Derived Artin stack and cotangent complex

Fix a characteristic-zero field $k$. The derived affine site used here is the opposite of the infinity-localization of simplicial commutative $k$-algebras at maps inducing isomorphisms on all homotopy groups. Its representable object associated to $A$ is denoted $\mathrm{Spec}A$; the affine-derived-scheme and cotangent constructions are those of Derived schemes and the cotangent complex of a morphism. For each use below of its cotangent constructions, assume the Axiom of Choice (AC) as stipulated there; this premise governs the cotangent and tangent-complex assertions, not the HAG geometricity conventions. For a discrete ring map $A\to B$, finite presentation means that $B\simeq A[x_1,\dots ,x_r]\mathrm{/}(f_1,\dots ,f_s)$ for finite $r,s$. It is formally smooth when, for every $A$-algebra $C$, nilpotent ideal $J\subset C$ (so $J^N=0$ for some $N$), and $A$-algebra map $B\to C\mathrm{/}J$, at least one $A$-algebra lift $B\to C$ exists. It is formally étale when that lift is unique. In this item, a discrete map is smooth if it is of finite presentation and formally smooth, and étale if it is of finite presentation and formally étale. A finite family of affine-scheme maps is jointly surjective when the union of its images is the target scheme. For a map $A\to B$ in the simplicial model, call $\mathrm{Spec}B\to \mathrm{Spec}A$ derived-étale when ${\pi }_{0}A\to {\pi }_{0}B$ is étale and, for every $i>0$, the canonical map ${\pi }_{i}A{\otimes }_{{\pi }_{0}A}{\pi }_{0}B⟶{\pi }_{i}B$ is an isomorphism. A family $\{\mathrm{Spec}{B}_{\alpha }\to \mathrm{Spec}A\}$ is a derived-étale cover when all its maps are derived-étale and a finite subfamily of the ordinary étale maps $\{\mathrm{Spec}{\pi }_0{B}_{\alpha }\to \mathrm{Spec}{\pi }_{0}A\}$ is jointly surjective. A derived-étale hypercover $U_{\bullet }\to U$ is an augmented simplicial derived affine for which, in each simplicial degree, the map to the corresponding matching object is a derived-étale cover. A derived stack is a functor $F$ from the opposite derived-affine infinity category to spaces that sends equivalences to equivalences, takes finite homotopy coproducts of derived affines to products of spaces, and satisfies $F(U)\simeq {\mathrm{holim}}_{[m]\in \mathrm{\Delta }}F(U_m)$ for every such hypercover. Explicitly, for every finite family $(U_i{)}_{i\in I}$, require the canonical map $F\text{\hspace{0.17em}⁣}\left({\coprod }_{i\in I}^h{U}_i\right)⟶{\prod }_{i\in I}F(U_i)$ to be an equivalence. The coproduct is the derived affine associated to the homotopy product of the corresponding algebras. The empty family is included: its coproduct is $\mathrm{\varnothing }=\mathrm{Spec}0$, and $F(\mathrm{\varnothing })\simeq \ast$, the one-point space. Effective epimorphy for a family of derived stacks means that its Čech nerve has colimit the target in this derived-stack infinity-category.

Use the connective derived HAG context of Toën--Vezzosi, Homotopical Algebraic Geometry II, §§2.2.2--2.2.3. Its affine class $\mathcal{P}$ consists of derived-smooth maps $\mathrm{Spec}B\to \mathrm{Spec}A$ for which ${\pi }_{0}B$ is smooth and finitely presented over ${\pi }_{0}A$ and the maps ${\pi }_{i}A{\otimes }_{{\pi }_{0}A}{\pi }_{0}B\to {\pi }_{i}B$ above are isomorphisms for all $i>0$. The derived-étale topology is quasi-compact; finite coproduct inclusions form a cover; and descent holds for its hypercovers. The pair $(\tau ,\mathcal{P})$ is the HAG context of HAG II Definition 1.3.2.13: covering maps lie in $\mathcal{P}$; $\mathcal{P}$ is closed under equivalences, composition, and homotopy base change; membership in $\mathcal{P}$ descends along derived-étale covers of the source; and the canonical maps from either summand to a homotopy coproduct over a common target lie in $\mathcal{P}$. HAG II Lemma 2.2.2.13 and Lemma 2.2.3.1 verify these conditions for the specified derived-étale site and smooth class. Corollary 2.2.2.11 identifies the affine smooth class with the stated ${\pi }_0$ and higher-homotopy condition.

Define a stack to be $(-1)$-geometric when it is representable by a derived affine. A morphism $f:X\to Y$ is $(-1)$-representable when for every derived affine $T\to Y$, $X{\times }_{Y}T$ is representable by a derived affine; it is $(-1)$-$\mathcal{P}$ when these affine base changes are affine derived-smooth over $T$. For $n\ge 0$, $f$ is $n$-representable when every such base change is $n$-geometric, and it is $n$-$\mathcal{P}$ when it is $n$-representable and each affine base change has an $n$-atlas whose affine charts map to $T$ by derived-smooth maps in $\mathcal{P}$. An $n$-atlas of $F$ is a family of derived affines $U_{\alpha }\to F$ such that each map is $(n-1)$-$\mathcal{P}$ and the family is an effective epimorphism. For $n\ge 0$, $F$ is $n$-geometric when its diagonal is $(n-1)$-representable and it has an $n$-atlas. A derived Artin stack is an $n$-geometric derived stack for some finite $n$. “Locally of finite presentation over $k$” means that the truncation is locally of finite presentation and the cotangent complex below is perfect. These are the n-geometric and smooth-atlas conventions used throughout this item.

For an atlas $U_{\bullet }\to F$, define $\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(F)$ as the descent-compatible limit of the categories $\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(U_i)$ with the pullback functors and their Cech coherences; compatible data from two atlases are identified by their common refinement. The required descent is the quasi-coherent descent theorem for derived affines described in the source references below. For stable modules use HAG II Definition 1.2.12.4 and Proposition 1.2.12.5, and its stack formulation, Theorem 1.3.7.7. Their hypotheses hold here: suspension of simplicial $k$-modules is fully faithful by the connective chain-complex model; derived-étale maps are strong and flat on ${\pi }_0$, hence flat in the module sense of HAG II Lemma 2.2.2.2. Homotopy Cartesian stable descent data mean precisely that all transition maps after derived extension of scalars are equivalences.

Let $f:F\to G$ be a morphism of derived stacks in this site, and let $x:\mathrm{Spec}A\to F$ be an affine test point. A coefficient module $N$ is connective when its homotopy groups in negative degrees vanish (equivalently, its cochain model has no positive cohomology). The split square-zero algebra $A\oplus N$ has the given $A$-module structure on $N$, product of two elements of $N$ equal to zero, and its canonical augmentation to $A$. Both the projection $A\oplus N\to A$ and its zero section $A\to A\oplus N$ are retained. Define the space of relative derivations at $x$ by ${\mathrm{D}\mathrm{e}\mathrm{r}}_{F\mathrm{/}G}(x,N)={\mathrm{hofib}}_{(x,0_{f(x)})}(F(A\oplus N)⟶F(A){\times }_{G(A)}^{h}G(A\oplus N))\mathrm{.}$ Here $0_{f(x)}$ is the lift of $f(x)$ induced by $A\to A\oplus N$; the homotopy fiber includes the specified identifications with $x$ and this zero lift, and all their higher homotopies. Equivalently it is the mapping space from $\mathrm{Spec}(A\oplus N)$ to $F$ under $\mathrm{Spec}A$ and over $G$, with those structure maps. For $G=\mathrm{Spec}k$, write ${\mathrm{D}\mathrm{e}\mathrm{r}}_F(x,N)$; the relative space is the homotopy fiber over zero of ${\mathrm{D}\mathrm{e}\mathrm{r}}_F(x,N)\to {\mathrm{D}\mathrm{e}\mathrm{r}}_G(f(x),N)$.

Say that $f$ has a relative cotangent complex if, at every affine test point $x$, there is a bounded-above stable $A$-module $L_{F\mathrm{/}G,x}$ with natural corepresenting equivalences, for every connective $A$-module $N$, ${\mathrm{Map}}_{{\mathrm{M}\mathrm{o}\mathrm{d}}_A}(L_{F\mathrm{/}G,x},N)\simeq {\mathrm{D}\mathrm{e}\mathrm{r}}_{F\mathrm{/}G}(x,N),$ and these representatives satisfy the following compatibility. For every derived algebra map $u:A\to B$ and point $y:\mathrm{Spec}B\to F$ identified with $x\circ \mathrm{Spec}u$, the canonical comparison $L_{F\mathrm{/}G,x}{\otimes }_A^{\mathbf{L}}B⟶L_{F\mathrm{/}G,y}$ is an equivalence, compatibly with compositions and all the specified homotopies. Bounded-above means $H^i(L_{F\mathrm{/}G,x})=0$ for all sufficiently large $i$; no uniform bound over all points is required. These data are required to form a quasi-coherent descent object. For derived Artin $F$, their object in $\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(F)$ is denoted $L_{F\mathrm{/}G}$, with $x^{\ast }{L}_{F\mathrm{/}G}=L_{F\mathrm{/}G,x}$. One descends these relative pointwise representatives on an atlas and its Čech nerve, not the absolute cotangent complexes of the atlas affines. In general $x^{\ast }{L}_{F\mathrm{/}G}$ is not $L_{A\mathrm{/}k}$.

The absolute cotangent complex is $L_{F\mathrm{/}k}=L_{F\mathrm{/}\mathrm{Spec}k}$. For a connective derived commutative $k$-algebra $R$ and a specified structure morphism $F\to \mathrm{Spec}R$, write $L_{F\mathrm{/}R}=L_{F\mathrm{/}\mathrm{Spec}R}$; affine tests and their zero lifts are then taken over $R$. In particular this defines the relative object for every discrete commutative $\mathbb{C}$-algebra $R$ when $k=\mathbb{C}$. On an affine morphism $\mathrm{Spec}A\to \mathrm{Spec}R$ it is the derived algebra cotangent complex $L_{A\mathrm{/}R}$, rather than the ordinary cotangent module unless the latter actually computes it. Existence of representatives and their descent is part of saying that $f$ has a cotangent complex; no such existence or perfectness is asserted for an arbitrary space-valued stack.

The relative convention has two compatibility formulas. If $F$ and $G$ have absolute cotangent complexes, the relative one is $L_{F\mathrm{/}G}\simeq \mathrm{cofib}(f^{\ast }{L}_{G\mathrm{/}k}⟶L_{F\mathrm{/}k})\mathrm{.}$ For a homotopy Cartesian square $F^{\mathrm{\prime }}=F{\times }_G^{h}H$, with projection $p:F^{\mathrm{\prime }}\to F$, the relative complex, whenever $L_{F\mathrm{/}G}$ exists, satisfies $L_{F^{\mathrm{\prime }}\mathrm{/}H}\simeq p^{\ast }{L}_{F\mathrm{/}G}\mathrm{.}$ All pullbacks and tensor products in these formulas are derived; no ordinary flatness assumption is substituted for the homotopy Cartesian hypothesis. The first formula follows by applying $\mathrm{Map}(-,N)$ to the cofiber on each affine test: it gives exactly the homotopy fiber of absolute derivation spaces above. For the second, evaluating the homotopy Cartesian square on $A\oplus N$ and $A$ identifies the relative derivation spaces on both sides. Corepresentability identifies the complexes, and the stipulated compatibilities descend these identifications. Thus, for $F_{R^{\mathrm{\prime }}}=F{\times }_{\mathrm{Spec}R}^h\mathrm{Spec}{R}^{\mathrm{\prime }}$, $L_{F_{R^{\mathrm{\prime }}}\mathrm{/}{R}^{\mathrm{\prime }}}\simeq L_{F\mathrm{/}R}{\otimes }_R^{\mathbf{L}}{R}^{\mathrm{\prime }}$, understood as pullback to $F_{R^{\mathrm{\prime }}}$.

If $L_{F\mathrm{/}G}$ is perfect (dualizable), its relative tangent complex is $T_{F\mathrm{/}G}:=R{\mathrm{Hom}}_{{\mathcal{O}}_F}(L_{F\mathrm{/}G},{\mathcal{O}}_F)$; use $T_{F\mathrm{/}k}$ and $T_{F\mathrm{/}R}$ for the corresponding bases. Without this perfectness hypothesis, no tangent dual is asserted in this item's interface. The simplicial commutative algebra presentation remains the all-characteristic model; this item makes no identification with commutative dg algebras outside the stated characteristic-zero setting.

Derived schemes and the cotangent complex of a morphism

Model categories and Quillen adjunctions

Categories fibred in groupoids over a site