}) = \mathrm{Vect}_Z^{\mathrm{ft}}$ is the (ordinary) category of finite free abelian groups and $\mathrm{Flat}(\mathcal{C}) = \mathrm{Flat}_Z$ is the (ordinary) category of flat abelian groups. This is a derived algebraic context, and the resulting monad $\mathrm{Flat}_Z \to \mathrm{Flat}_Z$ is the usual symmetric algebra monad. One thus obtains the $\infty$-categories $\mathrm{CAlg}^{\mathrm{an}} = \mathrm{CAlg}^{\mathrm{an}}(\mathrm{Mod}_Z)$ and $\mathrm{CAlg}^{\mathrm{der}} = \mathrm{CAlg}^{\mathrm{der}}(\mathrm{Mod}_Z)$ of animated and derived rings.

**Example A.11.** Let $\mathcal{C}$ be a derived algebraic context and let $\mathcal{J}$ be a small symmetric monoidal (ordinary) category. Then $\mathrm{Fun}(\mathcal{J}, \mathcal{C})$ is a derived algebraic context as well with respect to the Day convolution product and the $t$-structure in which a diagram is (co)connective if it is pointwise (co)connective in $\mathcal{C}$.

**Remark A.12.** There is a natural equivalence $\mathcal{LSym}(V) \otimes \mathcal{LSym}(W) \simeq \mathcal{LSym}(V \oplus W)$ for all $V, W \in \mathcal{C}$: this follows by right-left extension, using that there is such a natural equivalence for $V, W \in \mathrm{Flat}(\mathcal{C})$ by construction. Consequently, the coproduct of two derived algebras is simply given by their tensor product. Since $\mathrm{CAlg}^{\mathrm{der}}(\mathcal{C}) \to \mathcal{C}$ preserves sifted colimits by construction, it follows that pushouts of derived algebras are computed as relative tensor products, as usual.

**Remark A.13.** For every $M \in \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$, there is a natural map $(M^{\otimes n})_{h\Sigma_n} \to \mathrm{Sym}^n(M)$ from the coinvariants in $\mathcal{C}$. By right-left extension, this induces a map $\mathrm{Free}_{\mathcal{E}_\infty} \to \mathcal{LSym}$ from the $\mathcal{E}_\infty$-monad. In particular, any derived commutative algebra has an underlying $\mathcal{E}_\infty$-algebra and a module over a derived commutative algebra is simply a module over its underlying $\mathcal{E}_\infty$-algebra.
**Definition A.14.** For $A \in \mathrm{CAlg}^{\mathrm{der}}(\mathcal{C})$, we define the $\infty$-category of derived (commutative) $A$-algebras to be the under-category

$$ \mathrm{CAlg}_A^{\mathrm{der}}(\mathcal{C}) = \mathrm{CAlg}^{\mathrm{der}}(\mathcal{C})|_{A/}. $$

The forgetful functor $\mathrm{CAlg}_A^{\mathrm{der}}(\mathcal{C}) \to \mathrm{Mod}_A(\mathcal{C})$ is monadic and we will write $\mathbb{LSym}_A: \mathrm{Mod}_A(\mathcal{C}) \to \mathrm{Mod}_A(\mathcal{C})$ for the associated monad.

**Remark A.15.** If $A \in \mathrm{CAlg}^{\mathrm{der}}(\mathcal{C})$ is connective, then $\mathrm{Mod}_A(\mathcal{C})$ is stably projectively generated for the $t$-structure inherited from $\mathcal{C}$. The monad $\mathbb{LSym}_A$ is then derived from the restricted monad $\mathbb{LSym}_A: \mathrm{Flat}(\mathrm{Mod}_A(\mathcal{C})) \to \mathrm{Flat}(\mathrm{Mod}_A(\mathcal{C}))$.

The usual cotangent complex formalism for animated rings carries over to this setting. Below we will recall one way to see this, using an abstract method for constructing functors out of the $\infty$-category of $T$-algebras over some monad $T$ (see also [Hol23]).

**Proposition A.16.** Let $\mathcal{C}$ be a stably projectively generated stable $\infty$-category. Suppose that $T: \mathrm{Flat}(\mathcal{C}) \to \mathrm{Flat}(\mathcal{C})$ is a filtered colimit preserving and right-left extendable monad, and let $\mathbb{LT}: \mathcal{C} \to \mathcal{C}$ be the derived monad $T$. Let $\mathrm{Kl}_T(\mathrm{Flat}(\mathcal{C}))$ denote the full subcategory of $\mathrm{Alg}_T(\mathrm{Flat}(\mathcal{C}))$ spanned by the free algebras. Assume that

$$ F: \mathrm{Kl}_T(\mathrm{Flat}(\mathcal{C})) \to \mathcal{D} $$

is a functor such that

(1) the composite $F \circ \mathrm{Free}_T: \mathrm{Flat}(\mathcal{C}) \to \mathcal{D}$ preserves filtered colimits and is right-left extendable,

(2) $F$ preserves geometric realizations of simplicial objects in $\mathrm{Kl}_T(\mathrm{Flat}(\mathcal{C}))$ that become split after applying $\mathrm{Forget}_T$.

Then there exists a unique sifted colimit preserving functor $\mathbb{LF}: \mathrm{Alg}_T(\mathcal{C}) \to \mathcal{D}$ that extends $F$, together with a natural equivalence $\mathbb{LF} \circ \mathrm{Free} \simeq \mathbb{L}(F \circ \mathrm{Free})$.

**Proof.** Let $\mathrm{Fun}_{\mathrm{RL}}^{\omega}(\mathrm{Flat}(\mathcal{C}), \mathcal{D})$ denote the $\infty$-category of filtered-colimit preserving functors that are right-left extendable, and notice that there is a right action

$$ \mathrm{Fun}_{\mathrm{RL}}^{\omega}(\mathrm{Flat}(\mathcal{C}), \mathcal{D}) \cap \mathrm{Fun}_{\mathrm{RL}}^{\omega}(\mathrm{Flat}(\mathcal{C}), \mathrm{Flat}(\mathcal{C})). $$

In the above situation, we have an algebra object $T \in \mathrm{Fun}_{\mathrm{RL}}^{\omega}(\mathrm{Flat}(\mathcal{C}), \mathrm{Flat}(\mathcal{C}))$, and the functor $F \circ \mathrm{Free}_T \in \mathrm{Fun}_{\mathrm{RL}}^{\omega}(\mathrm{Flat}(\mathcal{C}), \mathcal{D})$ is a right module for $T$. Using the same logic as [Hol23, Proposition 2.2.15], taking right-left extension is lax monoidal and consequently, the functor $\mathbb{L}(F \circ \mathrm{Free}_T): \mathcal{C} \to \mathcal{D}$ is a right module for the derived monad $\mathbb{LT}$. On the other hand, $\mathrm{Forget}_T: \mathrm{Alg}_T(\mathcal{C}) \to \mathcal{C}$ is a left $T$-module. Since $\mathcal{D}$ admits sifted colimits, we can define $\mathbb{LF}$ by the formula:

$$ \mathbb{LF} := |\mathrm{Bar}(\mathbb{L}(F \circ \mathrm{Free}_T), T, \mathrm{Forget}_T)|: \mathrm{Alg}_T(\mathcal{C}) \to \mathcal{D}. $$

One readily verifies all the desired properties. $\square$

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

**Construction A.17** (Cotangent complex as a derived functor). Let $\mathcal{C}$ be a derived algebraic context and write $\mathrm{Mod}(\mathcal{C})$ for the $\infty$-category of tuples $(A, M)$ consisting of a derived commutative algebra $A$ and an $A$-module $M$ in $\mathcal{C}$. Notice that $\mathrm{Mod}$ arises as the $\infty$-category of algebras over a derived monad $T_{\mathrm{mod}}$ on $\mathcal{C}$, such that $\mathrm{Kl}_{T_{\mathrm{mod}}}(\mathcal{C})$ is simply the ordinary $\infty$-category of free derived algebras $A = \mathrm{Sym}(V)$ generated by $V \in \mathrm{Flat}(\mathcal{C})$, together with a free $A$-module $A \otimes W$ on $W \in \mathrm{Flat}(\mathcal{C})$. We now have:
(1) A functor between ordinary categories triv: $Kl_{T_{\text{mod}}}(\text{Flat}(\mathcal{C})) \to \text{CAlg}^{\text{der}}(\mathcal{C})$ sending each $(A, M)$ to the trivial square zero extension $A \oplus M$.

(2) A functor between ordinary categories $\Omega^1$: $Kl_{\text{Sym}}(\text{Flat}(\mathcal{C})) \to Kl_{T_{\text{mod}}}(\mathcal{C})$ defined by the universal property that $\text{Map}(\Omega_A^1, (B, M)) \simeq \text{Map}(A, B \oplus M)$. One readily verifies that such a universal object exists for each free algebra $A = \text{Sym}(V)$ and is given by the tuple $(\text{Sym}(V), \text{Sym}(V) \otimes V)$, which is indeed in $Kl_{T_{\text{mod}}}(\mathcal{C})$.

Both of these functors satisfy the conditions of Proposition A.16 and hence admit derived functors $L: \text{CAlg}^{\text{der}}(\mathcal{C}) \to \text{Mod}(\mathcal{C})$ and triv: $\text{Mod}(\mathcal{C}) \to \text{CAlg}^{\text{der}}(\mathcal{C})$. The universal property of $\Omega^1$ then implies that $L$ is left adjoint to triv. We refer to $L_A \in \text{Mod}_A(\mathcal{C})$ as the cotangent complex of $A$.

As usual, every map $f: A \to B$ in $\text{CAlg}^{\text{der}}(\mathcal{C})$ induces a map $(A, L_A) \to (B, L_B)$ in $\text{Mod}(\mathcal{C})$. Since the projection $\text{Mod}(\mathcal{C}) \to \text{CAlg}^{\text{der}}(\mathcal{C})$ is a cocartesian fibration, as $\mathcal{C} \in \text{CAlg}(\text{Pr}^L)$, this is equivalent to a map $f^*L_A \to L_B$ in $\text{Mod}_B$, whose cofibre is the relative cotangent complex $L_{B/A}$:

$$f^*L_A \longrightarrow L_B \longrightarrow L_{B/A}.$$

If $B = \text{LSym}_A(M)$ is the free derived $A$-algebra on $M \in \text{Mod}_A(\mathcal{C})$, then $L_{B/A} \simeq B \otimes_A M$.

We conclude with some observations about finite type conditions on maps between animated commutative algebras, well-known for animated rings.

**Definition A.18.** If $\mathcal{C}$ is a derived algebraic context and $A \in \text{CAlg}^{\text{an}}(\mathcal{C})$, then we will say that:

(1) a map $A \to B$ in $\text{CAlg}^{\text{an}}(\mathcal{C})$ is almost of finite presentation if $B$ is an almost compact object of $\text{CAlg}_A^{\text{an}}(\mathcal{C})$, that is, each $\tau_{\le n}B$ is a compact object of $\tau_{\le n}\text{CAlg}_A^{\text{an}}(\mathcal{C})$ [Lur17, Definition 7.2.4.8].

(2) an $A$-module $M \in \text{Mod}_A(\mathcal{C})$ is almost perfect if $M \in \text{Mod}_A(\mathcal{C})_{\ge k}$ and $M$ defines an almost compact object of $\text{Mod}_A(\mathcal{C})_{\ge k}$.

**Proposition A.19.** Let $\mathcal{C}$ be a derived algebraic context and $f: A \to B$ be a map of animated commutative algebras in $\mathcal{C}$. Then the following are equivalent:

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

(2) The map $\pi_0(A) \to \pi_0(B)$ of commutative algebras in $\mathcal{C}^{\heartsuit}$ is of finite presentation and $L_{B/A}$ is an almost perfect $B$-module.

In addition, the following assertions hold:

(3) If $B$ is almost perfect as an $A$-module, then $f$ is almost of finite presentation.

(4) If $f$ is almost of finite presentation and $\pi_0(A) \to \pi_0(B)$ is surjective, then $B$ is almost perfect as an $A$-module.

The proof of Proposition A.19 requires some preliminaries about cell decompositions and the resulting filtrations.

**Construction A.20.** Let $A \to B$ be a map of animated commutative algebras in $\mathcal{C}$. By a cell decomposition of $B$, we will mean a sequence of animated filtered $A$-algebras $A = B_{-1} \to B_0 \to B_1 \to \dots \to B$ constructed inductively according to the following procedure:

* Having constructed $B_n$, choose a map $\chi_n: M_n \to (B/B_n)[-1]$ from a connective $B_n$-module $M_n$ such that the cofibre is $(n+1)$-connective.
• Define $B_{n+1}$ as the pushout of derived filtered algebras

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

where the top map sends $M_n$ to zero. The map $B_n \to B$ then factors canonically over $\iota: B_n \to B_{n+1}$.

Let us record some properties of such cell decompositions:

(a) If $\pi_0(A) \to \pi_0(B)$ is surjective, one can take $B_0 = B_{-1} = A$.

(b) Each $B_{n+1}$ admits an increasing filtration whose associated graded is $\mathbb{LSym}_{B_n}(M_n[1])$: this filtration is the base change along $\chi$ of the filtration on $B_n$ obtained by applying $\mathbb{LSym}_{B_n}$ to the zero object with increasing filtration given by $F_0(0) = M_n$ and $F_i(0) = 0$ for $i > 0$. Here we use Example A.11 in the case $J = (\mathbb{Z}, \le)$.

(c) Each cofibre $B/B_n$ is $(n+1)$-connective and hence each $M_n$ is $n$-connective. To see this, we proceed by induction, the case $n = -1$ being evident. For the inductive step, note that the map $\chi_n$ factors as

$$ \chi_n: M_n[1] \to B_{n+1}/B_n \to B/B_n. $$

Since the cofibre of the composite is $(n+2)$-connective by construction, to show that the cofibre $B/B_{n+1}$ of the second map is $(n+2)$-connective it will suffice to show that $M_n[1] \to B_{n+1}/B_n$ has $(n+2)$-connective cofibre. Using the filtration from (b), this map decomposes as

$$ M_n[1] \simeq F_1(B_{n+1})/F_0(B_{n+1}) \to F_2(B_{n+1})/F_0(B_{n+1}) \to \dots \to B_{n+1}/F_0(B_{n+1}). $$

It therefore suffices to verify that the cofibre of each of these maps is $(n+2)$-connective. These cofibres are given by $\mathbb{LSym}_{B_n}^m(M_n[1])$ with $m \ge 2$, which fit into a cofibre sequence

$$ (M_n[1] \otimes_{B_n} \cdots \otimes_{B_n} M_n[1])_{h\Sigma_m} \to \mathbb{LSym}_{B_n}^m(M_n[1]) \to F_m(M_n[1]). $$

The result now follows because the outer terms are both $(n+2)$-connective. For the right term, this follows because $F_m$ is right-left extended from a functor on $\mathrm{Vect}^\mathrm{ft}(\mathcal{C})$ with 1-connective values.

(d) For each $n$, consider the natural $B_{n+1}$-linear maps

$$ B_{n+1} \otimes_{B_n} M_n[1] \to B_{n+1} \otimes_{B_n} (B_{n+1}/B_n) \to L_{B_{n+1}/B_n} $$

where the first map is induced by $\chi$ and the last map arises from the universal derivation $B_{n+1} \to B_{n+1} \oplus L_{B_{n+1}/B_n}$. The composite map is an equivalence of $(n+1)$-connective modules and the cofibre of the first map is $(n+2)$-connective, so that the cofibre of the natural map $B_{n+1} \otimes_{B_n} (B_{n+1}/B_n) \to L_{B_{n+1}/B_n}$ is $(n+3)$-connective. An inductive argument then shows that each $B \otimes_{B_n} (B/B_n) \to L_{B/B_n}$ has $(n+3)$-connective cofibre as well.

**Lemma A.21.** Let $\mathcal{C}$ be a derived algebraic context and let $\mathrm{Fun}(\langle\mathbb{Z}_{\ge 0}, =\rangle, \mathcal{C})$ be the $\infty$-category of non-negatively graded objects in $\mathcal{C}$, as in Example A.11. Given a graded algebra $A := A^\bullet$ and a graded module $M := M^\bullet$, the following hold:

(1) If $A^0 \otimes_A M \simeq 0$, then $M \simeq 0$.

(2) If $A$ and $M$ are connective and $A^0 \otimes_A M$ is an almost perfect $A^0$-module, then $M$ is almost perfect.
Proof. For (1), we prove the claim by induction on the weight. Clearly, $M^0 = 0$. Assuming that $M^i = 0$ for $i < n$, we have $M^n \simeq (A^0 \otimes_A M)^n$, so that $M^n = 0$ as well. For (2), note that

$$ \pi_0(A^0) \otimes_{\pi_0(A)} \pi_0(M) \cong \pi_0(A^0 \otimes_A M) $$

is the module of indecomposable associated to the graded $\pi_0(A)$-module $\pi_0(M)$. Since this module is finitely generated, it follows that $\pi_0(M)$ is a finitely generated $\pi_0(A)$-module. Lifting the generators provides a map $P \to M$ from a finite free graded $A$-module, which is surjective on $\pi_0$. The cofibre $N$ is then a 1-connective $A$-module such that $A^0 \otimes_A N$ is an almost perfect $A^0$-module. Proceeding by induction, one finds that $M$ is indeed almost perfect. $\square$

**Lemma A.22.** Let $\mathcal{C}$ be a derived algebraic context and let $\text{Fun}((\mathbb{Z}, \leq), \mathcal{C})$ be the $\infty$-category of objects in $\mathcal{C}$ with an unbounded increasing filtration, as in Example A.11. Let $A = F_*A$ be a connective filtered algebra which is complete, i.e. $\lim_{n \to -\infty} F_n A = 0$. Then a filtered $A$-module $M = F_*M$ is almost perfect if and only if it is complete and $\text{Gr}(M)$ is an almost perfect left $\text{Gr}(A)$-module.

Proof. If $M$ is almost perfect, then $\text{Gr}(M)$ is almost perfect because taking the associated graded preserves compact objects and is $t$-exact up to a shift. To see that $M$ is complete, let $n \geq 0$ and pick a perfect filtered $A$-module $N_n$ with a map $N_n \to M$ whose cofibre is $n$-connective. Since $A$ is complete, $N_n$ is complete and hence $\lim F^i M \simeq \lim F^i (M/N_n)$. By the Milnor sequence, the latter is $(n-1)$-connective. Since $n$ was arbitrary, we conclude that $\lim F^i M = 0$.

For the converse, it suffices that if $M$ is $n$-connective, there exists a cofibre sequence

$$ \bigoplus_{i=1}^{k} A \otimes F_{\geq w_i} V_i[n] \xrightarrow{f} M \longrightarrow M' $$

where $M'$ is $(n+1)$-connective and the first term consists of free filtered $A$-modules with $V_i \in \text{Vect}^{\text{ft}}(\mathcal{C})$, concentrated in filtration weight $\geq w_i$. Indeed, this implies that $M'$ is a complete $(n+1)$-connective module such that $\text{Gr}(M')$ is almost perfect; it follows by induction that each $\tau_{\leq m}(M)$ is equivalent to the $m$-truncation of a perfect $A$-module, so that $M$ is almost perfect.

Since $\text{Gr}(M)$ is an almost perfect $n$-connective $\text{Gr}(A)$-module, the desired cofibre sequence exists at the graded level. In particular, there exists a map $f': \bigoplus_{i=1}^n \text{Gr}(A) \otimes V_i(w_i)[n] \to \text{Gr}(M)$, where the domain is a sum of free $\text{Gr}(A)$-modules in various weights $w_i$. Since the $V_i \in \text{Vect}^{\text{ft}}(\mathcal{C})$ are projective in $\mathcal{C}_{\geq 0}$, we can lift $f'$ to a map $f$ of filtered modules. The cofibre $M'$ of $f$ is then a complete module with $(n+1)$-connective associated graded, and is thus itself $(n+1)$-connective by the Milnor sequence. $\square$

**Proof of Proposition A.19.** For (1) $\Rightarrow$ (2), note that $\pi_0(B)$ is a compact object in $\pi_0(A)$-algebras and hence finitely presented. The cotangent complex $L_{B/A}$ is almost perfect because for any filtered system of $k$-coconnective $B$-modules $M_\alpha$, the map $\text{colim} \text{Map}_B(L_{B/A}, M_\alpha) \to \text{Map}_B(L_{B/A}, \text{colim} M_\alpha)$ is the base change of the equivalence

$$ \text{colim} \text{Map}_{\text{Alg}^{\text{an}}_A(\mathcal{C})} (B, B \oplus M_\alpha) \to \text{Map}_{\text{Alg}^{\text{an}}_A(\mathcal{C})} (B, \text{colim}(B \oplus M_\alpha)). $$

For (2) $\Rightarrow$ (1), we claim that there exists a cell decomposition for $B$ with the property that each $B_n$ is almost of finite presentation over $A$. It then follows that $B$ is the colimit of a sequence of almost finitely presented algebras that stabilises upon $k$-truncation for any $k$. This implies that $A \to B$ almost finitely presented as well.

To construct this cell decomposition, we fix a finite presentation of $\pi_0(A) \to \pi_0(B)$. We can then choose a map $M_0 = A \otimes V_0 \to B$ from a free $A$-module on $V_0 \in \text{Vect}^{\text{ft}}(\mathcal{C})$ such that
$\pi_0(M_0) \to \pi_0(B)$ picks out the generators of $\pi_0(B)$. We then set $B_0 = \mathbb{L}\mathrm{Sym}_A(M_0)$. Next, we choose a map $M_1 = A \otimes V_1 \to \mathrm{fib}(B_0 \to B)$ from a free $B_0$-module on $V_1 \in \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$ such that $\pi_0(M_1) \to \pi_0(B_0) = \mathrm{Sym}_{\pi_0(A)}(\pi_0(M_0))$ picks out the relations of this presentation. The resulting $B_1$ is almost finitely presented and $\pi_0(B_1) = \pi_0(B)$. Now suppose we have constructed the desired cell decomposition up to $B_n$, for some $n \ge 1$. Since $A \to B_n$ is almost of finite presentation, the first part of the proof shows that $L_{B_n/A}$ is an almost perfect $B_n$-module. Consequently, $L_{B/B_n}$ is an almost perfect $B$-module. Since

$$B \otimes_{B_n} (B/B_n) \to L_{B/B_n}$$

has $(n+3)$-connective cofibre, it follows that we can choose an $n$-connective free $B_n$-module $M_n = B_n \otimes V_n[n]$ with $V_n \in \mathrm{Vect}^{\mathrm{ft}}(\mathcal{C})$, together with a map $\chi: M_n \to (B/B_n)[-1]$ that is surjective on $\pi_n$. The resulting map $B_n \to B_{n+1}$ is then almost finitely presented, so that the composite $A \to B_{n+1}$ is almost finitely presented as well.

For assertion (4), suppose that $f$ is almost finitely presented and that $\pi_0(A) \to \pi_0(B)$ is surjective, and consider the above cell decomposition. Then $B_0 = A$, and we claim that each $B_n$ is an almost perfect $A$-module. The colimit $B$ is then an almost perfect $A$-module as well, because the cofibre of each $B_n \to B_{n+1}$ is $(n+1)$-connective. Note that by part (b) of Construction A.20, $B_{n+1}$ comes with an increasing filtration by $B_n$-modules whose associated graded $\mathbb{L}\mathrm{Sym}_{B_n}(M_n[1])$ is almost perfect: indeed, each $\mathbb{L}\mathrm{Sym}_{B_n}^k(M_n[1])$ is almost perfect and (at least) $k$-connective, since $M_n$ was a connective almost perfect $B_n$-module. It follows that $B_{n+1}$ is an almost perfect $B_n$-module, so that an inductive argument shows that $B_n \in \mathrm{APerf}_A(\mathcal{C})$ (the case $n=0$ being evident).

Finally, let us prove assertion (3). Consider the cell decomposition for $B$ defined inductively by setting $B_{-1} = A$ and

$$M_n = (B/B_n)[-1].$$

We claim that $B$ is almost perfect as a $B_n$-module for each $B_n$. Assuming this, it follows by induction that each $B_n \to B_{n+1}$ is almost finitely presented, so that $A \to B_{n+1}$ is almost finitely presented as well. Since the sequence of truncations $\tau_{\le k}B_n$ is eventually constant, the colimit $B$ is then an almost finitely presented $A$-algebra as well.

We prove our claim by induction, the case $n = -1$ being evident. Assuming $B$ is an almost perfect $B_n$-module, we endow $B_{n+1}$ with the increasing filtration from part (b) of Construction A.20 and $B$ with the increasing filtration where $F_0B = B_n$ and $F_iB = B$ for $i \ge 1$. Then $F_*B$ is an $F_*B_{n+1}$-module, and it suffices to verify that it is almost perfect as a filtered module. By Lemma A.21, it suffices to verify that the associated graded is an almost perfect module over $\mathrm{Gr}(B_{n+1}) = \mathbb{L}\mathrm{Sym}_{B_n}(B/B_n)$. We then have a cofibre sequence of $\mathrm{Gr}(B_{n+1})$-modules $B_n(0) \to \mathrm{Gr}(B) \to (B/B_n)(1)$ of where the outer terms are concentrated in weight 0 and 1. By Lemma A.21, it therefore suffices to verify that the graded $B_n$-modules

$$B_n \otimes_{\mathbb{L}\mathrm{Sym}_{B_n}(B/B_n)} B_n \simeq \mathbb{L}\mathrm{Sym}_{B_n}(B/B_n[1]) \\ B_n \otimes_{\mathbb{L}\mathrm{Sym}_{B_n}(B/B_n)} (B/B_n) \simeq \mathbb{L}\mathrm{Sym}_{B_n}(B/B_n[1]) \otimes_{B_n} B/B_n$$

are almost perfect $B_n$-modules. This follows from the fact that $B/B_n$ is an almost perfect $B_n$-module and that $\mathbb{L}\mathrm{Sym}_{B_n}(B/B_n[1])$ is almost perfect, because each $\mathbb{L}\mathrm{Sym}_{B_n}^k(B/B_n[1])$ is almost perfect and $k$-connective. $\square$

### A.2. Pro-coherent modules.
As highlighted in [?], the usual $\infty$-category of quasi-coherent sheaves on a derived scheme (or prestack) is not very well-adapted to deformation theory. For schemes that are locally almost finitely presented over a field, it is better to consider their categories of
ind-coherent sheaves instead. In this situation, Serre duality provides a natural identification between ind-coherent sheaves and the dual notion of pro-coherent sheaves: on an almost finitely presented scheme $X$ over a field $k$, this is simply given by $\mathrm{Ind}(\mathrm{Coh}_X) \simeq \mathrm{Ind}(\mathrm{Coh}_X^{\mathrm{op}})$. To also deal with situations without Serre duality, where we cannot directly use the machinery of ind-coherent sheaves, let us introduce the following provisional modification of the $\infty$-category of pro-coherent sheaves over an affine derived scheme:

**Definition A.23.** Given an animated ring $A$, we will write $\mathrm{QC}_A^\nabla$ for the $\infty$-category of exact functors $F: \mathrm{Mod}_A \to \mathrm{Sp}$ satisfying the following two conditions:

(1) They are convergent, in the sense that they preserve limits of Postnikov towers in $\mathrm{Mod}_A$.

(2) They are almost finitely presented, in the sense that for each $n \in \mathbb{N}$ and each filtered diagram of $n$-coconnective $A$-modules $M_\alpha$, the map $\mathrm{colim}_\alpha F(M_\alpha) \to F(\mathrm{colim} M_\alpha)$ is an equivalence.

**Warning A.24.** For a general animated ring, the $\infty$-category $\mathrm{QC}_A^\nabla$ is not very manageable and one should view this as an ad-hoc definition. The definition will essentially only become useful when $A$ is a coherent animated ring: in that case $\mathrm{QC}_A^\nabla \simeq \mathrm{Ind}(\mathrm{Coh}_A^{\mathrm{op}})$ can be identified with the compactly generated $\infty$-category of pro-coherent sheaves on $A$.

Keeping this warning in mind, let us nonetheless record some formal properties of $\mathrm{QC}_A^\nabla$.

**Notation A.25.** If $\mathcal{C}$ is a stable $\infty$-category with a $t$-structure, we let $\mathcal{C}^+ = \bigcup_n \mathcal{C}_{\le n}$, $\mathcal{C}^- = \bigcup_n \mathcal{C}_{\ge n}$ and $\mathcal{C}^b = \mathcal{C}^+ \cap \mathcal{C}^-$ be the full subcategories of eventually coconnective, eventually connective and bounded objects [Lur17].

**Lemma A.26.** Let $A$ be an animated ring. Then the following $\infty$-categories are naturally equivalent:

(1) The $\infty$-category $\mathrm{QC}_A^\nabla$.

(2) The $\infty$-category of exact functors $\mathrm{Mod}_A^+ \to \mathrm{Sp}$ that are almost finitely presented.

(3) The $\infty$-category of left exact functors $F: \mathrm{Mod}_A^+ \to \mathcal{S}$ that are almost finitely presented.

(4) The $\infty$-category of functors $F: \mathrm{Mod}_{A, \ge 0}^+ \to \mathcal{S}$ that are locally almost finitely presented and preserve the terminal object and pullbacks along a $\pi_0$-surjection.

(5) The $\infty$-category of functors $F: \mathrm{Mod}_{A, \ge 0} \to \mathcal{S}$ preserving limits of almost eventually constant towers and satisfying the conditions from (4).

*Proof.* The equivalence between (1) and (2) is given by restriction and right Kan extension along $j: \mathrm{Mod}_A^+ \hookrightarrow \mathrm{Mod}_A$. Recall that the right Kan extension is given explicitly by $j_*F(M) = \lim F(\tau_{\le n}M)$. The equivalence between (2) and (3) is given by postcomposition with $\Omega^\infty$. The equivalence between (3) and (4) is given by restriction along the inclusion $i_1: \mathrm{Mod}_A^b \hookrightarrow \mathrm{Mod}_A^+$ and $i_2: \mathrm{Mod}_{A, \ge 0}^+ \hookleftarrow \mathrm{Mod}_A^b$ of bounded (resp. bounded connective) objects. The inverse is given by right Kan extension along $i_2$ followed by left Kan extension along $i_1$. The equivalence between (4) and (5) follows by restriction and right Kan extension along $\mathrm{Mod}_{A, \ge 0}^+ \hookrightarrow \mathrm{Mod}_{A, \ge 0}$. $\square$

**Lemma A.27.** Let $A$ be an animated ring and let $\kappa$ be a regular cardinal such that $\tau_{\le n}: \mathrm{Mod}_A \to \mathrm{Mod}_A$ preserves $\kappa$-compact objects. Then restriction and left Kan extension along $i: \mathrm{Mod}_A^{+;\kappa} \hookrightarrow \mathrm{Mod}_A^+$ induces an equivalence between $\mathrm{QC}_A^\nabla$ and the $\infty$-category of exact functors $F: \mathrm{Mod}_A^{+;\kappa} \to \mathrm{Sp}$ preserving colimits of $\kappa$-small filtered diagrams of $n$-coconnective objects, for each $n$.

*Proof.* Since $\tau_{\le n}$ preserves $\kappa$-small objects, for any $n$-coconnective $A$ module $M$ the inclusion $(\mathrm{Mod}_{A, \le n}^\kappa)_{/M} \hookrightarrow (\mathrm{Mod}_A^{+;\kappa})_{/M}$ is cofinal. Using that $(\mathrm{Mod}_{A, \le n}^\kappa)_{/M}$ is filtered, one readily sees that $i_1 i_* F \simeq F$ for any $F \in \mathrm{QC}_A^\nabla$. The identification of its essential image follows from the fact that any filtered colimit can be written as a $\kappa$-filtered colimit of $\kappa$-small filtered colimits. $\square$
**Corollary A.28.** Let A be an animated ring A. Then $QC_A^\vee$ is a stable presentable $\infty$-category.

*Proof.* By Lemma A.27, $QC_A^\vee \subseteq \text{Fun}(\text{Mod}_A^{+,\kappa}, \text{Sp})$ is the full subcategory of functors preserving certain types of colimits. It is hence presentable by the proof of [Lur09, Proposition 5.5.3.8]. □

**Corollary A.29.** Let $F: \text{Mod}_A \to \text{Sp}$ be an exact functor which is accessible and convergent and consider the $\infty$-category $(QC_A^\vee)_F$ of functors in $QC_A^\vee$ equipped with a natural transformation to $F$. This $\infty$-category admits a terminal object $F^{\text{aft}}$, which we will refer to as the almost finitely presented approximation to $F$.

*Proof.* Since $F$ is convergent and $\kappa$-accessible for some uncountable $\kappa$, it suffices to verify that there exists a universal object $F^{\text{aft}} \in QC_A^\vee$ equipped with a natural transformation to the restriction $F: \text{Mod}_A^{+,\kappa} \to \text{Sp}$. The previous proof shows that for large enough $\kappa$, the inclusion $QC_A^\vee \subseteq \text{Fun}(\text{Mod}_A^{+,\kappa}, \text{Sp})$ is accessible and preserves colimits, and hence admits a right adjoint. □

For any A, there is an adjoint pair

$$ \iota: \text{Mod}_A \xrightarrow{\phantom{\simeq}} QC_A^\vee : \upsilon $$

where the left adjoint sends an A-module $M$ to the exact functor $M \otimes_A (-): \text{Mod}_A^+ \to \text{Sp}$ and the right adjoint sends any $F: \text{Mod}_A^+ \to \text{Sp}$ to the A-module $\upsilon(F) = \lim_n F(\tau_{\le n} A)$.

**Observation A.30.** The $\infty$-category $QC_A^\vee$ comes equipped with a left complete t-structure whose connective part consists of those $F: \text{Mod}_A^+ \to \text{Sp}$ that are right t-exact. This induces an equivalence

$$ \text{Mod}_{A, \ge 0} \simeq QC_{A, \ge 0}^\vee $$

Indeed, the equivalences of Lemma A.26 identify $QC_{A, \ge 0}^\vee$ with the $\infty$-category of colimit-preserving functors $F: \text{Mod}_{A, \ge 0} \to \mathcal{S}$, which is equivalent to $\text{Mod}_{A, \ge 0}$ via sending such a functor $F$ to the A-module $F(A) \in \text{Mod}_{A, \ge 0}$.

Consequently, $\text{Mod}_A$ is the right completion of $QC_A^\vee$. In particular, we will typically identify eventually connective A-modules with the corresponding object in $QC_A^\vee$. If A is eventually coconnective, then $QC_A \to QC_A^\vee$ is fully faithful on all modules (not just the eventually connective ones).

Let A be an animated ring and $F \in QC_A^\vee$. For each $N \in \text{Mod}_A^+$, the spectrum $F(N)$ inherits a natural A-module structure from the A-module structure on $N$. More precisely, one can this describe this A-module structure on $F(N)$ by the exact functor $\text{Perf}(A)^{\text{op}} \to \text{Sp}$ sending $M$ to $F(M^\vee \otimes_A N)$. Using this, the $\infty$-category $QC_A^\vee$ is tensored over $\text{Mod}_A$ via

$$ (26) \qquad \otimes_A: QC_A^\vee \times \text{Mod}_A \longrightarrow QC_A^\vee; \quad (M \otimes_A F)(N) = M \otimes_A (F(N)) $$

for each $N \in \text{Mod}_A^+$. This preserves colimits in each variable and the functor $\iota: \text{Mod}_A \to QC_A^\vee$ is compatible with the tensoring.

**Definition A.31.** Let $M \in \text{Mod}_A$ be an A-module. Then the tensoring $M \otimes_A -: QC_A^\vee \to QC_A^\vee$ admits a right adjoint. We define the pro-coherent dual $M^\vee$ of M to be the value of this right adjoint on A.

**Example A.32.** If M is a perfect A-module, then the adjoint to $M \otimes_A -: QC_A^\vee \to QC_A^\vee$ is given by tensoring with the dual perfect module. Consequently, the pro-coherent dual of a perfect A-module is simply its usual A-linear dual. In terms of functors, one can also identify this with the functor $\text{Hom}_A(M, -): \text{Mod}_A \to \text{Sp}$. Using this and the fact that $(-)^\vee$ sends colimits in $\text{Mod}_A$ to limits
in QC$^{\vee}$, it follows that for any eventually connective $M \in \mathrm{Mod}_A^-$, its pro-coherent dual $M^{\vee}$ is the almost finitely presented approximation to $\mathrm{Hom}_A(M, -)$ in the sense of Corollary A.29.

We will be particularly interested in the pro-coherent duals of almost perfect A-modules: in this case, $M^{\vee}$ is given by $\mathrm{Hom}_A(M, -)$: $\mathrm{Mod}_A \to \mathrm{Sp}$, which is already almost finitely presented.

**Definition A.33.** We will say that a pro-coherent module $F \in \mathrm{QC}_A^{\vee}$ is dually almost perfect if $F: \mathrm{Mod}_A \to \mathrm{Sp}$ is corepresented by an almost perfect A-module. Taking pro-coherent duals therefore determines an equivalence

$$ (-)^{\vee}: \mathrm{APerf}_A^{\mathrm{op}} \xrightarrow{\sim} \mathrm{dAPerf}_A $$

between almost perfect A-modules and dually almost perfect pro-coherent A-modules.

**Lemma A.34.** Let $M \in \mathrm{Mod}_A$ be an A-module and let $M^{\vee}$ be its pro-coherent A-linear dual. Then pro-coherent duality determines an equivalence

$$ (27) \qquad (-)^{\vee}: ((\mathrm{APerf}_A)_{M/})^{\mathrm{op}} \xrightarrow{\sim} (\mathrm{dAPerf}_A)_{M^{\vee}}. $$

*Proof.* The functor (27) defines a map of right fibrations covering the equivalence $\mathrm{APerf}_A^{\mathrm{op}} \simeq \mathrm{dAPerf}_A$. It therefore suffices to verify that it induces an equivalence on fibres, i.e. that the map $\mathrm{Map}_{\mathrm{Mod}_A}(M, N) \to \mathrm{Map}_{\mathrm{QC}_A^{\vee}}(N^{\vee}, M^{\vee})$ is an equivalence for every $N \in \mathrm{APerf}_A$. To see this, recall that $N^{\vee}: \mathrm{Mod}_A \to \mathrm{Sp}$ is corepresented by $N$ and $M^{\vee}: \mathrm{Mod}_A \to \mathrm{Sp}$ is the almost finitely presented approximation to the functor corepresented by $M$, so that $\mathrm{Map}_{\mathrm{QC}_A^{\vee}}(N^{\vee}, M^{\vee})$ is equivalent to the space of maps between the functors corepresented by $N$ and $M$. Unravelling the definitions, one then sees that $\mathrm{Map}_{\mathrm{Mod}_A}(M, N) \to \mathrm{Map}_{\mathrm{QC}_A^{\vee}}(N^{\vee}, M^{\vee})$ is an equivalence by the Yoneda lemma. $\square$

**Definition A.35.** Let $A$ be an animated ring. For all $-\infty \le a \le b \le \infty$, we will say that an object $F \in \mathrm{QC}_A^{\vee}$ has tor-amplitude in $[a, b]$ if for any discrete A-module $M \in \mathrm{Mod}_A^{\vee}$,

$$ \pi_n F(M) \neq 0 \qquad \implies \qquad a \le n \le b. $$

Write $\mathrm{QC}_A^{\vee, [a, b]} \subseteq \mathrm{QC}_A^{\vee}$ for the full subcategory on the objects with tor-amplitude in $[a, b]$.

**Example A.36.** We have the following special cases:

(1) A pro-coherent module $F$ has tor-amplitude in $[0, \infty]$ if and only if $F$ is connective. In particular, $\mathrm{QC}_A^{\vee, [0, \infty]} \simeq \mathrm{Mod}_{A, \ge 0}$ is equivalent to the ordinary $\infty$-category of connective A-modules.

(2) $\mathrm{QC}_A^{\vee, [0, 0]} \simeq \mathrm{Flat}_A$ coincides with the full subcategory of $\mathrm{Mod}_{A, \ge 0}$ spanned by the flat A-modules.

(3) Let $F = M^{\vee}$ be dually almost perfect. Then $M^{\vee}$ has tor-amplitude in $[-b, -a]$ if and only if $M$ has tor-amplitude in $[a, b]$.

Let us now turn to the functoriality of $\mathrm{QC}_A^{\vee}$ in the animated ring $A$.

**Definition A.37.** Let $f: A \to B$ be a map of rings. We will write $f^*: \mathrm{QC}_A^{\vee} \to \mathrm{QC}_B^{\vee}: f_*$ for the adjoint pair whose left adjoint is given by precomposition with $f_*: \mathrm{Mod}_B^{\vee} \to \mathrm{Mod}_A^{\vee}$

Note that $f^*$ fits in a commuting square

$$ \begin{tikzcd} \mathrm{Mod}_A \arrow[r, "f^*"'] & \mathrm{Mod}_B \arrow[d, "\iota"] \\ \mathrm{QC}_A^{\vee} \arrow[r, "f^*"'] & \mathrm{QC}_B^{\vee}. \arrow[d, "\iota"'] \end{tikzcd} $$
In the presence of Serre duality, $f^*$ is Serre dual to the $f^1$-functor on ind-coherent sheaves [Gai11, Section 9.2.3]

**Lemma A.38.** If $f: A \to B$ has finite tor-amplitude, then the right adjoint $f_*$ preserves colimits. Furthermore, for any pullback square

$$ \begin{array}{ccc} \mathrm{Spec}(B') & \xrightarrow{f'} & \mathrm{Spec}(B) \\ \bigg\downarrow & & \bigg\downarrow \\ \mathrm{Spec}(A') & \xrightarrow{f} & \mathrm{Spec}(A). \end{array} $$

where $f$ has finite tor-amplitude, the Beck-Chevalley map $g^*f_* \to f'_*g'^*$ is an equivalence.

*Proof.* If $f$ has finite tor-amplitude, then $f_*$ is simply given by restriction along $f^*: \mathrm{Mod}_A^+ \to \mathrm{Mod}_B^+$. Using this, one sees that for every $F \in \mathrm{QC}_{A'}^\vee$ and $M \in \mathrm{Mod}_B^+$, the Beck-Chevalley map $(g^*f_*F)(M) \to (f'_*g'^*F)(M)$ is given by the image under $F$ of the Beck-Chevalley map $g^*f_*M \to f'_*g'^*M$, which is an equivalence. $\square$

**Corollary A.39.** The functor $\mathrm{QC}^\vee: \mathrm{CAlg}_R^{\mathrm{an}} \to \mathrm{Pr}^L$ satisfies descent with respect to universal descent morphisms of finite tor-amplitude, in the sense of [Lur16, Definition D.3.1.1]. In particular, it satisfies fppf descent.

*Proof.* Using that $A \mapsto \mathrm{Mod}_A$ preserves products, one readily sees that $\mathrm{QC}^\vee$ preserves finite products of animated rings. Next, let $f: A \to B$ be a universal descent morphism of finite tor-amplitude. Let $B^\bullet$ be the corresponding Čech nerve and consider the map $\theta: \mathrm{QC}_A^\vee \to \lim \mathrm{QC}_{B^\bullet}^\vee$. Using Lemma A.38 and [Lur17, Corollary 4.7.5.3], we see that $\theta$ is an equivalence if $f^*: \mathrm{QC}_A^\vee \to \mathrm{QC}_B^\vee$ is conservative. To see this, take $F \in \mathrm{QC}_A^\vee$ such that $f^*F \simeq 0$. The objects $M \in \mathrm{Mod}_A$ such that $F(M) \simeq 0$ form a stable subcategory of $\mathrm{Mod}_A$ that contains $f_*(\mathrm{Mod}_B)$ and is closed under retracts; since $f$ was a universal descent morphism, this means that it contains every $A$-module. $\square$

We now specialise to the case where $A$ is a coherent animated ring.

**Definition A.40.** An animated ring $A$ is said to be *coherent* if the standard $t$-structure on $\mathrm{Mod}_A$ restricts to a $t$-structure on $A\mathrm{Perf}_A$. Equivalently, $\pi_0(A)$ is coherent and each $\pi_n(A)$ is a finitely presented $\pi_0(A)$-module.

**Definition A.41.** Assume $A$ is a coherent animated ring. An almost perfect $A$-module $M$ is coherent if it eventually connective and coconnective. Equivalently, an $A$-module $M$ is coherent if all homotopy groups $\pi_n M$ are finitely presented over $\pi_0 A$, and only finitely many of $\pi_n M$ are non-zero. We denote $\mathrm{Coh}_A \subset A\mathrm{Perf}_A$ the full subcategory of coherent modules.

**Lemma A.42.** If $A \in \mathrm{CAlg}^{\mathrm{an}}$ is coherent, then $\mathrm{QC}_A^\vee$ is equivalent to the categories of:

(1) Exact functors $F: \mathrm{Coh}_A \to \mathrm{Sp}$.

(2) Functors $F: \mathrm{Coh}_{A, \ge 0} \to S$ that preserve the terminal object and pullbacks along $\pi_0$-surjections.

(3) Functors $F: A\mathrm{Perf}_{A, \ge 0} \to S$ that preserve almost eventually constant towers, the terminal object and pullbacks along $\pi_0$-surjections.

In particular, $\mathrm{QC}_A^\vee \simeq \mathrm{Ind}(\mathrm{Coh}_A^{\mathrm{op}})$ is compactly generated.

*Proof.* This follows from Lemma A.27 with $\kappa = \omega$, as well as the argument from Lemma A.26. $\square$
**Remark A.43.** Let $A$ be a coherent animated ring with dualising complex $\omega_A$. Then the $t$-structure on $\mathrm{QC}^\vee_A$ from Observation A.30 is Serre dual to the $t$-structure on $\mathrm{Ind}(\mathrm{Coh}_A)$ whose connective part is generated by $\omega_A$.

**Corollary A.44.** The following holds for a map of coherent animated rings $f: A \to B$:

(1) If $f$ has finite tor-amplitude, then the functor $f^*: \mathrm{QC}^\vee_A \to \mathrm{QC}^\vee_B$ coincides with $\mathrm{Ind}(f^*)$, for $f^*: \mathrm{Coh}^\mathrm{op}_A \to \mathrm{Coh}^\mathrm{op}_B$.

(2) If $f$ is finite, then $f^*$ has a left adjoint $f_!: \mathrm{QC}^\vee(B) \to \mathrm{QC}^\vee(A)$ given by the ind-completion of $f_*: \mathrm{Coh}^\mathrm{op}_B \to \mathrm{Coh}^\mathrm{op}_A$.

**Corollary A.45.** Let $A$ be a coherent animated ring. Then $\mathrm{QC}^\vee_A$ carries a unique closed symmetric monoidal structure whose restriction to $\mathrm{dAPerf}_A \simeq \mathrm{APerf}^\mathrm{op}_A$ is equivalent to the usual symmetric monoidal structure on almost perfect $A$-modules, i.e. $M^\vee \otimes_A N^\vee = (M \otimes_A N)^\vee$ for all $M, N \in \mathrm{APerf}_A$. Every map $f: A \to B$ between coherent animated rings induces a symmetric monoidal functor $f^*: \mathrm{QC}^\vee_A \to \mathrm{QC}^\vee_B$.

*Proof.* Using part (3) of Lemma A.42, it suffices to verify that $\mathrm{QC}^\vee_A \subseteq \mathrm{Fun}(\mathrm{APerf}_{A, \ge 0}, \mathcal{S})$ is a monoidal left Bousfield localisation with respect to the Day convolution product. This follows from the fact that if $F$ is a functor as in (3) and $M \in \mathrm{APerf}_{A, \ge 0}$, then $F(M \otimes_A -)$ satisfies the conditions from (3) as well. $\square$

**Remark A.46.** One readily verifies that $\iota: \mathrm{Mod}_A \to \mathrm{QC}^\vee_A$ is symmetric monoidal and that the tensoring from (26) is induced by this symmetric monoidal functor. It follows that the pro-coherent dual of an $A$-module $M$ from Definition A.31 coincides with the dual of $\iota(M)$ with respect to the symmetric monoidal structure from Corollary A.45.

# APPENDIX B. DERIVED ALGEBRAIC GEOMETRY RECOLLECTIONS

We recall the following terminology:

**Definition B.1 (Prestacks).** A prestack is an accessible functor $X: \mathrm{CAlg}^{\mathrm{an}} \to \mathcal{S}$ from animated rings to spaces. We will write $\mathrm{PrStk}$ for the $\infty$-category of prestacks. Given a prestack $S$, we let

(1) $\mathrm{PrStk}_S$ be the $\infty$-category of $S$-prestacks, that is, prestacks $X$ endowed with a map $X \to S$.

(2) $\mathrm{Aff}_S$ be the $\infty$-category of *affine derived schemes* with a map to $S$, i.e. all $\mathrm{Spec}(A) \to S$ where $\mathrm{Spec}(A)$ is the functor corepresented by an animated ring $A$.

The $\infty$-category of $S$-prestacks can then also be identified with the $\infty$-category of accessible functors $(\mathrm{Aff}_S)^{\mathrm{op}} \to \mathcal{S}$.

**Example B.2.** When $S = \mathrm{Spec}(R)$ is affine, $\mathrm{Aff}_S \simeq \mathrm{CAlg}^{\mathrm{an,op}}_R$ can be identified with the opposite of the $\infty$-category of animated commutative $R$-algebras. Consequently, a prestack $X$ over $R$ is simply an accessible functor $X: \mathrm{CAlg}^{\mathrm{an}}_R \to \mathcal{S}$; we will write $\mathrm{PrStk}_R$ for the resulting $\infty$-category of $R$-prestacks.

The purpose of this section is to briefly recall some of the results and constructions from $[?]$ in this setting.

B.1. **Finiteness and deformation-theoretic properties of prestacks.** We begin with a recollection of some of the conditions that one can impose on a prestack, following $[?]$ and [Lur16, Section 17]. As in $[?$, Volume 1, Chapter 2] and [Lur16, Section 17.4], the following finiteness condition will play a central role in this text:
**Definition B.3.** A map of prestacks $X \to S$ is said to be

• convergent if for all $A \in \mathrm{CAlg}^{\mathrm{an}}$, the square

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

is cartesian.

• locally almost finitely presented if it is convergent and satisfies the following condition for each $n \ge 0$: for each filtered diagram $A_\alpha: I \to \mathrm{CAlg}^{\mathrm{an}}_{\le n}$ of $n$-truncated animated rings with colimit $A$, the square

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

is cartesian. Equivalently, $X$ is convergent and for any $\mathrm{Spec} A \to S$ and each $n \ge 0$, the base change $X \times_S \mathrm{Spec}(A)$ restricts to a functor $\mathrm{CAlg}^{\mathrm{an}}_{A, \le n} \to S$ that preserves filtered colimits.

We will write $\mathrm{PrStk}_S^{\mathrm{laft}} \hookrightarrow \mathrm{PrStk}_S^{\mathrm{conv}} \hookrightarrow \mathrm{PrStk}_S$ for the full subcategories spanned by the $S$-prestacks that are locally of finite presentation, resp. convergent.

Next, let us recall the conditions on a prestack that guarantee that it has a well-behaved infinitesimal structure, using the following algebraic terminology:

**Definition B.4.** A map of animated rings $A \to B$ is said to be a *nilpotent extension* if $\pi_0(A) \to \pi_0(B)$ is surjective and its kernel is a nilpotent ideal.

**Lemma B.5** ([? , Volume 2, Chapter 1, Proposition 5.5.3]). Let $f: A \to B$ be a nilpotent extension. Then $f$ can be decomposed as the limit of an almost eventually constant tower

$$A = A_{\infty} \longrightarrow \dots \longrightarrow A_{n} \longrightarrow A_{n-1} \longrightarrow \dots \longrightarrow A_{0} = B$$

where each $A_n \to A_{n-1}$ is a square zero extension by a $\pi_0(B)$-module $M_n$ and for each $N$.

**Definition B.6** ([? , Volume 2, Chapter 1, Definition 7.1.2]). An $S$-prestack $X$ is said to *have deformation theory* (relative to $S$) if it is convergent (relative to $S$) and for each pullback diagram of animated rings on the left in which $B \to A$ (and hence $B' \to A'$) is a nilpotent extension, the square on the right is cartesian

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

Equivalently, for each $s: \mathrm{Spec}(A) \to S$, the fibre $X_s: \mathrm{CAlg}^{\mathrm{an}}_A \to S$ has deformation theory, i.e. preserves limits of Postnikov towers and pullbacks along nilpotent extensions of animated $A$-algebras.

**Lemma B.7.** Let $X$ be a convergent $S$-prestack. Then $X$ has deformation theory if and only if one of the following two equivalent conditions holds:
(1) For each square in $(\mathrm{Aff}_S)^{\mathrm{op}}$ opposite to a pullback square of animated rings

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

with $I \in \mathrm{QC}(A)_{\ge 0}$, its image under $X: (\mathrm{Aff}_S)^{\mathrm{op}} \to S$ is a pullback diagram of spaces.

(1') The condition of (1) holds for all squares such that $A, B$ and $I$ are eventually coconnective.

*Proof.* The equivalence between $X$ having deformation theory and condition (1) follows from Lemma B.5 by decomposing a general nilpotent extension as a tower of successive square-zero extensions. The equivalence between (1) and (1') follows from $X$ being convergent. $\square$

Finally, let us introduce two versions of the formal completion of a prestack, based on the following algebraic constructions:

**Definition B.8 (Reduction and pro-reduction).** Let $A \in \mathrm{CAlg}_R^{\mathrm{an}}$ be an animated $R$-algebra.

(1) The *reduction* $A_{\mathrm{red}}$ is the quotient of $\pi_0(A)$ by its nilradical.

(2) The *pro-reduction* $A_{\mathrm{prored}}$ to be the ind-ring

$$A_{\mathrm{prored}} = \text{``colim''} \pi_0(A)/I,$$

where the colimit is taken over all nilpotent ideals $I$ of $\pi_0(A)$.

For $S = \mathrm{Spec}(A)$ affine, we write $S_{\mathrm{red}} = \mathrm{Spec}(A_{\mathrm{red}})$ for the underlying reduced scheme. We moreover define a pro-scheme $S_{\mathrm{prored}}$ over $R$ as the formal limit

$$\mathrm{Spec}(A_{\mathrm{prored}}) = \text{``lim}_I \mathrm{Spec}(\pi_0(A)/I),$$

where $I$ again ranges over all nilpotent ideals of $\pi_0(A)$. Given another $R$-prestack $X$, we set

$$X(A_{\mathrm{prored}}) := \mathrm{Map}_{\mathrm{Pro}(\mathrm{PrStk})}(S_{\mathrm{prored}}, X) = \mathrm{colim}_I X(\pi_0(A)/I).$$

**Definition B.9.** Let $X \to Y$ be a map of prestacks over $R$.

(1) The *formal completion* $Y_X^{\wedge}$ is the $R$-prestack sending an animated $R$-algebra $A$ to

$$Y_X^{\wedge}(A) = X(A_{\mathrm{red}}) \times_{Y(A_{\mathrm{red}})} Y(A).$$

(2) The *formal infinitesimal completion* of $Y$ at $X$ is the $R$-prestack sending $A$ to

$$(X/Y)_{\mathrm{inf}}(A) = X(A_{\mathrm{prored}}) \times_{Y(A_{\mathrm{prored}})} Y(A).$$

**Remark B.10.** When $X \to Y$ is laft, then the canonical map $(X/Y)_{\mathrm{inf}} \to Y_X^{\wedge}$ is an equivalence, as $X(A_{\mathrm{red}}) \simeq X(A_{\mathrm{prored}}) \times_{Y(A_{\mathrm{prored}})} Y(A_{\mathrm{red}})$.

**Notation B.11.** For $Y = *$ the terminal prestack, we write $X_{\mathrm{inf}} = (X/*)_{\mathrm{inf}}$. In characteristic 0, the prestack $X_{\mathrm{inf}}$ is often called the *de Rham stack* of $X$.

**Definition B.12.** We will say that a map of $R$-prestacks $f: X \to Y$ is a *nil-isomorphism* if for every $A \in \mathrm{CAlg}_R^{\mathrm{an}}$, the map

$$X(A_{\mathrm{prored}}) \to Y(A_{\mathrm{prored}})$$

is an equivalence.

**Remark B.13.** When $X \to Y$ is laft, then it is a nil-isomorphism if and only if $X(A) \to Y(A)$ is an equivalence for all reduced commutative rings.
**Lemma B.14.** Completion satisfies the following properties:

(1) For any $X \to Y$, the map $Y_X^\wedge \to Y$ has deformation theory. If $X \to Y$ is laft, then $X \to Y_X^\wedge$ and $Y_X^\wedge \to Y$ are laft.

(2) Let $X' \to X$ be an étale map between derived schemes and let $X \to Y$ be a nil-isomorphism. Then the square

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

is cartesian.

*Proof.* The first part of (1) follows from the fact that for any nilpotent extension $A' \to A$ of animated rings, $Y_X^\wedge(A') \simeq Y_X^\wedge(A) \times_{Y(A)} Y(A')$. For the second part, it is enough to show that $Y_X^\wedge \to Y$ is laft. This is the base change of the map $*_{X'} \to *_{Y'}$, which is laft because for any filtered diagram of n-coconnective animated rings $A_\alpha$ with colimit $A$, colim $X(A_{\alpha,red}) \simeq$ colim $Y(A_{\alpha,red}) \times_{Y(A_{red})} X(A_{red})$. For (2), the fact that $X' \to X$ is laft implies that $Y_{X'}^\wedge \to Y$ is the base change of $X'_{\inf} \to X_{\inf}$ (Remark B.10). It now suffices to verify that the map $X' \to X'_{\inf} \times_{X_{\inf}} X$ is an equivalence. This follows from $X' \to X$ being étale, so that $X'(A) \simeq X(A) \times_{X(\pi_0 A/I)} X'(\pi_0 A/I)$ for any nilpotent ideal $I$. $\square$

## B.2. Pro-coherent sheaves.
The theory of pro-coherent sheaves on affine derived schemes from Section B.2 has an evident global analogue:

**Definition B.15.** For a prestack $X$, we define the stable presentable $\infty$-category $QC_X^\vee$ as

$$QC_X^\vee = \lim_{S \in \mathrm{Aff}/X} QC_{\mathcal{O}(S)}^\vee$$

where the limit is taken with respect to the $f^*$-functoriality. Every map $f: Y \to X$ between prestacks induces an adjoint pair $f^*: QC_X^\vee \hookrightarrow QC_Y^\vee: f_*$.

Let us point out that in the presence of Serre duality, this is Serre dual to the $\infty$-category $QC_X^!$ of ind-coherent sheaves on $X$ defined using $f^!$-functoriality [Gai11, 9.2.3]. The main features of $QC_A^\vee$ carry over to pro-coherent sheaves on prestacks by naturality:

(1) Since each $f^*: QC_A^\vee \to QC_B^\vee$ is right $t$-exact, for any prestack $X$ the $\infty$-category $QC_X^\vee$ admits a right complete $t$-structure such that all $f^*$-functors are right $t$-exact. Since the functor $\iota: \mathrm{Mod}_A \to QC_A^\vee$ intertwines the $f^*$-functors, there is a natural transformation $\iota: QC_X \to QC_X^\vee$ which induces an equivalence on connective objects.

(2) Each $QC_X^\vee$ is tensored and cotensored over $QC_X$ and $\iota$ is compatible with the tensoring.

(3) If $M$ is an almost perfect $A$-module, then the natural map $f^*(M^\vee) \to (f^*M)^\vee$ is an equivalence. Using this, one sees that for any prestack $X$, pro-coherent duality induces an equivalence

$$(-)^\vee: \mathrm{APerf}_X^{\mathrm{op}} \xrightarrow{\sim} \mathrm{dAPerf}_X$$

between the full subcategories of pro-coherent sheaves $F$ that are (dually) almost perfect in the sense that $f^*F$ is (dually) almost perfect for each $f: \mathrm{Spec}(A) \to X$.

The $\infty$-category $QC_X^\vee$ is particularly well-behaved when $X$ is locally coherent. In particular, we have the following generalisation of Lemma A.42 to derived schemes that are locally coherent, i.e. which admit an Zariski cover coherent affine open subschemes or equivalently, for which every affine open subscheme is coherent [Gla89, Theorem 2.4.2, Corollary 2.4.5].
**Proposition B.16.** Let $X$ be a $qcqs$ derived scheme which is locally coherent and write write $\mathfrak{A}$ for the poset of affine opens of $X$. Then the natural map

$$ \mathrm{Coh}_{X}^{\mathrm{op}} = \lim_{U \in \mathfrak{A}} \mathrm{Coh}_{U}^{\mathrm{op}} \longrightarrow \lim_{U \in \mathfrak{A}} \mathrm{QC}_{U}^{\vee} = \mathrm{QC}_{X}^{\vee} $$

induces an equivalence $\mathrm{Ind}(\mathrm{Coh}_{X}^{\mathrm{op}}) \simeq \mathrm{QC}_{X}^{\vee}$.

*Proof.* By the usual reduction principle for $qcqs$ schemes, it suffices to verify that for $U_0 \subseteq X \supseteq U_1$ an open cover such that the assertion holds on $U_0$, $U_1$ and $U_{01} = U_0 \cap U_1$, the assertion holds also for $X$. Since $\mathrm{QC}^{\vee}$ satisfies Zariski descent by Corollary A.39, it suffices to verify that $\mathrm{Ind}(\mathrm{Coh}^{\mathrm{op}})$ also satisfies Zariski descent.

To see this, let us write $j^*: \mathrm{Ind}(\mathrm{Coh}_{X}^{\mathrm{op}}) \hookrightarrow \mathrm{Ind}(\mathrm{Coh}_{U}^{\mathrm{op}})$: $j_*$ for the adjoint pair associated to an open inclusion $j: U \hookrightarrow X$. Since $j_*$ preserves colimits, this is an adjoint pair in $\mathrm{Pr}^{\mathrm{L}}$. As such, it is the dual with respect to the Lurie tensor product of the adjoint pair $j^*: \mathrm{Ind}(\mathrm{Coh}_{X}) \hookrightarrow \mathrm{Ind}(\mathrm{Coh}_{U})$: $j_*$ considered in [Gai11]. Using this duality, one sees the counit $j^*j_* \to \mathrm{id}$ is an equivalence [Gai11, Lemma 4.1.1] and that $\mathrm{Ind}(\mathrm{Coh}^{\mathrm{op}})$ satisfies base change for open inclusions [Gai11, Lemma 3.6.9]. It follows that

$$ (j_0^*, j_{01}^*, j_1^*): \mathrm{Ind}(\mathrm{Coh}_X^{\mathrm{op}}) \longrightarrow \mathrm{Ind}(\mathrm{Coh}_{U_0}^{\mathrm{op}}) \times_{\mathrm{Ind}(\mathrm{Coh}_{U_{01}}^{\mathrm{op}})} \mathrm{Ind}(\mathrm{Coh}_{U_1}^{\mathrm{op}}) $$

is an equivalence as in the proof of [Gai11, Proposition 4.2.1]: this functor has a colimit-preserving right adjoint sending a compatible triple $(F_0, F_1, F_{01})$ to the fibre product $j_{0*}F_0 \times_{j_{01}*F_{01}} j_{1*}F_1$. Using base change and the fact that $j_i^*j_{i*} \simeq \mathrm{id}$, one sees that the counit is an equivalence. To verify that the unit $F \to j_{0*}j_0^*F \times_{j_{01}*j_{01}^*F} j_{1*}j_1^*F$ is an equivalence, it suffices to treat the case where $F \in \mathrm{Coh}(X)^{\mathrm{op}}$. In that case, the result follows from descent for coherent sheaves. $\square$

**B.3. Tangent complex.** In this section, we will discuss the tangent complex of a map of prestacks $X \to S$ (if it exists). The main goal will be to establish a form of functoriality of the tangent complex, which is used in Section 5.2.

Let us start by recalling the pointwise definition of the tangent and cotangent complex of an $S$-prestack. Let $\pi: X \to S$ be an $S$-prestack and $x: \mathrm{Spec}(A) \to X$ a point, and consider the functor

$$ (28) \qquad \mathrm{Mod}_{A, \ge 0} \to S; \quad M \mapsto \left\{ \begin{array}{ccc} \mathrm{Spec}(A) & \xrightarrow{x} & X \\ \bigdownarrow & & \bigdownarrow \\ \mathrm{Spec}(A \oplus M) & \longrightarrow & \mathrm{Spec}(A) \xrightarrow{\pi(x)} & S \end{array} \right\} $$

We then have the following:

**Definition B.17.** Let $X$ be an $S$-prestack and let $x: \mathrm{Spec}(A) \to X$ be a point.

(1) We will say that $X$ admits a cotangent complex at $x$ (relative to $S$) if there exists a (necessarily unique) eventually connective module $L_{X/S,x} \in \mathrm{Mod}_{A, > \infty}$ corepresenting the functor (28).

(2) We will say that $X$ admits a (pro-coherent) tangent complex at $x$ (relative to $S$) if the functor (28) defines an object $T_{X/S,x} \in \mathrm{QC}_A^{\vee}$.

If $X$ is an $S$-prestack with deformation theory, then each map of rings $f: A \to B$ induces a natural equivalence

$$ X(A \oplus M) \times_{X(A)} \{x\} \simeq X(B \oplus f_*M) \times_{X(B)} \{f^*x\} $$
for each connective A-module M. This implies that $L_{X/S, f^*x} \simeq f^*L_{X/S, x}$ and $T_{X/S, f^*x} \simeq f^*T_{X/S, x}$ whenever X admits a (co)tangent complex at the point x. Consequently, if X has deformation theory and admits a (co)tangent complex at each of its points, then these (co)tangent complexes glue to global objects on X:

**Definition B.18.** Let X be an S-prestack with deformation theory.

(1) If X admits a cotangent complex at each point, then the cotangent complex $L_{X/S} \in \mathrm{QC}_X$ is the unique quasi-coherent sheaf such that $L_{X/S, x} \simeq x^*L_{X/S}$ for each point of X.

(2) If X admits a tangent complex at each point, then the tangent complex $T_{X/S} \in \mathrm{QC}_X^\nabla$ is the unique pro-coherent sheaf such that $T_{X/S, x} \simeq x^*T_{X/S}$ for each point of X.

**Example B.19.** Suppose that X is a laft S-prestack with deformation theory. Then X admits a relative tangent complex at each point.

Having a tangent complex at each point requires a finiteness condition on $X \to S$ similar to Example B.19 (but at the linear level). However, there are many prestacks for which one can make sense of their tangent complex, without requiring the existence of a tangent complex at each point. For example, for every affine derived R-scheme X, we can define the tangent complex $T_{X/R} = L_{X/R}^\nabla$ to be the pro-coherent dual of its cotangent complex. In order to define the tangent complex in such cases, and to make the functoriality of the tangent complex more explicit, we will need a somewhat technical argument.

**Construction B.20.** Let $\mathrm{Mod}_{\ge 0, *} \to \mathrm{Aff}$ be the cocartesian fibration classifying the functor sending $f: \mathrm{Spec}(A) \to \mathrm{Spec}(B)$ to the direct image $f_*: \mathrm{Mod}_{A, \ge 0} \to \mathrm{Mod}_{B, \ge 0}$ and let us consider the following pullback

<figure><img src="image_54.png" /></figure>

An object of $\mathcal{M}$ corresponds a map of prestacks $x: \mathrm{Spec}(A) \to X$ together with a connective A-module $M$. The functor $\mathcal{M} \to \mathrm{PrStk}$ sending $(\mathrm{Spec}(A) \to X, M) \mapsto X$ is a cocartesian fibration and each fibre $\mathcal{M}_X$ comes equipped with a cocartesian fibration $\mathcal{M}_X \to \mathrm{Aff}_X$ parametrising the $\infty$-category of connective modules at each point of X.

Let $\mathrm{Fun}_{/\mathrm{PrStk}}(\mathcal{M}, \mathcal{S}) \to \mathrm{PrStk}$ be the relative functor $\infty$-category over $\mathrm{PrStk}$; by [Lur09] this is a cartesian fibration, classified by the functor $\mathrm{PrStk}^{\mathrm{op}} \to \mathrm{Cat}_\infty$ sending a prestack $X$ to $\mathrm{Fun}(\mathcal{M}_X, \mathcal{S})$ and sending $f: X \to Y$ to restriction along $\mathcal{M}_X \to \mathcal{M}_Y$. Let us write

$$ \mathcal{D} \hookrightarrow \mathrm{Fun}_{/\mathrm{PrStk}}(\mathcal{M}, \mathcal{S}) $$

for the full subcategory of tuples $(X, F)$ consisting of a prestack $X$ and a functor $F: \mathcal{M}_X \to \mathcal{S}$ satisfying the following two properties:

(1) $F$ sends cocartesian arrows with respect to $\mathcal{M}_X \to \mathrm{Aff}_X$ to equivalences.

(2) For each $x: \mathrm{Spec}(A) \to X$, the restriction of $F$ to the fibre

$$ F_x: \mathcal{M}_{X,x} \simeq \mathrm{Mod}_{A, \ge 0} \to \mathcal{S} $$

is accessible and preserves the terminal object, pullbacks along $\pi_0$-surjections and limits of almost eventually constant towers.
Let us write $\mathcal{D}^{\text{aft}} \subseteq \mathcal{D}$ and $\mathcal{D}^{\text{corep}} \subseteq \mathcal{D}$ for the subcategories of tuples $(X, F)$ where, in addition to condition (2), each restriction $F_x$ is almost finitely presented, respectively corepresentable by an eventually connective $A$-module.

One readily verifies that for any map of prestacks $f: X \to Y$, restriction along $M_X \to M_Y$ preserves these conditions, so that for any cartesian arrow $(X, G) \to (Y, F)$ with $(Y, F) \in \mathcal{D}$, $(X, G)$ is in $\mathcal{D}$ as well (and similarly in the other cases). Consequently, we obtain a diagram of cartesian fibrations (and maps preserving cartesian arrows)

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

**Proposition B.21.** Each of the above three cartesian fibrations classifies a functor $\text{PrStk}^{\text{op}} \to \text{Cat}_{\infty}$ preserving limits. Furthermore, the cartesian fibration $\mathcal{D}^{\text{aft}} \to \text{PrStk}$ is classified by the functor $X \mapsto \text{QC}_X^{\vee}$ and the cartesian fibration $\mathcal{D}^{\text{corep}} \to \text{PrStk}$ is classified by the functor $X \mapsto (\text{QC}_X^{-})^{\text{op}}$.

*Proof.* Let $X$ be a prestack and consider the cocartesian fibration $M_X \to \text{Aff}_X$ that classifies the diagram of categories $\text{Aff}_X \to \text{Cat}_\infty$ sending a map of affines $f: \text{Spec}(A) \to \text{Spec}(B)$ over $X$ to $f_*: \text{Mod}_{A, \ge 0} \to \text{Mod}_{B, \ge 0}$. The localisation of $M_X$ at the cocartesian arrows classifies the colimit of this diagram of categories. Condition (1) therefore implies that $\mathcal{D}_X$ is a full subcategory of the limit

$$ \lim_{\text{Spec}(A) \in \text{Aff}_X} \text{Fun}(\text{Mod}_{A, \ge 0}, S) $$

where each morphism $f: \text{Spec}(A) \to \text{Spec}(B)$ in $\text{Aff}_X$ is sent to restriction along $f_*$. Using this, Condition (2) then identifies

$$ \mathcal{D}_X \simeq \lim_{\text{Spec}(A) \in \text{Aff}_X} \text{Fun}_{\text{rex,acc}}(\text{Mod}_{A, \ge 0}^{+}, S) $$

with the limit of the $\infty$-categories of accessible functors that are right exact (or equivalently, that preserve the terminal object and pullbacks along $\pi_0$-surjections). This identification is natural in $X$, so that $X \mapsto \mathcal{D}_X$ evidently preserves limits. The categories $\mathcal{D}_X^{\text{aft}}$ and $\mathcal{D}_X^{\text{corep}}$ are the limits over $\text{Aff}_X$ of the diagram of full subcategories of $\text{Fun}_{\text{rex,acc}}(\text{Mod}_{A, \ge 0}^{+}, S)$ spanned by almost finitely presented and corepresentable functors. It follows that there are natural equivalences $\mathcal{D}^{\text{aft}} \simeq \text{QC}_X^{\vee}$ and $\mathcal{D}_X^{\text{corep}} \simeq (\text{QC}_X^{-})^{\text{op}}$. $\square$

**Observation B.22.** Given a prestack $X$, the inclusion

$$ \text{QC}_X^{\vee} = \lim_{\text{Spec}(A) \to X} \text{QC}_A^{\vee} \longrightarrow \lim_{\text{Spec}(A) \to X} \text{Fun}_{\text{rex,acc}}(\text{Mod}_{A, \ge 0}^{+}, S) \simeq \mathcal{D}_X $$

preserves colimits. Since $\text{QC}_X^{\vee}$ is presentable, it follows that there is a right adjoint $\mathcal{D}_X \to \text{QC}_X^{\vee}$. Given $F \in \mathcal{D}_X$, we will refer to its image under this right adjoint $F^{\text{aft}}$ as the almost finitely presented approximation to $F$. Let us point out that the right adjoints $\mathcal{D}_X \to \text{QC}_X^{\vee}$ do not assemble into a right adjoint $\mathcal{D} \to \mathcal{D}^{\text{aft}}$.

Let us now use this construction to give a functorial description of the tangent complex.

**Construction B.23.** Let us write $\text{Fun}(\Delta^1, \text{PrStk})^{\text{def}} \subseteq \text{Fun}(\Delta^1, \text{PrStk})$ for the full subcategory of maps $X \to S$ that have deformation theory. We define a functor

$$ \Phi: \mathcal{M} \times_{\text{PrStk}} \text{Fun}(\Delta^1, \text{PrStk})^{\text{def}} \to S; $$
as follows. The domain of $\Phi$ is the fibre product of the cocartesian fibration $\mathcal{M} \to \text{PrStk}$ from Construction B.20 and the functor $\text{ev}_0$: $\text{Fun}(\Delta^1, \text{PrStk})^{\text{def}} \to \text{PrStk}$ taking the domain. We then define $\Phi$ by

$$ \Phi(x: \text{Spec}(A) \to X, M, X \to S) = \left\{ \begin{array}{c} \text{Spec}(A) \xrightarrow[x]{x} X \\ \text{Spec}(A \oplus M) \xrightarrow{\pi(x)} S \end{array} \right\}. $$

The functor $\Phi$ is adjoint to

<figure><img src="image_56.png" /></figure>

sending $X \to S$ to the restriction $\Phi(-, X \to S): \mathcal{M}_X \to \mathcal{S}$. The fact that $X \to S$ has deformation theory implies that $\Phi(-, X \to S)$ satisfies conditions (1) and (2) from Construction B.20. Consequently, we obtain a functor that we will denote by

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

If $X \to S$ is laft, then $T^{\text{pre}}(X/S)$ is contained in $\mathcal{D}^{\text{aft}}$. Under the equivalence $\mathcal{D}_X^{\text{aft}} \simeq \text{QC}^{\vee}(X)$, this corresponds to the tangent complex $T_{X/S}$ from Definition B.18. If $X \to S$ admits cotangent complexes at all points of $X$, then $T^{\text{pre}}(X/S) \in \mathcal{D}^{\text{corep}}$ and corresponds under the equivalence $\mathcal{D}_X^{\text{corep}} \simeq (\text{QC}_X^-)^{\text{op}}$ to the cotangent complex $L_{X/S}$ from Definition B.18.

From the formula for $\Phi$, one readily sees:

**Lemma B.24.** *The functor $T^{\text{pre}}$ preserves pullbacks.*

Since $\mathcal{D}^{\text{corep}} \subseteq \mathcal{D}$ is closed under pullbacks, this reproduces for example the well-known behaviour of the cotangent complex with respect to base change and composition.

Let us now use the functor $T^{\text{pre}}$ to define the tangent complex of a prestack with deformation theory:

**Definition B.25.** Let $X$ be an $S$-prestack with deformation theory. We define the (pro-coherent) tangent complex of $X$ over $S$ to be the almost finitely presented approximation $T_{X/S} = T_{X/S}^{\text{pre,aft}} \in \text{QC}_X^{\vee}$ from Observation B.22.

**Example B.26.** If $X$ is a laft $S$-prestack with deformation theory, this coincides with the pointwise definition of tangent complex from Definition B.18. Indeed, since $X \to S$ is laft, $T_{X/S}^{\text{pre}}$ is already contained in $\mathcal{D}_X^{\text{aft}} \subseteq \mathcal{D}_X$.

**Example B.27.** Suppose that $X$ is a (not necessarily laft) $S$-prestack with deformation theory and a cotangent complex at each point. Then there is an equivalence $T_{X/S} = L_{X/S}^{\vee}$ in $\text{QC}_X^{\vee}$, where $L_{X/S}^{\vee}$ is the pro-coherent linear dual of $L_{X/S} \in (\text{QC}_X^-)^{\text{op}}$ (see Example A.32). Since the right adjoint $\text{QC}_X^{\vee} \to \text{QC}_X$ preserves the linear dual of a quasi-coherent sheaf, one can think of $T_{X/S} \in \text{QC}_X^{\vee}$ as
a pro-coherent refinement of the coherator of the tangent sheaf on $X$, in the sense of [Sta19, Tag 08D6].

The tangent complex $T_{X/S}$ is much less well-behaved when $X \to S$ is not laft. For example, given a map of $S$-prestacks with deformation theory $X \to Y$, there is no direct map from $T_{X/S}$ to $f^*T_{Y/S}$. For instance, if $X$ and $Y$ admit a cotangent complex at each point, there is only a zig-zag of maps $L_{X/S}^\vee \to (f^*L_{Y/S})^\vee \leftarrow f^*(L_{Y/S}^\vee)$ because base change does not commute with duality without finiteness assumptions. Let us nonetheless record the following properties:

**Proposition B.28.** Let $f: X \to Y$ be a map of $S$-prestacks with deformation theory. Then the following hold.

(1) There is a fibre sequence $T_{X/Y} \to T_{X/S} \to (f^*T_{Y/S}^{\text{pre}})^{\text{aft}}$.

(2) There is a zig-zag of maps $T_{X/S} \to (f^*T_{Y/S}^{\text{pre}})^{\text{aft}} \leftarrow f^*T_{Y/S}$ in $\text{QC}_X^\vee$.

(3) If $f$ is a formally étale, then there is a map $f^*T_{Y/S} \to T_{X/S}$.

(4) $T_{X/S} \simeq T_{X/S_\lambda}$.

If $X$ and $Y$ are furthermore laft $S$-prestacks with deformation theory, then:

(5) There is a natural map $T_{X/S} \to f^*T_{Y/S}$ that fits into a fibre sequence $T_{X/Y} \to T_{X/S} \to f^*T_{Y/S}$ in $\text{QC}_X^\vee$.

(6) Given a morphism $S' \to S$, there is a natural equivalence $T_{X \times_S S'/S'} = p^*T_{X/S}$ where $p: X \times_S S' \to X$ is the evident projection.

*Proof.* The first three assertions follow from the natural fibre sequence $T_{X/Y}^{\text{pre}} \to T_{X/S}^{\text{pre}} \to f^*T_{Y/S}^{\text{pre}}$ in $\mathcal{D}_X$ by taking almost finitely presented approximations, using that $T_{X/Y}^{\text{pre}} \simeq 0$ if $X \to Y$ is formally étale. Likewise, (4) follows from the fact that $T_{X/S}^{\text{pre}} \simeq T_{X/S_\lambda}^{\text{pre}}$. Assertion (5) follows from the fact that $T_{Y/S} = T_{Y/S}^{\text{pre}}$ and (6) follows from the equivalence $T_{X \times_S S'/S'}^{\text{pre}} = p^*T_{X/S}^{\text{pre}}$, which follows from Lemma B.24. $\square$

The somewhat problematic behaviour of $T_{X/S}$ when $X \to S$ is not laft will not be a problem for us, because we are mainly interested in the $\infty$-category $(\text{QC}_X^\vee)_{T_{X/S}}$ of pro-coherent sheaves over $T_{X/S}$. This $\infty$-category is equivalent to the $\infty$-category $(\text{QC}_X^\vee)_{T_{X/S}^{\text{pre}}}$ of pro-coherent sheaves equipped with a map $F \to T_{X/S}^{\text{pre}}$ in $\mathcal{D}_X$, and $T_{X/S}^{\text{pre}}$ is very well-behaved.

For instance, each laft map $X \to Y$ of $S$-prestacks with deformation theory gives rise to a natural object $T_{X/Y} \in (\text{QC}_X^\vee)_{T_{X/S}}$, or equivalently, in $(\text{QC}_X^\vee)_{T_{X/S}^{\text{pre}}}$. Let us conclude with a functorial description of this construction:

**Construction B.29.** Let us fix a base prestack $S$ and write $\text{PrStk}_S^{\text{def,laft-map}}$ for the subcategory of $\text{PrStk}_S$ whose objects are prestacks with deformation theory over $S$ and whose morphisms are the laft morphisms between such $S$-prestacks. Let us write

$$ \mathcal{E} = \mathcal{D} \times_{\text{PrStk}} \text{PrStk}_S^{\text{def,laft-map}}, \qquad \text{QC}^\vee = \mathcal{D}^{\text{aft}} \times_{\text{PrStk}} \text{PrStk}_S^{\text{def,laft-map}} $$

for the evident restrictions. The cartesian fibration $\mathcal{E} \to \text{PrStk}_S^{\text{def,laft-map}}$ then admits a section $T^{\text{pre}}$ sending each $X \mapsto T_{X/S}^{\text{pre}}$. Taking fibrewise over-categories then produces a diagram that we
will denote by

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

At the level of the fibres over a prestack $X$, this is given by

$$ (\mathrm{QC}_X^{\vee})_{/T_{X/S}} \simeq (\mathrm{QC}_X^{\vee})_{\times_{\mathcal{D}_X}} (\mathcal{D}_X)_{/T_{X/S}^{\mathrm{pre}}} \longrightarrow (\mathcal{D}_X)_{/T_{X/S}^{\mathrm{pre}}}. $$

**Lemma B.30.** *The projection $\mathrm{QC}_X^{\vee} \to \mathrm{PrStk}_S^{\mathrm{def,laft-map}}$ is a cartesian fibration.*

*Proof.* Let us first consider the cartesian fibration $\pi: \mathcal{E} \to \mathrm{PrStk}_S^{\mathrm{def,laft-map}}$. This classifies a functor sending $X$ to $\mathcal{E}_X = \lim_{\mathrm{Spec}(A) \to X} \mathrm{Fun}_{\mathrm{ex,acc}}(\mathrm{Mod}_A^+, \mathrm{Sp})$ and $f: X \to Y$ to the evident functor between limits

$$ f^*: \mathcal{E}_Y = \lim_{\mathrm{Spec}(A) \to Y} \mathrm{Fun}_{\mathrm{ex,acc}}(\mathrm{Mod}_A^+, \mathrm{Sp}) \to \lim_{\mathrm{Spec}(A) \to X} \mathrm{Fun}_{\mathrm{ex,acc}}(\mathrm{Mod}_A^+, \mathrm{Sp}) = \mathcal{E}_X. $$

Note that each $\mathcal{E}_X$ is a stable $\infty$-category and that $f^*$ is exact. This implies that $\mathcal{E}$ admits finite limits and that $\pi$ admits a fully faithful right adjoint (taking the terminal object fibrewise). The projection $\mathcal{E}_{/T^{\mathrm{pre}}} \to \mathrm{PrStk}_S^{\mathrm{def,laft-map}}$ is then a cartesian fibration as well: indeed, given a map $f: X \to Y$ and $F \to T_{Y/S}^{\mathrm{pre}}$ in $\mathcal{E}_Y$, the cartesian lift of $f$ in $\mathcal{E}_{/T^{\mathrm{pre}}}$ corresponds to the following pullback square in $\mathcal{E}$:

<figure><img src="image_59.png" /></figure>

More explicitly, this is given by the pullback $T_{X/S}^{\mathrm{pre}} \times_{f^*T_{Y/X}^{\mathrm{pre}}} f^*F$ in the fibre $\mathcal{E}_X$.

Finally, it suffices that the full subcategory $\mathrm{QC}_X^{\vee} \subseteq \mathcal{E}_{/T^{\mathrm{pre}}}$ is stable under these cartesian arrows. For this, suppose that $F \in \mathrm{QC}_Y^{\vee} \subseteq \mathcal{E}_Y$ and consider the fibre sequence in $\mathcal{E}_X$

$$ T_{X/Y}^{\mathrm{pre}} \to T_{X/S}^{\mathrm{pre}} \times_{f^*T_{Y/X}^{\mathrm{pre}}} f^*F \to f^*F. $$

The base change functor $f^*$ sends $\mathrm{QC}_X^{\vee}$ to $\mathrm{QC}_X^{\vee}$, so $f^*F \in \mathrm{QC}_X^{\vee}$. Since $f: X \to Y$ was assumed to be laft, $T_{X/Y}^{\mathrm{pre}} = T_{X/Y} \in \mathrm{QC}_X^{\vee}$ as well so that the middle term is contained in $\mathrm{QC}_X^{\vee}$, as desired. $\square$

**Remark B.31.** Informally, Lemma B.30 asserts that there is a functor sending each $S$-prestack $X$ with deformation theory to the $\infty$-category $(\mathrm{QC}_X^{\vee})_{/T_{X/S}}$, and each map $f: X' \to X$ to a functor

$$ f^{\sharp}: (\mathrm{QC}_X^{\vee})_{/T_{X/S}} \longrightarrow (\mathrm{QC}_{X'}^{\vee})_{/T_{X'/S}}; \quad (F \to T_{X/S}) \longmapsto T_{X'/S} \times_{f^*(T_{X/S}^{\mathrm{pre}})_{\mathrm{alt}}} F. $$

In particular, when $f$ is formally étale, then $f^{\sharp}(F \to T_{X/S}) = f^*F$ by Proposition B.28.

If $f: X \to Y$ is a map of $S$-prestacks with deformation theory and a cotangent complex, Example B.27 identifies the resulting base change functor with $f^{\sharp}(F) = T_{X/S} \times_{(f^*L_{Y/S})^{\vee}} f^*F$, using the natural map $f^*(T_{Y/S}) = f^*(L_{Y/S}^{\vee}) \to (f^*L_{Y/S})^{\vee}$.
**Proposition B.32.** There is a map of cartesian fibrations preserving cartesian arrows

<figure><img src="image_60.png" /></figure>

sending each $X \to Y$ to $T_{X/Y} \to T_{X/S}$. Furthermore, the induced functors between fibres preserve limits.

In other words, for every square of laft maps between $S$-prestacks with deformation theory

<figure><img src="image_61.png" /></figure>

there is a natural map $T_{X'/Y'} \to f^{\sharp}(T_{X/Y})$ where $f^{\sharp}$ is as in Remark B.31. The fact that $T$ preserves cartesian arrows corresponds to this map being an equivalence if $Y' = Y$. We are mostly interested for in the situation of (locally coherent) derived schemes $X$ (over a base $S$): the above then provides a functor sending $X \to Y$ a map $T_{X/Y} \to L_{X/Y}^{\vee}$ in $QC_{X/}^{\vee}$, functorial with respect to laft maps of schemes $X' \to X$.

*Proof.* The desired functor arises formally from the functor $T^{\text{pre}}$ in (29), using that $T_{X/Y} = T_{X/Y}^{\text{pre}}$ if $X \to Y$ is laft (Example B.26). To see that it preserves cartesian arrows, note that $T^{\text{pre}}$ sends

<figure><img src="image_62.png" /></figure>

Here the left square defines a cartesian arrow covering $f$ in the domain. To see that its image is cartesian, it suffices to verify that the right square is cartesian in $\mathcal{E}$; this follows from the fact that the functor $T^{\text{pre}}$ from (29) preserves pullbacks by Lemma B.24. $\square$

We conclude by recording the following property of the induced functors $T_{X/-}$ between the fibres.

**Proposition B.33.** Let $X$ be an $S$-prestack with deformation theory and consider the composite

$$ \Psi: (\text{PrStk}_S^{\text{def,laft-map}})_{X/-} \xrightarrow{T_{X/-}} (\text{QC}_X^{\vee})_{T_{X/S}} \xrightarrow{\text{cofib}} \text{QC}_X^{\vee} $$

where the first functor is induced by the functor $T$ from Proposition B.32 and the second functor sends $F \to T_{X/S}$ to its cofibre. This functor admits a left adjoint, sending each $F \in QC_X^{\vee}$ to a map $X \to X_F$ that is furthermore a nil-isomorphism. In particular, the functor $T_{X/-}$ preserves limits.

*Proof.* Unravelling the definitions, $\Psi(f: X \to Y) = (f^*T_{Y/S}^{\text{pre}})^{\text{aft}}$. To describe its left adjoint, recall from Construction B.20 that each $F \in QC_X^{\vee}$ defines a functor $F: \mathcal{M}_X \to \mathcal{S}$ on the $\infty$-category of tuples $(x: \text{Spec}(A) \to X, M \in \text{Mod}_{A, \ge 0}^+)$. Let $\mathcal{F} \to \mathcal{M}_X$ be the (cocartesian) unstraightening of this functor. We then define

$$ X_F = \underset{(A,M) \in \mathcal{F}^{\text{op}}}{\text{colim}} \text{Spec}(A \oplus M) $$
as the colimit in convergent S-prestacks. There are natural maps $X \to X_F \to X$. The first map is the natural map from the colimit indexed by the subcategory of $\mathcal{F}^{\text{op}}$ on those $(A, M)$ where $M \simeq 0$ is the zero module. The second map is the colimit over the objects of $\mathcal{F}$, corresponding to tuples

$$ (x: \text{Spec}(A) \to X, M \in \text{Mod}_{A, \ge 0}^{+}, y \in F(M)), $$

of the maps $\text{Spec}(A \oplus M) \to \text{Spec}(A) \xrightarrow{x} X$. Given a map $f: X \to Y$ to an S-prestack with deformation theory, a map $X_F \to Y$ of S-prestacks under $X$ corresponds to a natural transformation $F \to f^*T_{Y/S}^{\text{pre}}$ between functors $\mathcal{M}_X \to S$, i.e. a map $F \to (f^*T_{Y/S}^{\text{pre}})^{\text{aft}} = \Psi(f: X \to Y)$ in $\text{QC}_X^{\vee}$.

Hence, it remains to show that $X \to X_F$ is indeed a laft nil-isomorphism and that $X_F$ has deformation theory. For this it suffices to check that for each $x: \text{Spec}(A) \to X$ with $A$ eventually coconnective, the base change $x^*X_F$ is a laft A-prestack with deformation theory that is nil-isomorphic to $\text{Spec}(A)$. This base change can be identified as follows. Let $\mathcal{F}_x$ denote the unstraightening of the restriction $x^*F: \text{Mod}_{A, \ge 0}^{+} \to S$ of $F$ to the fibre of $(\mathcal{M}_X) \times_{\text{Aff}/X} \{x\}$. Then

$$ (30) \qquad x^*X_F = \underset{M \in \mathcal{F}_x^{\text{op}}}{\text{colim}} \text{Spec}(A \oplus M) $$

where the colimit is taken in convergent A-prestacks. This is clearly nil-isomorphic to $\text{Spec}(A)$, as every $\text{Spec}(A \oplus M)$ is nil-isomorphic to $\text{Spec}(A)$. The category $\mathcal{F}_x^{\text{op}}$ is filtered, so that $x^*X_F$ has deformation theory, being a filtered colimit of A-prestacks with deformation theory.

To see that $x^*X_F$ is laft, observe that for any eventually coconnective animated A-algebra $B$, one can either compute the value $x^*X_F(B)$ using the colimit (30), or as follows. Let us write $\text{Sqz}_{B} = \text{Mod}_{A, \ge 0}^{+} \times_{\text{CAlg}_A^{\text{an}}} \text{CAlg}_{A/B}$ for the $\infty$-category of A-modules $M \in \text{Mod}_{A, \ge 0}^{+}$ equipped with a map of animated A-algebras $A \oplus M \to B$. Then

$$ x^*X_F(B) = \underset{M \in \text{Sqz}_B}{\text{colim}} x^*F(M). $$

If $B$ is $n$-coconnective, it suffices to take the colimit over the cofinal subcategory $\text{Sqz}_{\le n/B} \subseteq \text{Sqz}_B$ of $n$-coconnective A-modules $M$ equipped with a map $A \oplus M \to B$. Note that $\text{Sqz}_{\le n/B}$ is compactly generated with compact objects given by

$$ (M, A \oplus M \to B), $$

where $M$ is the $n$-truncation of a perfect connective A-module; let us refer to such $n$-coconnective modules as $n$-compact. Since $x^*F$ preserves colimits of filtered diagrams of $n$-coconnective A-modules, one can thus reduce the above colimit to the colimit over the full subcategory $\text{Sqz}_{\le n/B}^{\omega} \subseteq \text{Sqz}_{\le n/B}$ of $n$-compact A-modules $M$ with a map $A \oplus M \to B$. Equivalently, this means that

$$ x^*X_F(B) = \underset{M \in \mathcal{F}_{x, \le n}^{\omega, \text{op}}}{\text{colim}} \text{Map}_{\text{CAlg}_A^{\text{an}}}(A \oplus M, B) \simeq \underset{M \in \mathcal{F}_{x, \le n}^{\omega, \text{op}}}{\text{colim}} \text{Map}_{\tau_{\le n} \text{CAlg}_A^{\text{an}}}(\tau_{\le n} A \oplus M, B) $$

where $\mathcal{F}_{x, \le n}^{\omega} \subseteq \mathcal{F}_x$ is the full subcategory of tuples $(M, y \in x^*F(M))$ where $M$ is $n$-compact.

We conclude that the restriction of $x^*X_F$ to $\tau_{\le n} \text{CAlg}_A^{\text{an}}$ is equivalent to a colimit of representables $\text{Spec}(\tau_{\le n} A \oplus M)$ where $M$ is an $n$-compact A-module. By Lemma 2.20, each square zero extension $\tau_{\le n} A \oplus M$ by an $n$-compact A-module defines a compact object in $\tau_{\le n} \text{CAlg}_A^{\text{an}}$. It follows that $x^*X_F$ restricts to a functor on $\tau_{\le n} \text{CAlg}_A^{\text{an}}$ preserving filtered colimits, so that $x^*X_F$ is indeed laft. $\square$

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OXFORD UNIVERSITY, UNIVERSITÉ PARIS-SACLAY (CNRS)

Email address: brantner@maths.ox.ac.uk

NORTHWESTERN UNIVERSITY

Email address: kirill.magidson@northwestern.edu

INSTITUT DE MATHÉMATIQUES DE TOULOUSE, UNIVERSITÉ DE TOULOUSE

Email address: joost.nuiten@math.univ-toulouse.fr