A Darboux theorem for derived schemes with shifted symplectic structure

Abstract

We prove a Darboux theorem for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme $\boldsymbol{X}$ with symplectic form $\tilde{\omega }$ of degree $k$ is locally equivalent to $(\operatorname {\mathbf{Spec}}A,\omega )$ for $\operatorname {\mathbf{Spec}}A$ an affine derived scheme in which the cdga $A$ has Darboux-like coordinates with respect to which the symplectic form $\omega$ is standard, and in which the differential in $A$ is given by a Poisson bracket with a Hamiltonian function $\Phi$ of degree $k+1$.

When $k=-1$, this implies that a $-1$-shifted symplectic derived scheme $(\boldsymbol{X},\tilde{\omega })$ is Zariski locally equivalent to the derived critical locus $\operatorname {\mathbf{Crit}}(\Phi )$ of a regular function $\Phi :U\rightarrow {\mathbin{\mathbb{A}}}^1$ on a smooth scheme $U$. We use this to show that the classical scheme $X=t_0(\boldsymbol{X})$ has the structure of an algebraic d-critical locus, in the sense of Joyce.

In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications to categorified and motivic Donaldson–Thomas theory of Calabi–Yau 3-folds, and to defining new Donaldson–Thomas type invariants of Calabi–Yau 4-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.

1. Introduction

In the context of Toën and Vezzosi’s theory of derived algebraic geometry 272829, Pantev, Toën, Vaquié, and Vezzosi 2432 defined a notion of $k$-shifted symplectic structure $\omega$ on a derived scheme or stack ${\boldsymbol{X}}$, for $k\in {\mathbin{\mathbb{Z}}}$. If ${\boldsymbol{X}}$ is a derived scheme and $\omega$ a 0-shifted symplectic structure, then ${\boldsymbol{X}}=X$ is a smooth classical scheme and $\omega \in H^0(\Lambda ^2T^*X)$ a classical symplectic structure on $X$.

Pantev et al. 24 introduced a notion of Lagrangian $\boldsymbol{i}:\boldsymbol{L}\rightarrow {\boldsymbol{X}}$ in a $k$-shifted symplectic derived stack $({\boldsymbol{X}},\omega )$ and showed that the fibre product $\boldsymbol{L}\times _{\boldsymbol{X}}\boldsymbol{M}$ of Lagrangians $\boldsymbol{i}:\boldsymbol{L}\rightarrow {\boldsymbol{X}}$, $\boldsymbol{j}:\boldsymbol{M}\rightarrow {\boldsymbol{X}}$ is $(k-1)$-shifted symplectic. Thus, (derived) intersections $L\cap M$ of Lagrangians $L,M$ in a classical algebraic symplectic manifold $(S,\omega )$ are $-1$-shifted symplectic. They also proved that if $Y$ is a Calabi–Yau $m$-fold, then the derived moduli stacks $\boldsymbol{{\mathbin{\mathcal{M}}}}$ of (complexes of) coherent sheaves on $Y$ carry a natural $(2-m)$-shifted symplectic structure.

The main aim of this paper is to prove a Darboux theorem, Theorem 5.18 below, which says that if ${\boldsymbol{X}}$ is a derived scheme and $\tilde{\omega }$ a $k$-shifted symplectic structure on ${\boldsymbol{X}}$ for $k<0$ with $k\not \equiv 2\mod 4$, then $({\boldsymbol{X}},\tilde{\omega })$ is Zariski locally equivalent to $(\operatorname {\mathbf{Spec}}A,\omega )$, for $\operatorname {\mathbf{Spec}}A$ an affine derived scheme in which the commutative differential graded algebra (cdga) $A$ is smooth in degree $0$ and quasi-free in negative degrees, and that has Darboux-like coordinates $x^i_j,y^{k-i}_j$ with respect to which the symplectic form $\omega =\sum _{i,j}{\mathrm{d}_{dR}}x^i_j{\mathrm{d}_{dR}}y^{k-i}_j$ is standard, and in which the differential in $A$ is given by a Poisson bracket with a Hamiltonian function $\Phi$ of degree $k+1$.

When $k<0$ with $k\equiv 2\mod 4$ we give two statements, one Zariski local in ${\boldsymbol{X}}$ in which the symplectic form $\omega$ on $\operatorname {\mathbf{Spec}}A$ is standard except for the part in the degree $k/2$ variables, which depends on some functions $q_i$, and one étale local in ${\boldsymbol{X}}$ in which $\omega$ is entirely standard. In the case $k=-1$, Theorem 5.18 implies that a $-1$-shifted symplectic derived scheme $({\boldsymbol{X}},\tilde{\omega })$ is Zariski locally equivalent to the derived critical locus $\operatorname {\mathbf{Crit}}(\Phi )$ of a regular function $\Phi :U\rightarrow {\mathbin{\mathbb{A}}}^1$ on a smooth scheme $U$.

1.1. This series of papers

This is the second in a series of eight papers 16, 7, 9, 3, 8, 5, 17, with more to come. The previous paper 16 defined algebraic d-critical loci $(X,s)$, which are classical schemes $X$ with an extra (classical, not derived) geometric structure $s$ that records information on how $X$ may locally be written as a classical critical locus $\operatorname {Crit}(\Phi )$ of a regular function $\Phi :U\rightarrow {\mathbin{\mathbb{A}}}^1$ on a smooth scheme $U$.

Our second main result, Theorem 6.6 below, says that if $({\boldsymbol{X}},\omega )$ is a $-1$-shifted symplectic derived scheme, then the underlying classical scheme $X=t_0({\boldsymbol{X}})$ extends naturally to an algebraic d-critical locus $(X,s)$. That is, we define a truncation functor from $-1$-shifted symplectic derived schemes to algebraic d-critical loci.

The third and fourth papers 79 will show that if $(X,s)$ is an algebraic d-critical locus with an orientation, then we can define a natural perverse sheaf $P_{X,s}^\bullet$, a $\mathscr{D}$-module $D_{X,s}^\bullet$, a mixed Hodge module $M_{X,s}^\bullet$ (when $X$ is over ${\mathbin{\mathbb{C}}}$), and a motive $MF_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on $\operatorname {Crit}(\Phi :U\rightarrow {\mathbin{\mathbb{A}}}^1),$ then $P_{X,s}^\bullet ,D_{X,s}^\bullet , M_{X,s}^\bullet$ are locally modelled on the perverse sheaf, $\mathscr{D}$-module, and mixed Hodge module of vanishing cycles of $\Phi$, and $MF_{X,s}$ is locally modelled on the motivic vanishing cycle of $\Phi$.

Combining these with Theorem 6.6 and results of Pantev et al. 24 gives natural perverse sheaves, $\mathscr{D}$-modules, mixed Hodge modules, and motives on classical moduli schemes ${\mathbin{\mathcal{M}}}$ of simple (complexes of) coherent sheaves on a Calabi–Yau 3-fold with orientations, and on intersections $L\cap M$ of spin Lagrangians $L,M$ in an algebraic symplectic manifold $(S,\omega )$. These will have applications to categorified and motivic Donaldson–Thomas theory of Calabi–Yau 3-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.

The fifth paper 3 will extend the results of this paper and 79 from (derived) schemes to (derived) Artin stacks. The sixth 8 will prove a complex analytic analogue of Corollary 6.8 below, saying that the intersection $X=L\cap M$ of complex Lagrangians $L,M$ in a complex symplectic manifold $(S,\omega )$ extends naturally to a complex analytic d-critical locus $(X,s)$.

The seventh paper 5 uses Theorem 5.18 to show that any $-2$-shifted symplectic derived ${\mathbin{\mathbb{C}}}$-scheme $({\boldsymbol{X}},\omega )$ can be given the structure of a derived smooth manifold ${\boldsymbol{X}}_{\mathrm{dm}}$. If ${\boldsymbol{X}}_{\mathrm{dm}}$ is compact and oriented, it has a virtual cycle $[{\boldsymbol{X}}_{\mathrm{dm}}]$ in bordism or homology. Using this, we propose to define Donaldson–Thomas style invariants counting (semi)stable coherent sheaves on Calabi–Yau 4-folds.

The eighth paper 17 proves a Lagrangian neighbourhood theorem which gives local models for Lagrangians $\boldsymbol{L}$ in $k$-shifted symplectic derived ${\mathbin{\mathbb{K}}}$-schemes $({\boldsymbol{X}},\omega )$, relative to the Darboux form local models for $({\boldsymbol{X}},\omega )$ in Theorem 5.18.

1.2. Organisation of this paper

We begin with background material from algebra in §2, and from derived algebraic geometry in §3. Section 4, which is not particularly original, proves that any derived ${\mathbin{\mathbb{K}}}$-scheme ${\boldsymbol{X}}$ is near any point $x\in {\boldsymbol{X}}$ Zariski locally equivalent to $\operatorname {\mathbf{Spec}}A$ for $A$ a standard form cdga which is minimal at $x$, and explains how to compare two such local presentations $\operatorname {\mathbf{Spec}}A\simeq {\boldsymbol{X}}\simeq \operatorname {\mathbf{Spec}}B$.

The heart of the paper is §5, which defines our Darboux form local models $A,\omega$, and proves our main result, Theorem 5.18, that any $k$-shifted symplectic derived ${\mathbin{\mathbb{K}}}$-scheme $({\boldsymbol{X}},\tilde{\omega })$ for $k<0$ is locally of the form $(\operatorname {\mathbf{Spec}}A,\omega )$. Finally, §6 discusses algebraic d-critical loci from 16, and proves our second main result Theorem 6.6, defining a truncation functor from $-1$-shifted symplectic derived ${\mathbin{\mathbb{K}}}$-schemes to algebraic d-critical loci.

Bouaziz and Grojnowski 6 have independently proved their own Darboux theorem for $k$-shifted symplectic derived schemes for $k<0$, similar to Theorem 5.18, by a different method. See Joyce and Safronov 17, Rem. 2.15 for an explanation of how this paper and 6 are related.

1.3. Conventions

Throughout, ${\mathbin{\mathbb{K}}}$ will be an algebraically closed field with characteristic $0$. All cdgas will be graded in nonpositive degrees (i.e., they are connective cdgas). All classical ${\mathbin{\mathbb{K}}}$-schemes are assumed locally of finite type, and all derived ${\mathbin{\mathbb{K}}}$-schemes ${\boldsymbol{X}}$ are assumed to be locally finitely presented.

2. Background from algebra

We begin by reviewing some fairly standard facts about cdgas, and their cotangent complexes. Some references on cdgas in derived algebraic geometry are Toën and Vezzosi 28, §2.3, 29, §2, and Lurie 23, §7.1; on cdgas from other points of view are Gelfand and Manin 11, §5.3 and Hess 14; and on cotangent complexes are Toën and Vezzosi 28, §1.4, 27, §4.2.4–§4.2.5, §3.1.7, and Lurie 23, §7.3.

2.1. Commutative graded algebras

In this section we introduce general definitions and conventions about commutative graded algebras.

Definition 2.1.

Fix a ground field ${\mathbin{\mathbb{K}}}$ of characteristic $0$. Eventually we shall wish to standardise a quadratic form, at which point we need to assume in addition that ${\mathbin{\mathbb{K}}}$ is algebraically closed. We shall work with connective commutative graded algebras $A$ over ${\mathbin{\mathbb{K}}}$ so $A$ has a decomposition $A=\bigoplus _{i\leqslant 0} A^{i}$ (“connective” means concentrated in nonpositive degrees) and an associative product $m : A^{i} \otimes A^{j} \rightarrow A^{i+j}$ satisfying $fg=(-1)^{|f||g|}gf$ for homogeneous elements $f,g \in A$. Given a graded left module $M$, we consider it as a symmetric bimodule by setting

$$\begin{equation*} m \cdot f=(-1)^{|f||m|} f \cdot m \end{equation*}$$

for homogeneous elements $f \in A$, $m \in M$.

Define a derivation of degree $k$ from $A$ to a graded module $M$ to be a ${\mathbin{\mathbb{K}}}$-linear map $\delta : A \rightarrow M$ that is homogeneous of degree $k$ and satisfies

$$\begin{equation*} \delta (fg)=\delta (f)g+(-1)^{k|f|}f\delta (g). \end{equation*}$$

Just as for (ungraded) commutative algebras, there is a universal derivation into a (graded) module of Kähler differentials $\Omega ^1_A$, which can be constructed as $I/I^2$ for $I=\operatorname {Ker}(m: A\otimes A \rightarrow A)$. The universal derivation $\delta : A \rightarrow \Omega ^1_A$ is then computed as $\delta (a)=a\otimes 1-1 \otimes a \in I/I^2$. One checks that indeed $\delta$ is a universal degree $0$ derivation, so that ${}\circ \delta :\underline{\operatorname {Hom}}_A(\Omega ^1_A,M) \rightarrow \underline{\operatorname {Der}}(A,M)$ is an isomorphism of graded modules. Here the underline denotes that we take a sum over homogeneous morphisms of each degree.

In the particular case when $M=A,$ one sometimes refers to a derivation $X: A \rightarrow A$ of degree $k$ as a vector field of degree $k$. Define the graded Lie bracket of two homogeneous vector fields $X,Y$ by

$$\begin{equation*} [X,Y]:=XY-(-1)^{|X||Y|}YX. \end{equation*}$$

One checks that $[X,Y]$ is a homogeneous vector field of degree $|X|+|Y|$. On any commutative graded algebra, there is a canonical degree $0$ Euler vector field $E$ which acts on a homogeneous element $f \in A$ via $E(f)=|f|f$. In particular, it annihilates functions $f \in A^0$ of degree $0$.

Define the de Rham algebra of $A$ to be the free commutative graded algebra over $A$ on the graded module $\Omega ^1_A[1]$:

$$\begin{equation} \operatorname {DR}(A):=\operatorname {Sym}_A(\Omega ^1_A[1])=\textstyle \bigoplus _p\Lambda ^p \Omega ^1_A[p]. {\tag{2.1}} \end{equation}$$

We endow $\operatorname {DR}(A)$ with the de Rham operator ${\mathrm{d}_{dR}}$, which is the unique square-zero derivation of degree $-1$ on the commutative graded algebra $\operatorname {DR}(A)$ such that for $f \in A$, ${\mathrm{d}_{dR}}(f)=\delta (f)[1] \in \Omega ^1_A[1]$. Thus ${\mathrm{d}_{dR}}(fg)={\mathrm{d}_{dR}}(f)g+(-1)^{|f|}f\,{\mathrm{d}_{dR}}(g)$ for $f,g \in A$ and ${\mathrm{d}_{dR}}(\alpha \cdot \beta )={\mathrm{d}_{dR}}(\alpha )\beta +(-1)^{|\alpha |}\alpha {\mathrm{d}_{dR}}(\beta )$ for any two $\alpha , \beta \in \operatorname {DR}(A)$.

Remark 2.2.

The de Rham algebra $\operatorname {DR}(A)$ has two gradings, one induced by the grading on $A$ and on the module $\Omega ^1[1]$ and the other given by $p$ in the decomposition $\operatorname {DR}(A):=\operatorname {Sym}_A(\Omega ^1_A[1]) \cong \bigoplus _p \Lambda ^p \Omega ^1_A[p]$. We shall refer to the first grading as degree and the second grading as weight. Thus the de Rham operator ${\mathrm{d}_{dR}}$ has degree $-1$ and weight $+1$.

Note that the condition that ${\mathrm{d}_{dR}}$ be a derivation of degree $-1$ does not take into account the additional grading by weight, but nevertheless this convention does recover the usual de Rham complex in the classical case when the algebra $A$ is concentrated in degree $0$.

Example 2.3.

For us, a typical commutative graded algebra $A=\bigoplus _{i\leqslant 0} A^{i}$ will have $A^0$ smooth over ${\mathbin{\mathbb{K}}}$ and will be free over $A^0$ on graded variables $x^{-1}_1,\ldots , x^{-1}_{m_1}, x^{-2}_1,\ldots ,x^{-2}_{m_2}, \ldots ,x^{-n}_1,\ldots ,x^{-n}_{m_{n}}$ where $x^{-i}_j$ has degree $-i$. Localising $A^0$ if necessary, we may assume that there exist $x^0_1,\ldots ,x^0_{m_0} \in A^0$ so that

$$\begin{equation*} {\mathrm{d}_{dR}}x^0_1,\ldots ,{\mathrm{d}_{dR}}x^0_{m_0} \end{equation*}$$

form a basis of $\Omega ^1_{A^0}$.

The graded $A$-module $\Omega ^1_A[1]$ then has a basis

$$\begin{equation*} {\mathrm{d}_{dR}}x^0_1,\ldots ,{\mathrm{d}_{dR}}x^0_{m_0},{\mathrm{d}_{dR}}x^{-1}_1,\ldots , {\mathrm{d}_{dR}}x^{-1}_{m_{-1}},\ldots ,{\mathrm{d}_{dR}}x^{-n}_1,\ldots ,{\mathrm{d}_{dR}}x^{-n}_{m_{-n}}. \end{equation*}$$

Since $\operatorname {DR}(A)=\operatorname {Sym}_A(\Omega ^1_A[1])$ is free commutative on $\Omega ^1_A[1]$ over $A$ and since ${\mathrm{d}_{dR}}x^{-i}_{s}$ and ${\mathrm{d}_{dR}}x^{-j}_{t}$ have degrees $-i-1$ and $-j-1,$ respectively, we have

$$\begin{equation*} {\mathrm{d}_{dR}}x^{-i}_{s} {\mathrm{d}_{dR}}x^{-j}_{t}=(-1)^{(-i-1)(-j-1)}{\mathrm{d}_{dR}}x^{-j}_{t} {\mathrm{d}_{dR}}x^{-i}_{s}. \end{equation*}$$

Thus, for example, ${\mathrm{d}_{dR}}x^{-i}_{s}$ and ${\mathrm{d}_{dR}}x^{-i}_{t}$ anticommute when $-i$ is even and commute when $-i$ is odd. In particular, ${\mathrm{d}_{dR}}x^{-i}_{s} {\mathrm{d}_{dR}}x^{-i}_{s}= 0$ when $-i$ is even and ${\mathrm{d}_{dR}}x^{-i}_{s} {\mathrm{d}_{dR}}x^{-i}_{s}\neq 0$ when $-i$ is odd.

Definition 2.4.

Given a homogeneous vector field $X$ on $A$ of degree $|X|$, the contraction operator $\iota _X$ on $\operatorname {DR}(A)$ is defined to be the unique derivation of degree $|X|+1$ such that $\iota _Xf=0$ and $\iota _X{\mathrm{d}_{dR}}(f)=X(f)$ for all $f \in A$. We define the Lie derivative $L_X$ along a vector field $X$ by

$$\begin{equation*} L_X=[\iota _X,{\mathrm{d}_{dR}}]=\iota _X{\mathrm{d}_{dR}}-(-1)^{|X|+1}{\mathrm{d}_{dR}}\iota _X= \iota _X{\mathrm{d}_{dR}}+(-1)^{|X|}{\mathrm{d}_{dR}}\iota _X. \end{equation*}$$

It is a derivation of $\operatorname {DR}(A)$ of degree $|X|$. In particular, the Lie derivative along $E$ is of degree $0$. Given $f \in A$, we have

$$\begin{equation*} L_Ef=\iota _E{\mathrm{d}_{dR}}f=E(f)=|f|f, \qquad L_E{\mathrm{d}_{dR}}f=|f|{\mathrm{d}_{dR}}f. \end{equation*}$$

In particular, the de Rham differential ${\mathrm{d}_{dR}}: A \rightarrow \Omega ^1_A[1]$ is injective except in degree $0$. Furthermore, we see that for a homogeneous form $\alpha \in \Lambda ^p\Omega ^1_A[p],$

$$\begin{equation*} L_E\alpha =\iota _E{\mathrm{d}_{dR}}\alpha +{\mathrm{d}_{dR}}\iota _E \alpha =(|\alpha |+p)\alpha . \end{equation*}$$

If $\alpha$ is in addition de Rham closed, then ${\mathrm{d}_{dR}}\iota _E \alpha =(|\alpha |+p)\alpha$, so if $|\alpha |+p \neq 0$, then $\alpha$ is in fact de Rham exact:

$$\begin{equation*} \alpha ={\mathrm{d}_{dR}}\Bigl (\frac{\iota _E\alpha }{|\alpha |+p}\Bigr ). \end{equation*}$$

The top degree of $\Lambda ^p\Omega ^1_A[p]$ is $-p$ and the elements of top degree are of the form $\alpha \in \Lambda ^p\Omega ^1_{A^0}[p]$, so a de Rham closed form $\alpha$ can fail to be exact only if it lives on $A^0 \subset A$.

The following relations between derivations of $\operatorname {DR}(A)$ can be checked by noting that both sides of an equation have the same degree and act in the same way on elements of weight $0$ and weight $1$ in $\operatorname {DR}(A)$.

Lemma 2.5.

Let $X,Y$ be homogeneous vector fields on $A$. Then we have the following equalities of derivations on $\operatorname {DR}(A)\!:$

$$\begin{equation*} [{\mathrm{d}_{dR}},L_X]=0,\quad [\iota _X,\iota _Y]=0,\quad [L_X,\iota _Y]=\iota _{[X,Y]},\quad [L_X,L_Y]=L_{[X,Y]}. \end{equation*}$$

Remark 2.6.

The last statement ensures that the map $X \mapsto L_X$ determines a dg-Lie algebra homomorphism $\underline{\operatorname {Der}}(A) \rightarrow \underline{\operatorname {Der}}(\operatorname {DR}(A))$.

2.2. Commutative differential graded algebras

Definition 2.7.

A commutative differential graded algebra or cdga $(A,{\mathrm{d}})$ is a commutative graded algebra $A$ over ${\mathbin{\mathbb{K}}}$, as in §2, endowed with a square-zero derivation ${\mathrm{d}}$ of degree 1. Usually we write $A$ rather than $(A,{\mathrm{d}})$, leaving ${\mathrm{d}}$ implicit. Note that the cohomology $H^*(A)$ of $A$ with respect to the differential ${\mathrm{d}}$ is a commutative graded algebra. Note too that by default all of our cdgas are connective (that is, concentrated in nonpositive cohomological degrees).

A morphism of cdgas is a map of complexes $f: A \rightarrow B$ respecting units and multiplication. It is a quasi-isomorphism or weak equivalence if the underlying map of complexes is so, it is a fibration if it is a degree-wise surjection, and it is a cofibration if it has the left lifting property with respect to trivial cofibrations. A standard argument (see for instance Goerss and Schemmerhorn 12, Th. 3.6) shows that in characteristic $0$ this choice of weak equivalences and fibrations define a model structure on cdgas.

We shall assume that all cdgas in this paper are (homotopically) of finite presentation over the ground field ${\mathbin{\mathbb{K}}}$. As for finitely presented classical ${\mathbin{\mathbb{K}}}$-algebras, this can be formulated in terms of the preservation of filtered (homotopy) colimits. For the precise notion, see Toën and Vezzosi 28, §1.2.3. A classical ${\mathbin{\mathbb{K}}}$-algebra that is of finite presentation in the classical sense need not be of finite presentation when considered as a cdga. Indeed, Theorem 4.1 will show that a cdga $A$ of finite presentation admits a very strong form of finite resolution.

In much of the paper, we will work with cdgas of the following form:

Example 2.8.

We will explain how to inductively construct a sequence of cdgas $A(0),A(1),\ldots ,A(n)$, where $A(0)$ is a smooth ${\mathbin{\mathbb{K}}}$-algebra, and $A(k)$ has underlying commutative graded algebra free over $A(0)$ on generators of degrees $-1,\ldots ,-k$.

Begin with a commutative algebra $A(0)$ smooth over ${\mathbin{\mathbb{K}}}$. To make Proposition 2.12 below hold (which says that the cotangent complex ${\mathbin{\mathbb{L}}}_{A(k)}$ has a simple description) we assume the cotangent module $\Omega ^1_{A(0)}$ is a free $A(0)$-module, which can always be achieved by Zariski localising $A(0)$. Choose a free $A(0)$-module $M^{-1}$ of finite rank together with a map $\pi ^{-1} :M^{-1} \rightarrow A(0)$. Define a cdga $A(1)$ whose underlying commutative graded algebra is free over $A(0)$ with generators given by $M^{-1}$ in degree $-1$ and with differential ${\mathrm{d}}$ determined by the map $\pi ^{-1}: M^{-1} \rightarrow A(0)$. By construction, we have $H^0(A(1)) = A(0)/I$, where the ideal $I\subseteq A(0)$ is the image of the map $\pi ^{-1}:M^{-1}\rightarrow A(0)$.

Note that $A(1)$ fits in a homotopy pushout diagram of cdgas

$$\begin{equation*} \vcenter{\img[][148pt][43pt][{$\xymatrix@R=14pt@C=90pt{*+[r]{\Sym_{A(0)}(M^{-1})} \ar[r]_(0.7){0_*} \ar[d]^{\pi^{-1}_*} &amp; *+[l]{A(0)} \ar[d] \\ *+[r]{A(0)} \ar[r]^(0.7){f^{-1}} &amp; *+[l]{A(1),\!\!{}} }$}]{Images/img4dc547bb5daa2f38bf2fea6f9a0229b9.svg}} \end{equation*}$$

with morphisms $\pi ^{-1}_*,0_*$ induced by $\pi ^{-1},0:M^{-1}\rightarrow A(0)$. Write $f^{-1}:A(0)\rightarrow A(1)$ for the resulting map of algebras.

Next, choose a free $A(1)$-module $M^{-2}$ of finite rank together with a map $\pi ^{-2}: M^{-2}[1] \rightarrow A(1)$. Define a cdga $A(2)$ whose underlying commutative graded algebra is free over $A(1)$ with generators given by $M^{-2}$ in degree $-2$ and with differential ${\mathrm{d}}$ determined by the map $\pi ^{-2}: M^{-2}[1] \rightarrow A(1)$. Write $f^{-2}$ for the resulting map of algebras $A(1)\rightarrow A(2)$.

As the underlying commutative graded algebra of $A(1)$ was free over $A(0)$ on generators of degree $-1$, the underlying commutative graded algebra of $A(2)$ is free over $A(0)$ on generators of degrees $-1,-2$. Since $A(2)$ is obtained from $A(1)$ by adding generators in degree $-2$, we have $H^0(A(1))\cong H^0(A(2)) \cong A(0)/I$.

Note that $A(2)$ fits in a homotopy pushout diagram of cdgas

$$\begin{equation*} \vcenter{\img[][154pt][43pt][{$\xymatrix@R=14pt@C=90pt{*+[r]{\Sym_{A(1)}(M^{-2}[1])} \ar[r]_(0.7){0_*} \ar[d]^{\pi^{-2}_*} &amp; *+[l]{A(1)} \ar[d] \\ *+[r]{A(1)} \ar[r]^(0.7){f^{-2}} &amp; *+[l]{A(2),\!\!{}} }$}]{Images/img95fa9ab8c7f88b786cd21717168e82ec.svg}} \end{equation*}$$

with morphisms $\pi ^{-2}_*,0_*$ induced by $\pi ^{-2},0:M^{-2}[1]\rightarrow A(1)$.

Continuing in this manner inductively, we define a cdga $A(n)=A$ with $A^0=A(0)$ and $H^0(A)=A(0)/I$, whose underlying commutative graded algebra is free over $A(0)$ on generators of degrees $-1,\ldots ,-n$.

Definition 2.9.

A cdga $A=A(n)$ over ${\mathbin{\mathbb{K}}}$ constructed inductively as in Example 2.8 from a smooth ${\mathbin{\mathbb{K}}}$-algebra $A(0)$ with $\Omega ^1_{A(0)}$ free of finite rank over $A(0)$, and free finite rank modules $M^{-1},M^{-2},\ldots , M^{-n}$ over $A(0),\ldots ,A(n-1)$, will be called a standard form cdga.

Equivalently, $A$ is of standard form if it is finitely presented, and $A(0)$ is smooth with $\Omega _{A(0)}$ a free $A(0)$-module, and the underlying graded algebra of $A$ is freely generated over $A(0)$ by finitely many generators in negative degrees.

If $A$ is of standard form, we will call a cdga $A'$ a localisation of $A$ if $A'=A\otimes _{A^0}A^0[f^{-1}]$ for $f\in A^0$; that is, $A'$ is obtained by inverting $f$ in $A$. Then $A'$ is also of standard form, with $A^{\prime \, 0}\cong A^0[f^{-1}]$. If $p\in \operatorname {Spec}H^0(A)$ with $f(p)\ne 0$, we call $A'$ a localisation of $A$ at $p$.

Standard form cdgas are a mild generalisation of connective semifree cdgas, which are standard form cdgas in which $A(0)$ is a free commutative ${\mathbin{\mathbb{K}}}$-algebra. Semifree cdgas are among the cofibrant objects in the model structure on $\mathop{\mathbf{cdga}_{\mathbin{\mathbb{K}}}^{\leqslant 0}}$, and Proposition 2.12 below is standard for semifree cdgas.

2.3. Cotangent complexes of cdgas

Definition 2.10.

Let $(A,d)$ be a cdga. Then as in Definition 2.1, to the underlying commutative graded algebra $A$ we associate the module of Kähler differentials $\Omega ^1_A$ with universal degree 0 derivation $\delta :A\rightarrow \Omega ^1_A$, and the de Rham algebra $\operatorname {DR}(A)=\operatorname {Sym}_A\bigl (\Omega ^1_A[1]\bigr )$ in 2.1, with degree $-1$ de Rham differential ${\mathrm{d}_{dR}}:\operatorname {DR}(A)\rightarrow \operatorname {DR}(A)$.

The differential ${\mathrm{d}}$ on $A$ induces a unique differential on $\Omega ^1_A$, also denoted ${\mathrm{d}}$, satisfying ${\mathrm{d}}\circ \delta =\delta \circ {\mathrm{d}}: A\rightarrow \Omega ^1_A$, and making $(\Omega ^1_A,{\mathrm{d}})$ into a dg-module. According to our sign conventions, the differential ${\mathrm{d}}$ on $\Omega ^1_A[1]$ is that on $\Omega ^1_A$ multiplied by $-1$, so ${\mathrm{d}}$ on $\Omega ^1_A[1]$ anticommutes with the de Rham operator ${\mathrm{d}_{dR}}:A\rightarrow \Omega ^1[1]$. We extend the differential ${\mathrm{d}}$ uniquely to all of $\operatorname {DR}(A)$ by requiring it to be a derivation of degree $1$ with respect to the multiplication on $\operatorname {DR}(A)$.

When $(A,{\mathrm{d}})$ is sufficiently nice (as in Example 2.8), then the Kähler differentials $(\Omega ^1_A,{\mathrm{d}})$ give a model for the cotangent complex ${\mathbin{\mathbb{L}}}_{(A,{\mathrm{d}})}$ of $(A,{\mathrm{d}})$. In practice, we shall always work with such cdgas, so we shall freely identify ${\mathbin{\mathbb{L}}}_{(A,{\mathrm{d}})}$ and $(\Omega ^1_A,{\mathrm{d}})$. Usually, when dealing with cdgas and their cotangent complexes, we leave ${\mathrm{d}}$ implicit and write ${\mathbin{\mathbb{L}}}_A,\Omega ^1_A$ rather than ${\mathbin{\mathbb{L}}}_{(A,{\mathrm{d}})}$ and $(\Omega ^1_A,{\mathrm{d}})$.

Similarly, given a map $A \rightarrow B$ of cdgas, we can define the relative Kähler differentials $\Omega ^1_{B/A}$, and when the map $A \rightarrow B$ is nice enough (for example, $B$ is obtained from $A$ adding free generators of some degree and imposing a differential, as in Example 2.8), then the relative Kähler differentials give a model for the relative cotangent complex ${\mathbin{\mathbb{L}}}_{B/A}$.

We recall some basic facts about cotangent complexes:

(i)

Given a map of cdgas $\alpha :A \rightarrow B$, there is an induced map ${\mathbin{\mathbb{L}}}_\alpha :{\mathbin{\mathbb{L}}}_A \rightarrow {\mathbin{\mathbb{L}}}_B$ of $A$-modules, and we have a (homotopy) fibre sequence$$\begin{equation*} \vcenter{\img[][202pt][16pt][{$\xymatrix@C=60pt{\bL_A \ot_A B \ar[r]^(0.55){\bL_\al\ot\id_B} &amp; \bL_B \ar[r] &amp; \bL_{B/A}.}$}]{Images/img99cc51794d0d718c96f8323834a0ddcc.svg}}{} \end{equation*}$$

(ii)

Given a (homotopy) pushout square of cdgas$$\begin{equation*} \vcenter{\img[][106pt][34pt][{$\xymatrix@R=12pt@C=80pt{*+[r]{A} \ar[r] \ar[d] &amp; *+[l]{B} \ar[d] \\ *+[r]{A'} \ar[r] &amp; *+[l]{B',\!\!{}} }$}]{Images/img9c8d6ac27e49edc52f45e081e385f70c.svg}} \end{equation*}$$

we have a base change equivalence$$\begin{equation} {\mathbin{\mathbb{L}}}_{B'/A'} \simeq {\mathbin{\mathbb{L}}}_{B/A} \otimes _B B'. {\tag{2.2}} \end{equation}$$

(iii)

If $B=\operatorname {Sym}_A(M)$ for a module $M$ over a cdga $A$, with inclusion morphism $\iota :A\hookrightarrow B=A\oplus M\oplus S^2M\oplus \cdots$, we have a canonical equivalence$$\begin{equation*} {\mathbin{\mathbb{L}}}_{B/A} \simeq B \otimes _A M. \end{equation*}$$

Also, any $A$-module morphism $\rho :M\rightarrow A$ (such as $\rho =0$) induces a unique cdga morphism $\rho _*:B\rightarrow A$ with $\rho _*\vert _{\operatorname {Sym}^0(M)}=\operatorname {id}_A$ and $\rho _*\vert _{\operatorname {Sym}^1(M)}=\rho$. Then we have a canonical equivalence$$\begin{equation} {\mathbin{\mathbb{L}}}_{A/B}\simeq M[1]. {\tag{2.3}} \end{equation}$$

Example 2.11.

In Example 2.8 we constructed inductively a sequence of cdgas $A(0) \rightarrow A(1) \rightarrow \cdots \rightarrow A(n)=A$, where $A(k)$ was obtained from $A(k-1)$ by adjoining a module of generators $M^{-k}$ in degree $-k$ and imposing a differential. The cdga $A(k)$ then fits into a (homotopy) pushout diagram

$$\begin{equation} \begin{gathered} \vcenter{\img[][252pt][45pt][{$\xymatrix@R=16pt@C=140pt{*+[r]{B(k-1):=\Sym_{A(k-1)}(M^{-k}[k-1])} \ar[r]_(0.76){0_*} \ar[d]^{\pi^{-k}_*} &amp; *+[l]{A(k-1)} \ar[d] \\ *+[r]{A(k-1)} \ar[r]^(0.76){f^{-k}} &amp; *+[l]{A(k),} }$}]{Images/imgb7feb80cfc4ac8f18ef096377af6ed8a.svg}} \end{gathered} {\tag{2.4}} \end{equation}$$

in which $\pi ^{-k}_*,0_*$ are induced by maps $\pi ^{-k},0:M^{-k}[k-1]\rightarrow A(k-1)$.

We will describe the relative cotangent complexes ${\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}$ and hence, inductively, the cotangent complex ${\mathbin{\mathbb{L}}}_A$. By equation 2.3, the relative cotangent complex ${\mathbin{\mathbb{L}}}_{A(k-1)/B(k-1)}$ for $0_*:B(k-1)\rightarrow A(k-1)$ in 2.4 is given by

$$\begin{equation*} {\mathbin{\mathbb{L}}}_{A(k-1)/B(k-1)}\simeq M^{-k}[k]. \end{equation*}$$

Thus by base change 2.2, the relative cotangent complex ${\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}$ of $f^{-k}:A(k-1) \rightarrow A(k)$ in 2.4 satisfies

$$\begin{equation*} {\mathbin{\mathbb{L}}}_{A(k)/A(k-1)} \simeq A(k) \otimes _{A(k-1)}M^{-k}[k]. \end{equation*}$$

Note in particular that when $i > -k$, we have

$$\begin{equation} H^{i}({\mathbin{\mathbb{L}}}_{A(k)/A(k-1)})=0. {\tag{2.5}} \end{equation}$$

Commutative diagrams of the following form, in which the rows and columns are fibre sequences, are very useful for computing the cohomology of ${\mathbin{\mathbb{L}}}_A$:

$$\begin{equation} \begin{gathered} \vcenter{\img[][267pt][101pt][{$\xymatrix@C=35pt@R=14pt{&amp;&amp; A \ot_{A(k)}\bL_{A(k)/A(k-1)} \ar[d] \\ A \ot_{A(k-1)}\bL_{A(k-1)} \ar[r]\ar[d] &amp; \bL_A \ar[r]\ar@{=}[d] &amp; \bL_{A/A(k-1) \ar[d]} \\ A \ot_{A(k)}\bL_{A(k)} \ar[r]\ar[d] &amp; \bL_A \ar[r] &amp; \bL_{A/A(k)} \\ A \ot_{A(k)}\bL_{A(k)/A(k-1)}. }$}]{Images/img1ac4300416e8570d2409a7ed8249a846.svg}} \end{gathered} {\tag{2.6}} \end{equation}$$

Proposition 2.12.

Let $A=A(n)$ be a standard form cdga constructed inductively as in Example 2.8. Then the restriction of the cotangent complex ${\mathbin{\mathbb{L}}}_A$ to $\operatorname {Spec}H^0(A)$ is naturally represented as a complex of free $H^0(A)$-modules

$$\begin{equation} \begin{gathered} \vcenter{\img[][300pt][15pt][{$\xymatrix@C=29pt{ 0 \ar[r] &amp; V^{-n} \ar[r]^{\Xd^{-n}} &amp; V^{1-n} \ar[r]^{\Xd^{1-n}} &amp; \cdots\ar[r]^{\Xd^{-2}} &amp; V^{-1} \ar[r]^{\Xd^{-1}} &amp; V^0 \ar[r] &amp; 0, }$}]{Images/img27f014da98ca2ec57a7473c2fcadfa26.svg}} \end{gathered} {\tag{2.7}} \end{equation}$$

where $V^{-k}=H^{-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}\bigr )$ is in degree $-k$ and the differential ${\mathrm{d}}^{-k}:V^{-k}\rightarrow V^{1-k}$ is identified with the composition

$$\begin{equation*} H^{-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}\bigr )\longrightarrow H^{1-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k-1)} \bigr )\longrightarrow H^{1-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k-1)/A(k-2)}\bigr ), \end{equation*}$$

in which $H^{-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}\bigr )\rightarrow H^{1-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k-1)}\bigr )$ is induced by the fibre sequence

$$\begin{equation*} A(k) \otimes _{A(k-1)} {\mathbin{\mathbb{L}}}_{A(k-1)}\longrightarrow {\mathbin{\mathbb{L}}}_{A(k)} \longrightarrow {\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}, \end{equation*}$$

and $H^{1-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k-1)}\bigr )\rightarrow H^{1-k}\bigl ( {\mathbin{\mathbb{L}}}_{A(k-1)/A(k-2)}\bigr )$ is induced by the fibre sequence

$$\begin{equation*} A(k-1)\otimes _{A(k-2)}{\mathbin{\mathbb{L}}}_{A(k-2)}\longrightarrow {\mathbin{\mathbb{L}}}_{A(k-1)}\longrightarrow {\mathbin{\mathbb{L}}}_{A(k-1)/A(k-2)}. \end{equation*}$$

Proof.

The proof is almost immediate by induction on $n$. When $n=0$, ${\mathbin{\mathbb{L}}}_{A(0)}=\Omega ^1_{A(0)}$ is free of finite rank over $A(0)$ by definition of standard form cdgas, so $V^0=\Omega ^1_{A(0)}\otimes _{A(0)}H^0(A)$ is free of finite rank over $H^0(A)$. For $n=1$, tensor the fibre sequence $A(1) \otimes _{A(0)} {\mathbin{\mathbb{L}}}_{A(0)} \rightarrow {\mathbin{\mathbb{L}}}_{A(1)} \rightarrow {\mathbin{\mathbb{L}}}_{A(1)/A(0)}$ with $H^0(A)$ to get a fibre sequence $V^0 \rightarrow H^0(A)\otimes _{A(1)} {\mathbin{\mathbb{L}}}_{A(1)} \rightarrow V^{-1}[1]$ in which the connecting morphism $V^{-1} \rightarrow V^0$ is as claimed.

Similarly, assuming we have proved the proposition for ${\mathbin{\mathbb{L}}}_{A(n-1)}$, tensor the fibre sequence $A \otimes _{A(n-1)} {\mathbin{\mathbb{L}}}_{A(n-1)}\rightarrow {\mathbin{\mathbb{L}}}_A \rightarrow {\mathbin{\mathbb{L}}}_{A/A(n-1)}$ with $H^0(A)$ to get a fibre sequence $H^0(A)\otimes _{A(n-1)} {\mathbin{\mathbb{L}}}_{A(n-1)} \rightarrow H^0(A) \otimes _A {\mathbin{\mathbb{L}}}_A \rightarrow V^{-n}[n]$ in which the connecting morphism is as claimed.

■

Definition 2.13.

Let $A$ be a standard form cdga constructed as in Example 2.8. We call $A$ minimal at $p \in \operatorname {Spec}H^0(A)$ if ${\mathrm{d}}^{-k}\vert _p=0$ for $k=1,\ldots ,n$, for $d^{-k}$ the internal differential in ${\mathbin{\mathbb{L}}}_A$ as in 2.7. That is, the compositions

$$\begin{equation} H^{-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}\bigr ) \longrightarrow H^{1-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k-1)}\bigr )\longrightarrow H^{1-k}\bigl ({\mathbin{\mathbb{L}}}_{A(k-1)/A(k-2)}\bigr ) {\tag{2.8}} \end{equation}$$

in the cotangent complexes restricted to $\operatorname {Spec}H^0(A)$ vanish at $p$ for all $k$.

3. Background from derived algebraic geometry

Next we outline the background material from derived algebraic geometry that we need, aiming in particular to explain the key notion from Pantev, Toën, Vaquié, and Vezzosi 24 of a $k$-shifted symplectic structure on a derived ${\mathbin{\mathbb{K}}}$-scheme, which is central to our paper. There are two main frameworks for derived algebraic geometry in the literature, due to Toën and Vezzosi 272829 and Lurie 2223, which are broadly interchangeable in characteristic $0$. Following our principal reference 24, we use the Toën–Vezzosi version.

To understand this paper (except for a few technical details), one does not really need to study derived algebraic geometry in any depth or to know what a derived stack is. The main point for us is that a derived ${\mathbin{\mathbb{K}}}$-scheme is a geometric space locally modelled on $\operatorname {\mathbf{Spec}}A$ for $A$ a cdga over ${\mathbin{\mathbb{K}}}$, just as a classical ${\mathbin{\mathbb{K}}}$-scheme is a space locally modelled on $\operatorname {Spec}A$ for $A$ a commutative ${\mathbin{\mathbb{K}}}$-algebra; though the meaning of “locally modelled” is more subtle in the derived than the classical setting. Readers who are comfortable with this description of derived schemes can omit §3.1 and §3.2.

As in §2, all cdgas in this paper are connective (graded in nonpositive degrees), as these are the allowed local models in derived algebraic geometry.

3.1. Derived stacks

We will use Toën and Vezzosi’s theory of derived algebraic geometry 272829. We give a brief outline to fix notation. Fix a base field ${\mathbin{\mathbb{K}}}$, of characteristic $0$. In 27, §3, 28, §2.1, Toën and Vezzosi define an $\infty$-category of (higher) ${\mathbin{\mathbb{K}}}$-stacks $\operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$. Objects $X$ in $\operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$ are $\infty$-functors

$$\begin{equation*} X:\{\text{commutative ${\mathbin{\mathbb{K}}}$-algebras}\}\longrightarrow \{\text{simplicial sets}\} \end{equation*}$$

satisfying sheaf-type conditions. They also define a full $\infty$-subcategory $\operatorname {\mathbf{Art}}_{\mathbin{\mathbb{K}}}\subset \operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$ of (higher) Artin ${\mathbin{\mathbb{K}}}$-stacks, with better geometric properties.

Classical ${\mathbin{\mathbb{K}}}$-schemes and algebraic ${\mathbin{\mathbb{K}}}$-spaces may be written as functors

$$\begin{equation*} X:\{\text{commutative ${\mathbin{\mathbb{K}}}$-algebras}\}\longrightarrow \{\text{sets}\}, \end{equation*}$$

and classical Artin ${\mathbin{\mathbb{K}}}$-stacks may be written as functors

$$\begin{equation*} X:\{\text{commutative ${\mathbin{\mathbb{K}}}$-algebras}\}\longrightarrow \{\text{groupoids}\}. \end{equation*}$$

As simplicial sets (an $\infty$-category) generalise both sets (a 1-category) and groupoids (a 2-category), higher ${\mathbin{\mathbb{K}}}$-stacks generalise ${\mathbin{\mathbb{K}}}$-schemes, algebraic ${\mathbin{\mathbb{K}}}$-spaces, and Artin ${\mathbin{\mathbb{K}}}$-stacks.

Toën and Vezzosi define the $\infty$-category $\operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$ of derived ${\mathbin{\mathbb{K}}}$-stacks (or $D^-$-stacks) 28, Def. 2.2.2.14, 27, Def. 4.2. Objects ${\boldsymbol{X}}$ in $\operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$ are $\infty$-functors

$$\begin{equation*} {\boldsymbol{X}}:\{\text{simplicial commutative ${\mathbin{\mathbb{K}}}$-algebras}\}\longrightarrow \{\text{simplicial sets}\} \end{equation*}$$

satisfying sheaf-type conditions. They also define a full $\infty$-subcategory $\operatorname {\mathbf{dArt}}_{\mathbin{\mathbb{K}}}\subset \operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$ of derived Artin ${\mathbin{\mathbb{K}}}$-stacks, with better geometric properties.

There is a truncation functor $t_0:\operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}\rightarrow \operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$ from derived stacks to (higher) stacks, which maps $t_0:\operatorname {\mathbf{dArt}}_{\mathbin{\mathbb{K}}}\rightarrow \operatorname {\mathbf{Art}}_{\mathbin{\mathbb{K}}}$, and a fully faithful left adjoint inclusion functor $i:\operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}\rightarrow \operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$ mapping $\operatorname {\mathbf{Art}}_{\mathbin{\mathbb{K}}}\rightarrow \operatorname {\mathbf{dArt}}_{\mathbin{\mathbb{K}}}$. As $i$ is fully faithful, we can regard it as embedding $\operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}},\operatorname {\mathbf{Art}}_{\mathbin{\mathbb{K}}}$ as full subcategories of $\operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}, \operatorname {\mathbf{dArt}}_{\mathbin{\mathbb{K}}}$. Thus, we can regard classical ${\mathbin{\mathbb{K}}}$-schemes and Artin ${\mathbin{\mathbb{K}}}$-stacks as examples of derived ${\mathbin{\mathbb{K}}}$-stacks.

The adjoint property of $i,t_0$ implies that for any $X\in \operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$ there is a natural morphism $X\rightarrow t_0\circ i(X)$, which is an equivalence as $i$ is fully faithful, and for any ${\boldsymbol{X}}\in \operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$ there is a natural morphism $i\circ t_0({\boldsymbol{X}})\rightarrow {\boldsymbol{X}}$, which we may regard as embedding the classical truncation $X=t_0({\boldsymbol{X}})$ of ${\boldsymbol{X}}$ as a substack of ${\boldsymbol{X}}$. On notation: we generally write derived schemes and stacks ${\boldsymbol{X}},{\boldsymbol{Y}},\ldots$ and their morphisms $\boldsymbol{f},\boldsymbol{g},\ldots$ in bold, and classical schemes, stacks, and higher stacks $X,Y,\ldots$ and morphisms $f,g,\ldots$ not in bold, and we will write $X=t_0({\boldsymbol{X}})$, $f=t_0(\boldsymbol{f})$, and so on, for classical truncations of derived ${\boldsymbol{X}},\boldsymbol{f},\ldots .$

3.2. Derived schemes and cdgas

Toën and Vezzosi 272829 base their derived algebraic geometry on simplicial commutative ${\mathbin{\mathbb{K}}}$-algebras, but we prefer to work with commutative differential graded ${\mathbin{\mathbb{K}}}$-algebras (cdgas). As in 23, §8.1.4 there is a normalisation functor

$$\begin{equation*} N:\{\text{simplicial commutative ${\mathbin{\mathbb{K}}}$-algebras}\}\longrightarrow \{\text{cdgas over ${\mathbin{\mathbb{K}}}$}\} \end{equation*}$$

which is an equivalence of $\infty$-categories, since ${\mathbin{\mathbb{K}}}$ has characteristic $0$ by our assumption in §1. So, working with simplicial commutative ${\mathbin{\mathbb{K}}}$-algebras and with cdgas over ${\mathbin{\mathbb{K}}}$ are essentially equivalent.

There is a spectrum functor

$$\begin{equation*} \operatorname {Spec}:\{\text{commutative ${\mathbin{\mathbb{K}}}$-algebras}\}\longrightarrow \operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}. \end{equation*}$$

An object $X$ in $\operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$ is called an affine ${\mathbin{\mathbb{K}}}$-scheme if it is equivalent to $\operatorname {Spec}A$ for some commutative ${\mathbin{\mathbb{K}}}$-algebra $A$, and a ${\mathbin{\mathbb{K}}}$-scheme if it may be covered by Zariski open $Y\subseteq X$ with $Y$ an affine ${\mathbin{\mathbb{K}}}$-scheme. Write $\operatorname {\mathbf{Sch}}_{\mathbin{\mathbb{K}}}$ for the full ($\infty$-)subcategory of ${\mathbin{\mathbb{K}}}$-schemes in $\operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$.

Similarly, there is a spectrum functor

$$\begin{equation*} \operatorname {\mathbf{Spec}}:\{\text{commutative differential graded ${\mathbin{\mathbb{K}}}$-algebras}\}\longrightarrow \operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}. \end{equation*}$$

A derived ${\mathbin{\mathbb{K}}}$-stack ${\boldsymbol{X}}$ is called an affine derived ${\mathbin{\mathbb{K}}}$-scheme if ${\boldsymbol{X}}$ is equivalent in $\operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$ to $\operatorname {\mathbf{Spec}}A$ for some cdga $A$ over ${\mathbin{\mathbb{K}}}$. (This is true if and only if ${\boldsymbol{X}}$ is equivalent to $\operatorname {\mathbf{Spec}}{}^\Delta (A^\Delta )$ for some simplicial commutative ${\mathbin{\mathbb{K}}}$-algebra $A^\Delta$, as the normalisation functor $N$ is an equivalence.) As in Toën 27, §4.2, a derived ${\mathbin{\mathbb{K}}}$-stack ${\boldsymbol{X}}$ is called a derived ${\mathbin{\mathbb{K}}}$-scheme if it may be covered by Zariski open ${\boldsymbol{Y}}\subseteq {\boldsymbol{X}}$ with ${\boldsymbol{Y}}$ an affine derived ${\mathbin{\mathbb{K}}}$-scheme. Write $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$ for the full $\infty$-subcategory of derived ${\mathbin{\mathbb{K}}}$-schemes in $\operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$. Then $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}\subset \operatorname {\mathbf{dArt}}_{\mathbin{\mathbb{K}}}\subset \operatorname {\mathbf{dSt}}_{\mathbin{\mathbb{K}}}$.

If $A$ is a cdga over ${\mathbin{\mathbb{K}}}$, there is an equivalence $t_0\circ \operatorname {\mathbf{Spec}}A\simeq \operatorname {Spec}H^0(A)$ in $\operatorname {\mathbf{St}}_{\mathbin{\mathbb{K}}}$. Hence, if ${\boldsymbol{X}}$ is an affine derived ${\mathbin{\mathbb{K}}}$-scheme, then $X=t_0({\boldsymbol{X}})$ is an affine ${\mathbin{\mathbb{K}}}$-scheme, and if ${\boldsymbol{X}}$ is a derived ${\mathbin{\mathbb{K}}}$-scheme, then $X=t_0({\boldsymbol{X}})$ is a ${\mathbin{\mathbb{K}}}$-scheme, and the truncation functor maps $t_0:\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}\rightarrow \operatorname {\mathbf{Sch}}_{\mathbin{\mathbb{K}}}$. Also the inclusion functor maps $i:\operatorname {\mathbf{Sch}}_{\mathbin{\mathbb{K}}}\rightarrow \operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$. As in 27, §4.2, one can show that if ${\boldsymbol{X}}$ is a derived Artin ${\mathbin{\mathbb{K}}}$-stack and $t_0({\boldsymbol{X}})$ is a ${\mathbin{\mathbb{K}}}$-scheme, then ${\boldsymbol{X}}$ is a derived ${\mathbin{\mathbb{K}}}$-scheme.

Let ${\boldsymbol{X}}$ be a derived ${\mathbin{\mathbb{K}}}$-scheme, and let $X=t_0({\boldsymbol{X}})$ be the corresponding classical truncation, a classical ${\mathbin{\mathbb{K}}}$-scheme. Then as in §3.1 there is a natural inclusion morphism $i_{\boldsymbol{X}}:X\rightarrow {\boldsymbol{X}}$, which embeds $X$ as a derived ${\mathbin{\mathbb{K}}}$-subscheme in ${\boldsymbol{X}}$. A good analogy in classical algebraic geometry is this: Let $Y$ be a nonreduced scheme, and let $Y^{\mathrm{red}}$ be the corresponding reduced scheme. Then there is a natural inclusion $Y^{\mathrm{red}}\hookrightarrow Y$ of $Y^{\mathrm{red}}$ as a subscheme of $Y$, and we can think of $Y$ as an infinitesimal thickening of $Y^{\mathrm{red}}$. In a similar way, we can regard a derived ${\mathbin{\mathbb{K}}}$-scheme ${\boldsymbol{X}}$ as an infinitesimal thickening of its classical ${\mathbin{\mathbb{K}}}$-scheme $X=t_0({\boldsymbol{X}})$.

We shall assume throughout this paper that any derived ${\mathbin{\mathbb{K}}}$-scheme ${\boldsymbol{X}}$ is locally finitely presented, which means that it can be covered by Zariski open affine ${\boldsymbol{Y}}\simeq \operatorname {\mathbf{Spec}}A$, where $A$ is a cdga over ${\mathbin{\mathbb{K}}}$ of finite presentation.

Points $x$ of a derived ${\mathbin{\mathbb{K}}}$-scheme ${\boldsymbol{X}}$ are the same as points $x$ of $X=t_0({\boldsymbol{X}})$. If $A$ is a standard form cdga, as in Example 2.8, and $A'=A\otimes _{A^0}A^0[f^{-1}]$ is a localisation of $A$, as in Definition 2.9, then $\operatorname {\mathbf{Spec}}A'$ is the Zariski open subset of $\operatorname {\mathbf{Spec}}A$ where $f\ne 0$. If $A'$ is a localisation of $A$ at $p\in \operatorname {Spec}(H^0(A))=t_0(\operatorname {\mathbf{Spec}}A)$, then $\operatorname {\mathbf{Spec}}A'$ is a Zariski open neighbourhood of $p$ in $\operatorname {\mathbf{Spec}}A$.

A morphism $\boldsymbol{f}:{\boldsymbol{X}}\rightarrow {\boldsymbol{Y}}$ of derived ${\mathbin{\mathbb{K}}}$-schemes is called étale if it is Zariski locally modelled on $\operatorname {\mathbf{Spec}}\phi :\operatorname {\mathbf{Spec}}A\rightarrow \operatorname {\mathbf{Spec}}B$, for $\phi :B\rightarrow A$ a morphism of cdgas over ${\mathbin{\mathbb{K}}}$ such that $H^0(\phi ):H^0(B)\rightarrow H^0(A)$ is étale in the classical sense and $H^i(\phi ):H^i(B)\rightarrow H^i(A)$ induces an isomorphism $H^i(B)\rightarrow H^i(A)\otimes _{H^0(A)}H^0(B)$ for all $i<0$. In particular, $t_0(\boldsymbol{f}):t_0({\boldsymbol{X}})\rightarrow t_0({\boldsymbol{Y}})$ is an étale morphism of classical schemes.

Since the $\infty$-category of affine derived ${\mathbin{\mathbb{K}}}$-schemes is a localisation of the $\infty$-category of cdgas over ${\mathbin{\mathbb{K}}}$ at quasi-isomorphisms, morphisms of affine derived ${\mathbin{\mathbb{K}}}$-schemes need not lift to morphisms of the corresponding cdgas. We can ask:

Question 3.1.

Suppose $A,B$ are standard form cdgas over ${\mathbin{\mathbb{K}}},$ as in Example 2.8, and $\boldsymbol{f}:\operatorname {\mathbf{Spec}}A\rightarrow \operatorname {\mathbf{Spec}}B$ is a morphism in the $\infty$-category $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$. Then:

(a)

does there exist a morphism of cdgas $\alpha :B\rightarrow A$ (that is, a strict morphism of cdgas, not a morphism in the $\infty$-category) with $\boldsymbol{f}\simeq \operatorname {\mathbf{Spec}}\alpha$?

(b)

for each $p\in \operatorname {Spec}H^0(A),$ does there exist a localisation $A'$ of $A$ at $p$ and a (strict) morphism of cdgas $\alpha :B\rightarrow A'$ with $\boldsymbol{f}\vert _{\operatorname {\mathbf{Spec}}A'}\simeq \operatorname {\mathbf{Spec}}\alpha$?

One can show that:

(i)

For general $A,B$, the answers to Question 3.1(a),(b) may both be no.

(ii)

If $A$ is general, but for $B$, the smooth ${\mathbin{\mathbb{K}}}$-scheme $\operatorname {Spec}B^0$ is isomorphic to an affine space ${\mathbin{\mathbb{A}}}^n$, then the answers to Questions 3.1(a),(b) are both yes. Indeed, the condition that $\operatorname {Spec}B^0$ be isomorphic to an affine space ensures that $B$ is a cofibrant as a cdga, and therefore every map $B \rightarrow A$ in the $\infty$-category of of cdgas is represented by a strict map out of $B$.

(iii)

If $A$ is general, but for $B$ the smooth ${\mathbin{\mathbb{K}}}$-scheme $\operatorname {Spec}B^0$ is isomorphic to a Zariski open set in an affine space ${\mathbin{\mathbb{A}}}^n$, then the answer to Question 3.1(a) may be no, but to Question 3.1(b) is yes.

(iv)

For general $A,B$, the answer to Question 3.1(b) at $p$ is yes if and only if there exists a localisation $A^{\prime \, 0}=A^0[g^{-1}]$ of $A^0$ at $p$ and a morphism of ${\mathbin{\mathbb{K}}}$-algebras $\alpha ^0:B^0\rightarrow A^{\prime \, 0}$ such that the following commutes:$$\begin{equation*} \vcenter{\img[][259pt][42pt][{$\xymatrix@C=180pt@R=13pt{ *+[r]{\Spec\bigl(H^0(A)[g^{-1}]\bigr)} \ar[r]_(0.56){t_0(\bs f)\vert_{\Spec(H^0(A)[g^{-1}])}} \ar[d]^{\mathrm{inc}} &amp; *+[l]{\Spec H^0(B)} \ar[d]_{\mathrm{inc}} \\ *+[r]{\Spec A^{\prime\, 0}} \ar[r]^(0.56){\Spec\al^0} &amp; *+[l]{\Spec B^0.\!{}} }$}]{Images/img4e266cdb0ef23ee91f9356d14c22ead0.svg}} \end{equation*}$$

That is, if $\boldsymbol{f}$ lifts to a morphism $B^0\rightarrow A^0$, then (possibly after further localisation) it lifts to a morphism $B^k\rightarrow A^k$ for all $k$.

Remark 3.2.

Our notion of standard form cdgas is a compromise. We chose to start the induction in Example 2.8 with a smooth ${\mathbin{\mathbb{K}}}$-algebra $A^0$. This ensured that cotangent complexes of ${\mathbin{\mathbb{L}}}_A$ behave well, as in §2.3. We could instead have adopted one of the stronger conditions that $\operatorname {Spec}A^0$ is isomorphic to ${\mathbin{\mathbb{A}}}^n$ or to a Zariski open subset of ${\mathbin{\mathbb{A}}}^n$. This would give better lifting properties of morphisms of derived ${\mathbin{\mathbb{K}}}$-schemes, as in (ii) and (iii) above. However, requiring $\operatorname {Spec}A^0\subseteq {\mathbin{\mathbb{A}}}^n$ to be open does not fit well with the notion of $A$ minimal in Definition 2.13, and Theorem 4.1 would be false for these stronger notions of standard form cdga.

3.3. Cotangent complexes of derived schemes and stacks

We discuss cotangent complexes of derived schemes and stacks, following Toën and Vezzosi 28, §1.4, 27, §4.2.4–§4.2.5, 31, and Lurie 22, §3.4. We will restrict our attention to derived Artin ${\mathbin{\mathbb{K}}}$-stacks ${\boldsymbol{X}}$, rather than general derived ${\mathbin{\mathbb{K}}}$-stacks, which ensures in particular that cotangent complexes exist.

Let ${\boldsymbol{X}}$ be a derived Artin ${\mathbin{\mathbb{K}}}$-stack. In classical algebraic geometry, cotangent complexes ${\mathbin{\mathbb{L}}}_X$ lie in $D(\operatorname {qcoh}(X))$, but in derived algebraic geometry they lie in a dg category $L_{\operatorname {qcoh}}({\boldsymbol{X}})$ defined by Toën 27, §3.1.7, §4.2.4, which is a generalisation of $D(\operatorname {qcoh}(X))$. Here are some properties of these:

(i)

If ${\boldsymbol{X}}$ is a derived Artin ${\mathbin{\mathbb{K}}}$-stack, then $L_{\operatorname {qcoh}}({\boldsymbol{X}})$ is a pretriangulated dg category with a t-structure whose heart is the abelian category $\operatorname {qcoh}(t_{0}({\boldsymbol{X}}))$ of quasi-coherent sheaves on the classical truncation $t_{0}({\boldsymbol{X}})$ of ${\boldsymbol{X}}$. In particular, if ${\boldsymbol{X}}=i(X)$ is a classical Artin ${\mathbin{\mathbb{K}}}$-stack or ${\mathbin{\mathbb{K}}}$-scheme, then $L_{\operatorname {qcoh}}(X)\simeq D(\operatorname {qcoh}(X))$, but in general $L_{\operatorname {qcoh}}({\boldsymbol{X}})\not \simeq D(\operatorname {qcoh}(t_{0}({\boldsymbol{X}})))$.

(ii)

If $\boldsymbol{f}:{\boldsymbol{X}}\rightarrow {\boldsymbol{Y}}$ is a morphism of derived Artin ${\mathbin{\mathbb{K}}}$-stacks, it induces a pullback functor $\boldsymbol{f}^*:L_{\operatorname {qcoh}}({\boldsymbol{Y}})\rightarrow L_{\operatorname {qcoh}}({\boldsymbol{X}})$, analogous to the left derived pullback functor $f^*:D(\operatorname {qcoh}(Y))\rightarrow D(\operatorname {qcoh}(X))$ in the classical case.

(iii)

If $A$ is a cdga over ${\mathbin{\mathbb{K}}}$ and ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ is the corresponding derived affine ${\mathbin{\mathbb{K}}}$-scheme, then $L_{\operatorname {qcoh}}({\boldsymbol{X}})\simeq \text{dgmod-}A$ is the pre-triangulated dg category of dg-modules over $A$. In contrast, $D(\operatorname {qcoh}({\boldsymbol{X}}))\simeq D(\text{mod-}H^0(A))$, which depends only on the classical truncation $X=t_0({\boldsymbol{X}})$.

(iv)

There is a notion of when a complex ${\mathbin{\mathcal{E}}}^\bullet$ in $L_{\operatorname {qcoh}}({\boldsymbol{X}})$ is perfect. When ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ is affine, as in (iii), perfect complexes in $L_{\operatorname {qcoh}}({\boldsymbol{X}})$ correspond to perfect dg-modules in $\text{dgmod-}A$, that is, to modules which may be built from finitely many copies of $A[k]$ for $k\in {\mathbin{\mathbb{Z}}}$ by repeated extensions and splitting of idempotents.

If ${\boldsymbol{X}}$ is a derived Artin ${\mathbin{\mathbb{K}}}$-stack, then Toën and Vezzosi 27, §4.2.5, 28, §1.4 or Lurie 22, §3.2 define an (absolute) cotangent complex ${\mathbin{\mathbb{L}}}_{\boldsymbol{X}}$ in $L_{\operatorname {qcoh}}({\boldsymbol{X}})$. If $\boldsymbol{f}:{\boldsymbol{X}}\rightarrow {\boldsymbol{Y}}$ is a morphism of derived Artin stacks, they construct a morphism ${\mathbin{\mathbb{L}}}_{\boldsymbol{f}}:\boldsymbol{f}^*({\mathbin{\mathbb{L}}}_{\boldsymbol{Y}})\rightarrow {\mathbin{\mathbb{L}}}_{\boldsymbol{X}}$ in $L_{\operatorname {qcoh}}({\boldsymbol{X}})$, and the (relative) cotangent complex ${\mathbin{\mathbb{L}}}_{{\boldsymbol{X}}/{\boldsymbol{Y}}}$ is defined to be the cone on this, giving the distinguished triangle

$$\begin{equation*} \vcenter{\img[][228pt][16pt][{$\xymatrix@C=35pt{ \bs f^*(\bL_\bY) \ar[r]^(0.55){\bL_{\bs f}} &amp; \bL_\bX\ar[r] &amp; \bL_{\bX/\bY} \ar[r] &amp; \bs f^*(\bL_\bY)[1]. }$}]{Images/imgdb025cae1cae630d5717a55aa1cce754.svg}} \end{equation*}$$

Here are some properties of these:

(a)

If ${\boldsymbol{X}}$ is a derived ${\mathbin{\mathbb{K}}}$-scheme, then $h^k({\mathbin{\mathbb{L}}}_{\boldsymbol{X}})=0$ for $k>0$. Also, if $X=t_0({\boldsymbol{X}})$ is the corresponding classical ${\mathbin{\mathbb{K}}}$-scheme, then $h^0({\mathbin{\mathbb{L}}}_{\boldsymbol{X}})\cong h^0({\mathbin{\mathbb{L}}}_X)\cong \Omega _X$ under the equivalence $\operatorname {qcoh}({\boldsymbol{X}})\simeq \operatorname {qcoh}(X)$.

(b)

Let ${\boldsymbol{X}}\,{\buildrel \boldsymbol{f}\over \longrightarrow }\,{\boldsymbol{Y}}\, {\buildrel \boldsymbol{g}\over \longrightarrow }\,{\boldsymbol{Z}}$ be morphisms of derived Artin ${\mathbin{\mathbb{K}}}$-stacks. Then by Lurie 22, Prop. 3.2.12 there is a distinguished triangle in $L_{\operatorname {qcoh}}({\boldsymbol{X}})$,$$\begin{equation*} \vcenter{\img[][237pt][12pt][{$\xymatrix@C=30pt{ \bs f^*(\bL_{\bY/\bZ}) \ar[r] &amp; \bL_{\bX/\bZ} \ar[r] &amp; \bL_{\bX/\bY} \ar[r] &amp; \bs f^*(\bL_{\bY/\bZ})[1]. }$}]{Images/img529f4135b5b47fdd618176aabafc52e9.svg}} \end{equation*}$$

(c)

Suppose we have a Cartesian diagram of derived Artin ${\mathbin{\mathbb{K}}}$-stacks,$$\begin{equation*} \vcenter{\img[][114pt][34pt][{$\xymatrix@C=90pt@R=14pt{ *+[r]{\bW} \ar[r]_(0.3){\bs f} \ar[d]^{\bs e} &amp; *+[l]{\bY} \ar[d]_{\bs h} \\ *+[r]{\bX} \ar[r]^(0.7){\bs g} &amp; *+[l]{\bZ.\!\!{}} }$}]{Images/imgaf12f0b67b942b2fd6fd27c11036703d.svg}} \end{equation*}$$

Then by Toën and Vezzosi 28, Lems. 1.4.1.12 and 1.4.1.16 or Lurie 22, Prop. 3.2.10 we have base change isomorphisms$$\begin{equation} {\mathbin{\mathbb{L}}}_{{\boldsymbol{W}}/{\boldsymbol{Y}}}\cong \boldsymbol{e}^*({\mathbin{\mathbb{L}}}_{{\boldsymbol{X}}/{\boldsymbol{Z}}}),\qquad {\mathbin{\mathbb{L}}}_{{\boldsymbol{W}}/{\boldsymbol{X}}}\cong \boldsymbol{f}^*({\mathbin{\mathbb{L}}}_{{\boldsymbol{Y}}/{\boldsymbol{Z}}}). {\tag{3.1}} \end{equation}$$

Note that the analogous result for classical ${\mathbin{\mathbb{K}}}$-schemes requires $g$ or $h$ to be flat, but 3.1 holds with no flatness assumption.

(d)

Suppose ${\boldsymbol{X}}$ is a locally finitely presented derived Artin ${\mathbin{\mathbb{K}}}$-stack 28, §2.2.3 (also called locally of finite presentation in 27, §3.1.1, and fp-smooth in 29, §4.4). Then ${\mathbin{\mathbb{L}}}_{\boldsymbol{X}}$ is a perfect complex in $L_{\operatorname {qcoh}}({\boldsymbol{X}})$.

If $\boldsymbol{f}:{\boldsymbol{X}}\rightarrow {\boldsymbol{Y}}$ is a morphism of locally finitely presented derived Artin ${\mathbin{\mathbb{K}}}$-stacks, then ${\mathbin{\mathbb{L}}}_{{\boldsymbol{X}}/{\boldsymbol{Y}}}$ is perfect.

(e)

Suppose ${\boldsymbol{X}}$ is a derived ${\mathbin{\mathbb{K}}}$-scheme, $X=t_0({\boldsymbol{X}})$ is its classical truncation, and $i_{\boldsymbol{X}}:X\hookrightarrow {\boldsymbol{X}}$ is the natural inclusion. Then Schürg, Toën, and Vezzosi 25, Prop. 1.2 show that $h^k({\mathbin{\mathbb{L}}}_{X/{\boldsymbol{X}}})=0$ for $k\geqslant -1$, and they deduce that ${\mathbin{\mathbb{L}}}_{i_{\boldsymbol{X}}}:i_{\boldsymbol{X}}^*({\mathbin{\mathbb{L}}}_{\boldsymbol{X}})\rightarrow {\mathbin{\mathbb{L}}}_X$ in $D(\operatorname {qcoh}(X))$ is an obstruction theory on $X$ in the sense of Behrend and Fantechi 2.

Now suppose $A$ is a cdga over ${\mathbin{\mathbb{K}}}$ and ${\boldsymbol{X}}$ is a derived ${\mathbin{\mathbb{K}}}$-scheme with ${\boldsymbol{X}}\simeq \operatorname {\mathbf{Spec}}A$ in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$. Then we have an equivalence of triangulated categories $L_{\operatorname {qcoh}}({\boldsymbol{X}})\simeq D(\operatorname {mod}A)$, where $D(\operatorname {mod}A)$ is the derived category of dg-modules over $A$. This equivalence identifies cotangent complexes ${\mathbin{\mathbb{L}}}_{\boldsymbol{X}}\simeq {\mathbin{\mathbb{L}}}_A$. If $A$ is of standard form, then as in §2.3 the Kähler differentials $\Omega ^1_A$ are a model for ${\mathbin{\mathbb{L}}}_A$ in $D(\operatorname {mod}A)$, and Proposition 2.12 gives a simple explicit description of $\Omega ^1_A$. Thus, if ${\boldsymbol{X}}$ is a derived ${\mathbin{\mathbb{K}}}$-scheme with ${\boldsymbol{X}}\simeq \operatorname {\mathbf{Spec}}A$ for $A$ a standard form cdga, we can understand ${\mathbin{\mathbb{L}}}_{\boldsymbol{X}}$ well. We will use this to do computations with $k$-shifted $p$-forms and $k$-shifted closed $p$-forms on ${\boldsymbol{X}}$, as in §3.4.

3.4. $k$-shifted symplectic structures on derived stacks

We now outline the main ideas of our principal reference Pantev, Toën, Vaquié, and Vezzosi 24. Let $X$ be a classical smooth ${\mathbin{\mathbb{K}}}$-scheme. Its tangent and cotangent bundles $TX,T^*X$ are vector bundles on $X$. A $p$-form on $X$ is a global section $\omega$ of $\Lambda ^pT^*X$. There is a de Rham differential ${\mathrm{d}_{dR}}:\Lambda ^pT^*X\rightarrow \Lambda ^{p+1}T^*X$, which is a morphism of sheaves of ${\mathbin{\mathbb{K}}}$-vector spaces but not of sheaves of ${\mathbin{\mathcal{O}}}_X$-modules, and induces ${\mathrm{d}_{dR}}:H^0(\Lambda ^pT^*X)\rightarrow H^0(\Lambda ^{p+1}T^*X)$. A closed $p$-form is $\omega \in H^0(\Lambda ^pT^*X)$ with ${\mathrm{d}_{dR}}\omega =0$. A symplectic form on $X$ is a closed 2-form $\omega$ such that the induced morphism $\omega :TX\rightarrow T^*X$ is an isomorphism.

The derived loop stack ${\mathbin{\mathcal{L}}}X$ of $X$ is the fibre product $X\times _{\Delta _X,X\times X,\Delta _X}X$ in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$, where $\Delta _X:X\rightarrow X\times X$ is the diagonal map. When $X$ is smooth, ${\mathbin{\mathcal{L}}}X$ is a quasi-smooth derived ${\mathbin{\mathbb{K}}}$-scheme. It is interpreted in derived algebraic geometry as ${\mathbin{\mathcal{L}}}X={\mathbin{\mathbb{R}}}{\mathbf{Map}}({\mathcal{S}}^1,X)$, the moduli stack of loops in $X$. Here the circle ${\mathcal{S}}^1$ may be thought of as the simplicial complex $\vcenter{\img[][28pt][16pt][{$\xymatrix@C=15pt{\bu\ar@&lt;.3ex&gt;@/^4pt/[r] \ar@&lt;-.3ex&gt;@/_4pt/[r] &amp; \bu}$}]{Images/img73789a2ae5ef4f1ef0e83a51357f96ec.svg}}$, so a map from ${\mathcal{S}}^1$ to $X$ means two points $x,y$ in $X$, corresponding to the two vertices “•” in ${\mathcal{S}}^1$, plus two relations $x=y$, $x=y$ corresponding to the two edges “$\rightarrow$” in ${\mathcal{S}}^1$. This agrees with ${\mathbin{\mathcal{L}}}X$, as points of $X\times _{\Delta _X,X\times X,\Delta _X}X$ correspond to points $x,y$ in $X$ satisfying the relation $(x,x)=(y,y)$ in $X\times X$. Note that in derived algebraic geometry, imposing a relation twice is not the same as imposing it once, as the derived structure sheaf records the relation in its simplicial structure.

Consider the projection $p: {\mathbin{\mathcal{L}}}X=X\times _{X\times X}X \rightarrow X$, where $X$ is a classical smooth ${\mathbin{\mathbb{K}}}$-scheme. By the Hochschild–Kostant–Rosenberg theorem, there is a decomposition $p_{*}({\mathbin{\mathcal{O}}}_{{\mathbin{\mathcal{L}}}X})\cong \bigoplus _{p} \Lambda ^pT^*X[p]$ in $D(\operatorname {qcoh}(X))$. Thus, $p$-forms on $X$ may be interpreted as “functions on the loop space ${\mathbin{\mathcal{L}}}X$ of weight $p\,$”, while closed $p$-forms can be interpreted as “${\mathcal{S}}^1$-invariant functions on the loop space ${\mathbin{\mathcal{L}}}X$ of weight $p$” by identifying the ${\mathcal{S}}^{1}$-action with the de Rham differential. See Toën and Vezzosi 30 and Ben-Zvi and Nadler 4.

Now let ${\boldsymbol{X}}$ be a locally finitely presented derived Artin ${\mathbin{\mathbb{K}}}$-stack. The aim of Pantev et al. 24 is to define good notions of $p$-forms, closed $p$-forms, and symplectic structures on ${\boldsymbol{X}}$, and show that these occur naturally on certain derived moduli stacks. The rough idea is as above: we form the derived loop stack ${\mathbin{\mathcal{L}}}{\boldsymbol{X}}:={\boldsymbol{X}}\times _{\Delta _{\boldsymbol{X}},{\boldsymbol{X}}\times {\boldsymbol{X}},\Delta _{\boldsymbol{X}}}{\boldsymbol{X}}$ in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$ and then (closed) $p$-forms on $X$ are called “(${\mathcal{S}}^1$-invariant) functions on ${\mathbin{\mathcal{L}}}{\boldsymbol{X}}$ of weight $p$”.

However, the problem is more complicated than the classical smooth case in two respects. First, as the $p$-forms $\Lambda ^p{\mathbin{\mathbb{L}}}_{\boldsymbol{X}}$ are a complex, rather than a vector bundle, one should consider cohomology classes $H^k({\boldsymbol{X}},\Lambda ^p{\mathbin{\mathbb{L}}}_{\boldsymbol{X}})$ for $k\in {\mathbin{\mathbb{Z}}}$ rather than just global sections $\omega \in H^0({\boldsymbol{X}},\Lambda ^pT^*{\boldsymbol{X}})$. This leads to the idea of a $p$-form of degree $k$ on ${\boldsymbol{X}}$ for $p\geqslant 0$ and $k\in {\mathbin{\mathbb{Z}}}$, which is roughly an element of $H^k({\boldsymbol{X}},\Lambda ^p{\mathbin{\mathbb{L}}}_{\boldsymbol{X}})$. Essentially, ${\mathbin{\mathcal{O}}}_{{\mathbin{\mathcal{L}}}{\boldsymbol{X}}}$ has two gradings, the cohomological grading $k$ and the grading by weight $p$.

Secondly, as the ${\mathcal{S}}^1$-action on ${\mathbin{\mathcal{L}}}{\boldsymbol{X}}={\mathbin{\mathbb{R}}}{\mathbf{Map}}({\mathcal{S}}^1,{\boldsymbol{X}})$ by rotating the domain ${\mathcal{S}}^1$ is only up to homotopy, to be ${\mathcal{S}}^1$-invariant is not a property of a function on ${\mathbin{\mathcal{L}}}{\boldsymbol{X}}$, but an extra structure. As an analogy, if an algebraic ${\mathbin{\mathbb{K}}}$-group $G$ acts on a ${\mathbin{\mathbb{K}}}$-scheme $V$, then for a vector bundle $E\rightarrow V$ to be $G$-invariant is not really a property of $E$: the right question to ask is whether the $G$-action on $V$ lifts to a $G$-action on $E$, and there can be many such lifts, each a different way for $E$ to be $G$-invariant. Similarly, the definition of “closed $p$-form on ${\boldsymbol{X}}$” in 24 is not just a $p$-form satisfying an extra condition, but a $p$-form with extra data, a closing structure, satisfying some conditions.

In fact, for technical reasons, for a general locally finitely presented derived Artin ${\mathbin{\mathbb{K}}}$-stack ${\boldsymbol{X}}$, Pantev et al. 24 do not define $p$-forms as elements of $H^{-p}({\mathbin{\mathcal{O}}}_{{\mathbin{\mathcal{L}}}{\boldsymbol{X}}})$, and so on, as we have sketched above. Instead, they first define explicit notions of (closed) $p$-forms on affine derived ${\mathbin{\mathbb{K}}}$-schemes ${\boldsymbol{Y}}=\operatorname {\mathbf{Spec}}A$ which are spectra of cdgas $A$, and show these satisfy étale descent. Then for general ${\boldsymbol{X}}$, $k$-shifted (closed) $p$-forms are defined as a mapping stack; basically, a $k$-shifted (closed) $p$-form $\omega$ on ${\boldsymbol{X}}$ is the functorial choice for all ${\boldsymbol{Y}},\boldsymbol{f}$ of a $k$-shifted (closed) $p$-form $\boldsymbol{f}^*(\omega )$ on ${\boldsymbol{Y}}$ whenever ${\boldsymbol{Y}}=\operatorname {\mathbf{Spec}}A$ is affine and $\boldsymbol{f}:{\boldsymbol{Y}}\rightarrow {\boldsymbol{X}}$ is a morphism.

The families (simplicial sets) of $p$-forms and of closed $p$-forms of degree $k$ on ${\boldsymbol{X}}$ are written ${\mathbin{\mathcal{A}}}^p_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)$ and ${\mathbin{\mathcal{A}}}^{p,\operatorname {cl}}_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)$, respectively. There is a morphism $\pi :{\mathbin{\mathcal{A}}}^{p,\operatorname {cl}}_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)\rightarrow {\mathbin{\mathcal{A}}}^p_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)$, which is in general neither injective nor surjective.

A 2-form $\omega ^0$ of degree $k$ on ${\boldsymbol{X}}$ induces a morphism $\omega ^0:{\mathbin{\mathbb{T}}}_{\boldsymbol{X}}\rightarrow {\mathbin{\mathbb{L}}}_{\boldsymbol{X}}[k]$ in $L_{\operatorname {qcoh}}({\boldsymbol{X}})$. We call $\omega ^0$ nondegenerate if $\omega ^0:{\mathbin{\mathbb{T}}}_{\boldsymbol{X}}\rightarrow {\mathbin{\mathbb{L}}}_{\boldsymbol{X}}[k]$ is an equivalence. As in 24, Def. 0.2, a closed 2-form $\omega$ of degree $k$ on ${\boldsymbol{X}}$ for $k\in {\mathbin{\mathbb{Z}}}$ is called a $k$-shifted symplectic structure if the corresponding 2-form $\omega ^0=\pi (\omega )$ is nondegenerate.

A 0-shifted symplectic structure on a classical ${\mathbin{\mathbb{K}}}$-scheme is a equivalent to a classical symplectic structure. Pantev et al. 24 construct $k$-shifted symplectic structures on several classes of derived moduli stacks. In particular, if $Y$ is a Calabi–Yau $m$-fold and $\boldsymbol{{\mathbin{\mathcal{M}}}}$ a derived moduli stack of coherent sheaves or perfect complexes on $Y$, then $\boldsymbol{{\mathbin{\mathcal{M}}}}$ has a $(2-m)$-shifted symplectic structure.

4. Local models for derived schemes

The next theorem, based on Lurie 23, Th. 8.4.3.18 and proved in §4.1, says every derived ${\mathbin{\mathbb{K}}}$-scheme ${\boldsymbol{X}}$ is Zariski locally modelled on $\operatorname {\mathbf{Spec}}A$ for $A$ a minimal standard form cdga. Recall that all derived ${\mathbin{\mathbb{K}}}$-schemes in this paper are assumed locally finitely presented. A morphism $\boldsymbol{f}:{\boldsymbol{X}}\rightarrow {\boldsymbol{Y}}$ in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$ is called a Zariski open inclusion if $\boldsymbol{f}$ is an equivalence from ${\boldsymbol{X}}$ to a Zariski open derived ${\mathbin{\mathbb{K}}}$-subscheme ${\boldsymbol{Y}}'\subseteq {\boldsymbol{Y}}$. A morphism of cdgas $\alpha :B\rightarrow A$ is a Zariski open inclusion if $\operatorname {\mathbf{Spec}}\alpha :\operatorname {\mathbf{Spec}}A\rightarrow \operatorname {\mathbf{Spec}}B$ is a Zariski open inclusion.

Theorem 4.1.

Let ${\boldsymbol{X}}$ be a locally finitely presented derived ${\mathbin{\mathbb{K}}}$-scheme, and let $x\in {\boldsymbol{X}}$. Then there exist a standard form cdga $A$ over ${\mathbin{\mathbb{K}}}$ which is minimal at a point $p\in \operatorname {Spec}H^0(A),$ in the sense of Example 2.8 and Definitions 2.9 and 2.13, and a morphism $\boldsymbol{f}:\operatorname {\mathbf{Spec}}A\rightarrow {\boldsymbol{X}}$ in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$ which is a Zariski open inclusion with $\boldsymbol{f}(p)=x$.

We think of $A,\boldsymbol{f}$ in Theorem 4.1 as being like a coordinate system on ${\boldsymbol{X}}$ near $x$. As well as being able to choose coordinates near any point, we want to be able to compare different coordinate systems on their overlaps. That is, given local equivalences $\boldsymbol{f}:\operatorname {\mathbf{Spec}}A\rightarrow {\boldsymbol{X}}$, $\boldsymbol{g}:\operatorname {\mathbf{Spec}}B\rightarrow {\boldsymbol{X}}$, we would like to compare the cdgas $A,B$ on the overlap of their images in ${\boldsymbol{X}}$.

This is related to the discussion after Question 3.1 in §3.2: for general $A,B$ we cannot (even locally) find a cdga morphism $\alpha :B\rightarrow A$ with $\boldsymbol{f}\simeq \boldsymbol{g}\circ \operatorname {\mathbf{Spec}}\alpha$. However, the next theorem, proved in §4.2, shows we can find a third cdga $C$ and open inclusions $\alpha :A\rightarrow C$, $\beta :B\rightarrow C$ with $\boldsymbol{f}\circ \operatorname {\mathbf{Spec}}\alpha \simeq \boldsymbol{g}\circ \operatorname {\mathbf{Spec}}\beta$.

Theorem 4.2.

Let ${\boldsymbol{X}}$ be a locally finitely presented derived ${\mathbin{\mathbb{K}}}$-scheme, let $A,B$ be standard form cdgas over ${\mathbin{\mathbb{K}}},$ and let $\boldsymbol{f}:\operatorname {\mathbf{Spec}}A\rightarrow {\boldsymbol{X}},$ $\boldsymbol{g}:\operatorname {\mathbf{Spec}}B\rightarrow {\boldsymbol{X}}$ be Zariski open inclusions in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$. Suppose $p\in \operatorname {Spec}H^0(A)$ and $q\in \operatorname {Spec}H^0(B)$ with $\boldsymbol{f}(p)=\boldsymbol{g}(q)$ in ${\boldsymbol{X}}$. Then there exist a standard form cdga $C$ over ${\mathbin{\mathbb{K}}}$ which is minimal at $r$ in $\operatorname {Spec}H^0(C)$ and morphisms of cdgas $\alpha :A\rightarrow C,$ $\beta :B\rightarrow C$ which are Zariski open inclusions, such that $\operatorname {\mathbf{Spec}}\alpha :r\mapsto p,$ $\operatorname {\mathbf{Spec}}\beta :r\mapsto q,$ and $\boldsymbol{f}\circ \operatorname {\mathbf{Spec}}\alpha \simeq \boldsymbol{g}\circ \operatorname {\mathbf{Spec}}\beta$ as morphisms $\operatorname {\mathbf{Spec}}C\rightarrow {\boldsymbol{X}}$ in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$.

If instead $\boldsymbol{f},\boldsymbol{g}$ are étale rather than Zariski open inclusions, the same holds with $\alpha ,\beta$ étale rather than Zariski open inclusions.

4.1. Proof of Theorem 4.1

The proof is a variation on Lurie 23, Th. 8.4.3.18. We give an outline, referring the reader to 23 for more details.

As ${\boldsymbol{X}}$ is a locally finitely presented derived ${\mathbin{\mathbb{K}}}$-scheme, it is covered by affine finitely presented derived ${\mathbin{\mathbb{K}}}$-subschemes. So we can choose an open neighbourhood ${\boldsymbol{Y}}$ of $x$ in ${\boldsymbol{X}}$, a cdga $B$ of finite presentation over ${\mathbin{\mathbb{K}}}$, and an equivalence $\boldsymbol{g}:\operatorname {\mathbf{Spec}}B\rightarrow {\boldsymbol{Y}}\subseteq {\boldsymbol{X}}$. There is then a unique $q\in \operatorname {Spec}H^0(B)$ with $\boldsymbol{g}(q)=x$.

Since $B$ is of finite presentation, the cotangent complex ${\mathbin{\mathbb{L}}}_B$ has finite Tor-amplitude, say in the interval $[-n,0]$. We will show that a localisation $B'$ of $B$ at $q$ is equivalent to a standard form cdga $A=A(n)$ over ${\mathbin{\mathbb{K}}}$, constructed inductively in a sequence $A(0),A(1),\ldots ,A(n)$ as in Example 2.8, with $A$ minimal at the point $p\in \operatorname {Spec}H^0(A)$ corresponding to $q\in \operatorname {Spec}H^0(B')\subseteq \operatorname {Spec}H^0(B)$. For the inductive construction, we do not fix a particular model for the cdga $B$, but understand it as an object in the $\infty$-category of cdgas. Similarly, when we assert the existence of a map $A(k) \rightarrow B$ from some level of the inductive construction to $B$, this map should be understood as a map in the $\infty$-category of cdgas.

Let $m_0=\dim \Omega ^1_{H^0(B)}\vert _q$, the embedding dimension of $\operatorname {Spec}H^0(B)$ at $q$. Localising $H^0(B)$ at $q$ if necessary, we can find a smooth algebra $A(0)$ of dimension $m_0$, an ideal $I\subset A(0)$, and an isomorphism of algebras $A(0)/I\rightarrow H^0(B)$ such that the induced surjection of modules $\Omega ^1_{A(0)}\otimes _{A(0)} H^0(B)\rightarrow \Omega ^1_{H^0(B)}$ is an isomorphism at $q$. Geometrically, we have chosen an embedding of $\operatorname {Spec}H^0(B)$ into a smooth scheme $\operatorname {Spec}A(0)$ that is minimal at $q$. Let $p\in \operatorname {Spec}A(0)$ be the image of $q\in \operatorname {Spec}H^0(B)$.

Since $B$ is the homotopy limit of its Postnikov tower $\cdots \rightarrow \tau _{\geq -1}B\rightarrow \tau _{\geq 0}B\simeq H^{0}(B)$ in which each map is a square-zero extension of cdgas 23, Prop. 7.1.3.19, and since $A(0)$ is smooth and hence maps out of it can be lifted along square-zero extensions, we can lift the surjection $A(0)\rightarrow H^0(B)$ along the canonical map $B\rightarrow H^0(B)$ to obtain a map $\alpha ^0:A(0)\rightarrow B$ which is a surjection on $H^0$. Consider the fibre sequence $\operatorname {fib}(\alpha ^0)\rightarrow A(0)\rightarrow B$. One can show that there is an isomorphism $H^0(B)\otimes _{A(0)}H^0(\operatorname {fib}(\alpha ^0))\cong H^{-1}({\mathbin{\mathbb{L}}}_{B/A(0)})$. Furthermore, we see that there is a surjection $H^0(\operatorname {fib}(\alpha ^0))\twoheadrightarrow I$ and hence a surjection $H^{-1}({\mathbin{\mathbb{L}}}_{B/A(0)})=H^0(B)\otimes _{A(0)} H^0(\operatorname {fib}(\alpha ^0)) \twoheadrightarrow I/I^2$.

Localising if necessary and using Nakayama’s lemma, we may choose a free finite rank $A(0)$-module $M^{-1}$ together with a surjection $M^{-1}\rightarrow H^{-1}({\mathbin{\mathbb{L}}}_{B/A(0)})$ that is an isomorphism at $p$. We therefore obtain a surjection $M^{-1} \rightarrow I/I^2$ which can be lifted through the map $I\rightarrow I/I^2$. Localising again if necessary and using Nakayama’s lemma, we may assume the lift $M^{-1}\rightarrow I$ is surjective.

Using this choice of $M^{-1}$ together with the map $M^{-1}\rightarrow I \subset A(0)$, we define a cdga $A(1)$ as in Example 2.8. Note that by construction, the induced map $\alpha ^{-1}:A(1)\rightarrow B$ induces an isomorphism $H^0(A(1))\rightarrow H^0(B)$.

Now consider the fibre sequence $B\otimes _{A(1)}{\mathbin{\mathbb{L}}}_{A(1)/A(0)}\rightarrow {\mathbin{\mathbb{L}}}_{B/A(0)}\rightarrow {\mathbin{\mathbb{L}}}_{B/A(1)}$. By construction, ${\mathbin{\mathbb{L}}}_{A(1)/A(0)} \simeq A(1)\otimes _{A(0)}M^{-1}[1]$, and the map $H^0(B) \otimes _{A(0)}M^{-1}=H^{-1}({\mathbin{\mathbb{L}}}_{A(1)/A(0)})\rightarrow H^{-1}({\mathbin{\mathbb{L}}}_{B/A(0)})$ is surjective, so $H^i({\mathbin{\mathbb{L}}}_{B/A(1)})$ vanishes for $i=0,-1$. One can then deduce that there are isomorphisms

$$\begin{equation*} H^{-1}(\operatorname {fib}(\alpha ^{-1})) \cong H^{-2}(\operatorname {cof}(\alpha ^{-1}))\cong H^{-2}({\mathbin{\mathbb{L}}}_{B/A(1)}). \end{equation*}$$

Localising if necessary, we can choose a free $A(1)$-module $M^{-2}$ of finite rank with a surjection $H^0(B) \otimes _{H^0(A(1))}H^0(M^{-2}) \rightarrow H^{-1}(\operatorname {fib}(\alpha ^{-1})) \cong H^{-2}({\mathbin{\mathbb{L}}}_{B/A(1)})$ that is an isomorphism at $p$. Choosing a lift $M^{-2}[1] \rightarrow \operatorname {fib}(\alpha ^{-1}) \rightarrow A(1)$, we construct $A(2)$ from $A(1)$ as in Example 2.8.

Continuing in this manner, we construct a sequence of cdgas

$$\begin{equation*} A(0) \longrightarrow A(1) \longrightarrow \cdots \longrightarrow A(k-1) \longrightarrow A(k) \longrightarrow \cdots \longrightarrow A(n-1) \end{equation*}$$

so that ${\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}\simeq A(k) \otimes _{A(k-1)} M^{-k}[k]$ is a free $A(k)$-module and the natural map $H^{-k}({\mathbin{\mathbb{L}}}_{A(k)/A(k-1)}) \rightarrow H^{-k}({\mathbin{\mathbb{L}}}_{B/A(k-1)})$ is surjective and is an isomorphism at $p$.

By induction on $k$, we see that ${\mathbin{\mathbb{L}}}_{A(k)}$ has Tor-amplitude in $[-k,0]$. Considering the fibre sequence $B\otimes _{A(n-1)}{\mathbin{\mathbb{L}}}_{A(n-1)} \rightarrow {\mathbin{\mathbb{L}}}_B\rightarrow {\mathbin{\mathbb{L}}}_{B/A(n-1)}$ and bearing in mind that ${\mathbin{\mathbb{L}}}_B$ has Tor-amplitude in $[-n,0]$ by assumption, we see that ${\mathbin{\mathbb{L}}}_{B/A(n-1)}$ has Tor-amplitude in $[-n,0]$. But by 2.5, $H^i({\mathbin{\mathbb{L}}}_{B/A(n-1)})=0$ for $i>-n$, so we see that ${\mathbin{\mathbb{L}}}_{B/A(n-1)}[-n]$ is a projective $B$-module of finite rank. Localising if necessary, we may assume it is free.

Finally, choose a free $A(n-1)$-module $M^{-n}$ with ${\mathbin{\mathbb{L}}}_{B/A(n-1)} \simeq B\otimes _{A(n-1)}M^{-n}[n]$. Then $H^0(M^{-n})\cong H^{-n}({\mathbin{\mathbb{L}}}_{B/A(n-1)}) \cong H^{1-n}(\operatorname {fib}(\alpha ^{1-n}))$, so using $M^{-n}[n-1]\rightarrow A(n-1)$ to build $A(n)$, we have a fibre sequence ${\mathbin{\mathbb{L}}}_{A(n)/A(n-1)}\rightarrow {\mathbin{\mathbb{L}}}_{B/A(n-1)}\rightarrow {\mathbin{\mathbb{L}}}_{B/A(n)}$ in which the first arrow is an equivalence. Thus ${\mathbin{\mathbb{L}}}_{B/A(n)}\simeq 0$, and since the map $\alpha ^{-n}:A(n)\rightarrow B$ induces an isomorphism on $H^0$, it must be an equivalence. Set $A=A(n)$, so that $A$ is a standard form cdga over ${\mathbin{\mathbb{K}}}$. As $\operatorname {\mathbf{Spec}}\alpha ^{-n}:\operatorname {\mathbf{Spec}}B\rightarrow \operatorname {\mathbf{Spec}}A$ is an equivalence in $\operatorname {\mathbf{dSch}}_{\mathbin{\mathbb{K}}}$ with $\operatorname {\mathbf{Spec}}\alpha ^{-n}:q\mapsto p$, there exists a quasi-inverse $\boldsymbol{h}:\operatorname {\mathbf{Spec}}A\rightarrow \operatorname {\mathbf{Spec}}B$ for $\operatorname {\mathbf{Spec}}\alpha ^{-n}$, with $\boldsymbol{h}(p)=q$. Then $\boldsymbol{f}=\boldsymbol{g}\circ \boldsymbol{h}:\operatorname {\mathbf{Spec}}A\rightarrow {\boldsymbol{X}}$ is a Zariski open inclusion with $\boldsymbol{f}(p)=\boldsymbol{g}\circ \boldsymbol{h}(p)=\boldsymbol{g}(q)=x$, as we have to prove.

It remains to show that $A$ is minimal at $p$. For this we must check that for each $k$, the composition 2.8 is zero at $p$. Using the commutative diagram 2.6, and 2.6 with $k-1$ in place of $k$, we have a commutative diagram

$$\begin{equation*} \vcenter{\img[][239pt][73pt][{$\xymatrix@R=12pt@C=130pt{*+[r]{H^{-k}(\bL_{A(k)/A(k-1)})} \ar@{=}[r] \ar[d] &amp; *+[l]{H^{-k}(\bL_{A(k)/A(k-1)})} \ar[d] \\ *+[r]{H^{-k}(\bL_{B/A(k-1)})} \ar[r] \ar[d] &amp; *+[l]{H^{1-k}(\bL_{A(k-1)})} \ar[d] \\ *+[r]{H^{1-k}(\bL_{A(k-1)/A(k-2)})} \ar@{=}[r] &amp; *+[l]{H^{1-k}(\bL_{A(k-1)/A(k-2)}),\!\!{}}}$}]{Images/img8e741cb8ac2b186cfc50af7aea7bcad5.svg}} \end{equation*}$$

where the right-hand column is 2.8. It is therefore enough to see that composition in the left-hand column is zero at $p$.

But by construction, the first map in 2.8 is an isomorphism at $p$, so we need the second map to be zero at $p$. But again by construction, we have an exact sequence $H^{-k}({\mathbin{\mathbb{L}}}_{B/A(k-1)})\rightarrow H^{1-k}({\mathbin{\mathbb{L}}}_{A(k-1)/A(k-2)}) \rightarrow H^{1-k}({\mathbin{\mathbb{L}}}_{B/A(k-2)})$ in which the second map is an isomorphism at $p$, and hence $H^{-k}({\mathbin{\mathbb{L}}}_{B/A(k-1)}) \rightarrow H^{1-k}({\mathbin{\mathbb{L}}}_{A(k-1)/A(k-2)})$ is indeed zero at $p$. This completes the proof.

4.2. Proof of Theorem 4.2

Given a derived scheme ${\boldsymbol{X}}$ with a point $x\in {\boldsymbol{X}}$ and Zariski open neighbourhoods $\boldsymbol{f}:{\boldsymbol{U}}=\operatorname {\mathbf{Spec}}A\rightarrow {\boldsymbol{X}}$ and $\boldsymbol{g}:{\boldsymbol{V}}=\operatorname {\mathbf{Spec}}B\rightarrow {\boldsymbol{X}}$ of $x$, we may assume that $A$ and $B$ are given as standard form cdgas. We want to show that we can find a Zariski open neighbourhood $\boldsymbol{h}:{\boldsymbol{W}}=\operatorname {\mathbf{Spec}}C\rightarrow {\boldsymbol{X}}$ of $x$ contained in ${\boldsymbol{U}}$ and ${\boldsymbol{V}}$, where $C$ is a standard form cdga minimal at $r\in \operatorname {Spec}H^0(C)$ with $\boldsymbol{h}(r)=x$ and such that in the homotopy commutative diagram

$$\begin{equation} \begin{gathered} \vcenter{\img[][300pt][69pt][{$\xymatrix@!0@C=65pt@R=30pt{ &amp;&amp; \bW=\bSpec C \ar[dd]_{\bs h} \ar[dl]^(0.4){\bs c} \ar[dr]_(0.4){\bs d} \\ &amp; \bU=\bSpec A \ar[dl] \ar[dr]_(0.4){\bs f} &amp;&amp; \bV=\bSpec B \ar[dl]^(0.4){\bs g} \ar[dr] \\ \Spec A(0) &amp;&amp; \bX&amp;&amp; \Spec B(0), }$}]{Images/imgb08a0d7af33b89953152c687ddcc2300.svg}} \end{gathered} {\tag{4.1}} \end{equation}$$

the maps $\boldsymbol{c}$ and $\boldsymbol{d}$ are induced by maps of cdgas $\alpha :A\rightarrow C$ and $\beta :B\rightarrow C$.

To begin with, choose a Zariski open immersion of an affine derived scheme ${\boldsymbol{W}}\rightarrow {\boldsymbol{U}}\times _{\boldsymbol{X}}{\boldsymbol{V}}\rightarrow {\boldsymbol{X}}$ whose image contains $x$. From the above commutative diagram, we have maps ${\boldsymbol{W}}\rightarrow \operatorname {\mathbf{Spec}}A\rightarrow \operatorname {Spec}A(0)$ and ${\boldsymbol{W}}\rightarrow \operatorname {\mathbf{Spec}}B\rightarrow \operatorname {Spec}B(0)$ in which the first arrows in each sequence are open immersions and the second arrows are closed immersions on the underlying classical schemes. The induced map ${\boldsymbol{W}}\rightarrow \operatorname {Spec}A(0)\times \operatorname {Spec}B(0)$ is therefore a locally closed immersion on the underlying classical scheme $W=t_0({\boldsymbol{W}})$. Localising if necessary, we may assume that in fact ${\boldsymbol{W}}\rightarrow \operatorname {Spec}A(0)\times \operatorname {Spec}B(0)$ factors via a closed immersion $W\hookrightarrow \operatorname {Spec}C(0)$ into a smooth, affine, locally closed ${\mathbin{\mathbb{K}}}$-subscheme $\operatorname {Spec}C(0)\subseteq \operatorname {Spec}A(0)\times \operatorname {Spec}B(0)$ of minimal dimension at $r:=(p,q)$.

Proceeding as in the proof of Theorem 4.1, we may build a standard form model $C$ for ${\boldsymbol{W}}=\operatorname {Spec}C$ that is free over $C(0)$ and minimal at $x$. Since $A$ is free over $A(0)$ the map ${\boldsymbol{W}}=\operatorname {\mathbf{Spec}}C\rightarrow \operatorname {Spec}C(0)\rightarrow \operatorname {Spec}A(0)$ factors through $\boldsymbol{c}:\operatorname {\mathbf{Spec}}C\rightarrow \operatorname {\mathbf{Spec}}A$, and the latter is induced by an actual map of cdgas $\alpha :A\rightarrow C$. Similarly, the map $\boldsymbol{d}:\operatorname {\mathbf{Spec}}C\rightarrow \operatorname {\mathbf{Spec}}B$ is induced by an actual map of cdgas $\beta :B\rightarrow C$, since $B$ is free over $B(0)$. This proves the first part of Theorem 4.2.

For the second part, if instead $\boldsymbol{f}:{\boldsymbol{U}}=\operatorname {\mathbf{Spec}}A\rightarrow {\boldsymbol{X}}$ and $\boldsymbol{g}:{\boldsymbol{V}}=\operatorname {\mathbf{Spec}}B\rightarrow {\boldsymbol{X}}$ are étale neighbourhoods of $x$, we apply the same fibre product construction. Since étale maps are stable under pullbacks, $\boldsymbol{c}$ and $\boldsymbol{d}$ are now étale, rather than Zariski open inclusions. However, the induced map ${\boldsymbol{W}}\rightarrow \operatorname {Spec}A(0)\times \operatorname {Spec}B(0)$ is still a locally closed immersion on $W=t_0({\boldsymbol{W}})$. So in the same way we obtain a standard form model $C$ for ${\boldsymbol{W}}=\operatorname {\mathbf{Spec}}C$ that is free over $C(0)$ and minimal at $r=(p,q)$ with étale maps $\boldsymbol{c},\boldsymbol{d}$ that are induced by actual maps of cdgas.

5. A derived Darboux theorem

Section 5.1 explains what is meant by a $k$-shifted symplectic structure $\omega =(\omega ^0,\omega ^1,\omega ^2,\ldots )$ on an affine derived ${\mathbin{\mathbb{K}}}$-scheme $\operatorname {\mathbf{Spec}}A$ for $A$ of standard form and $k<0$, expanding on §3.4, and §5.2 proves that up to equivalence we can take $\omega =({\mathrm{d}_{dR}}\phi ,0,0,\ldots )$. Sections 5.3 and 5.4 define standard models for $k$-shifted symplectic structures on $\operatorname {\mathbf{Spec}}A$, which we call Darboux form.

Our main result Theorem 5.18, stated in §5.5 and proved in §5.6, says that every $k$-shifted symplectic derived ${\mathbin{\mathbb{K}}}$-scheme $({\boldsymbol{X}},\tilde{\omega })$ for $k<0$ is locally equivalent to $(\operatorname {\mathbf{Spec}}A,\omega )$ for $A,\omega$ in Darboux form. Section 5.7 explains how to compare different Darboux form presentations of $({\boldsymbol{X}},\tilde{\omega })$ on their overlaps.

5.1. $k$-shifted symplectic structures on $\operatorname {\mathbf{Spec}}A$

Let $A=A(n)$ be a standard form cdga over ${\mathbin{\mathbb{K}}}$, as in Example 2.8, constructed from a smooth ${\mathbin{\mathbb{K}}}$-algebra $A(0)$ and free finite rank modules $M^{-1}, M^{-2},\ldots , M^{-n}$ over $A(0),\ldots ,A(n-1)$. Write ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ for the corresponding affine derived ${\mathbin{\mathbb{K}}}$-scheme. We will explain in more detail how the material of §3.4 works out for ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$.

As in §3.3 there is an equivalence $L_{\operatorname {qcoh}}({\boldsymbol{X}})\simeq D(\operatorname {mod}A)$ which identifies ${\mathbin{\mathbb{L}}}_{\boldsymbol{X}}\simeq {\mathbin{\mathbb{L}}}_A$, and as $A$ is of standard form, as in §2.3 the Kähler differentials $\bigl (\Omega ^1_A,{\mathrm{d}}\bigr )$ are a model for ${\mathbin{\mathbb{L}}}_A$ in $D(\operatorname {mod}A)$, and Proposition 2.12 gives an explicit description of $\bigl (\Omega ^1_A,{\mathrm{d}}\bigr )$. Write $\Omega ^1_A=\bigoplus _{k=-\infty }^0(\Omega ^1_A)^k$ for the decomposition of $\Omega ^1_A$ into graded pieces, so that the differential maps ${\mathrm{d}}:(\Omega ^1_A)^k\rightarrow (\Omega ^1_A)^{k+1}$.

As in §2.1 and §2.3, the de Rham algebra of $A$ is

$$\begin{equation*} \operatorname {DR}(A):=\operatorname {Sym}_A(\Omega ^1_A[1])=\bigoplus _{p=0}^\infty \Lambda ^p \Omega ^1_A[p]= \bigoplus _{p=0}^\infty \bigoplus _{k=-\infty }^0(\Lambda ^p\Omega ^1_A)^k[p]. \end{equation*}$$

It has two gradings, degree and weight, where the component $(\Lambda ^p\Omega ^1_A)^k[p]$ has degree $k-p$ and weight $p$. It has two differentials, the internal differential ${\mathrm{d}}:\operatorname {DR}(A)\rightarrow \operatorname {DR}(A)$ of degree 1 and weight 0, so that ${\mathrm{d}}$ maps

$$\begin{equation*} {\mathrm{d}}:(\Lambda ^p\Omega ^1_A)^k[p]\longrightarrow (\Lambda ^p\Omega ^1_A)^{k+1}[p], \end{equation*}$$

and the de Rham differential ${\mathrm{d}_{dR}}:\operatorname {DR}(A)\rightarrow \operatorname {DR}(A)$ of degree $-1$ and weight 1, so that ${\mathrm{d}_{dR}}$ maps

$$\begin{equation*} {\mathrm{d}_{dR}}:(\Lambda ^p \Omega ^1_A)^k[p]\longrightarrow (\Lambda ^{p+1}\Omega ^1_A)^k[p+1]. \end{equation*}$$

They satisfy ${\mathrm{d}}\circ {\mathrm{d}}={\mathrm{d}_{dR}}\circ {\mathrm{d}_{dR}}={\mathrm{d}}\circ {\mathrm{d}_{dR}}+{\mathrm{d}_{dR}}\circ {\mathrm{d}}=0$. The multiplication “$\,\cdot \,$” on $\operatorname {DR}(A)$ maps

$$\begin{equation*} (\Lambda ^p \Omega ^1_A)^k[p]\times (\Lambda ^p \Omega ^1_A)^l[q]\longrightarrow (\Lambda ^{p+q}\Omega ^1_A)^{k+l}[p+q]. \end{equation*}$$

In 24, Def. 1.7, Pantev et al. define a simplicial set ${\mathbin{\mathcal{A}}}^p_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)$ of $p$-forms of degree $k\in {\mathbin{\mathbb{Z}}}$ on the derived ${\mathbin{\mathbb{K}}}$-scheme ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ by

$$\begin{equation} {\mathbin{\mathcal{A}}}^p_{\boldsymbol{X}}(Y,k)=\big \vert \Lambda ^p{\mathbin{\mathbb{L}}}_A[k] \big \vert . {\tag{5.1}} \end{equation}$$

In our case we may take $\Lambda ^p{\mathbin{\mathbb{L}}}_A=\Lambda ^p\Omega ^1_A$. If $E^\bullet$ is a complex of ${\mathbin{\mathbb{K}}}$-modules, then $\vert E^\bullet \vert$ means: take the truncation $\tau _{\leqslant 0}(E^\bullet )$, and turn it into a simplicial set via the Dold–Kan correspondence. To avoid dealing with simplicial sets, note that this implies that $\pi _0\bigl (\vert E^\bullet \vert \bigr )\cong H^0\bigl (\tau _{\leqslant 0}(E^\bullet )\bigr )\cong H^0\bigl (E^\bullet \bigr )$. Thus 5.1 yields

$$\begin{equation} \pi _0\bigl ({\mathbin{\mathcal{A}}}^p_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)\bigr )\cong H^k\bigl (\Lambda ^p\Omega ^1_A,{\mathrm{d}}\bigr ) =H^{k-p}\bigl (\Lambda ^p\Omega ^1_A[p],{\mathrm{d}}\bigr ). {\tag{5.2}} \end{equation}$$

So, (connected components of the simplicial set of) $p$-forms of degree $k$ on ${\boldsymbol{X}}$ are just $k$-cohomology classes of the complex $\bigl (\Lambda ^p\Omega ^1_A,{\mathrm{d}}\bigr )$. We prefer to deal with explicit representatives, rather than cohomology classes. So we define:

Definition 5.1.

In the situation above with ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$, a $p$-form of degree $k$ on ${\boldsymbol{X}}$ for $p\geqslant 0$ and $k\leqslant 0$ is $\omega ^0\in (\Lambda ^p\Omega ^1_A)^k$ with ${\mathrm{d}}\omega ^0=0$ in $(\Lambda ^p \Omega ^1_A)^{k+1}$. Two $p$-forms $\omega ^0,\tilde{\omega }^0$ of degree $k$ are equivalent, written $\omega ^0\sim \tilde{\omega }^0$, if there exists $\alpha ^0\in (\Lambda ^p \Omega ^1_A)^{k-1}$ with $\omega ^0-\tilde{\omega }^0={\mathrm{d}}\alpha ^0$.

Then equivalence classes $[\omega ^0]$ of $p$-forms $\omega ^0$ of degree $k$ on ${\boldsymbol{X}}$ in the sense of Definition 5.1 correspond to connected components of the simplicial set of $p$-forms of degree $k$ on ${\boldsymbol{X}}$ in the sense of Pantev et al. 24. The reason for including the superscripts $0$ in $\omega ^0,\alpha ^0$ will become clear in Definition 5.2.

Letting ${\mathbin{\mathbb{T}}}_A=\underline{\operatorname {Hom}}_A(\Omega ^1_A,A)=\underline{\operatorname {Der}}(A)$ denote the tangent complex of $A$, a $2$-form $\omega ^0$ of degree $k$ on ${\boldsymbol{X}}$ defines an antisymmetric morphism

$$\begin{equation} \omega ^0:{\mathbin{\mathbb{T}}}_A\longrightarrow \Omega ^1_A[k], {\tag{5.3}} \end{equation}$$

via $X\mapsto \iota _X\omega ^0$ for $X \in \underline{\operatorname {Der}}(A)$. The $2$-form $\omega ^0$ is said to be nondegenerate if this induced map is a quasi-isomorphism.

The definition of the simplicial set ${\mathbin{\mathcal{A}}}^{p,\operatorname {cl}}_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)$ of closed $p$-forms of degree $k\in {\mathbin{\mathbb{Z}}}$ on ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ in Pantev et al. 24, Def. 1.7 yields

$$\begin{equation} {\mathbin{\mathcal{A}}}^{p,\operatorname {cl}}_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)=\big \vert \textstyle \prod _{i\geqslant 0}\Lambda ^{p+i\,}{\mathbin{\mathbb{L}}}_A[k-i] \big \vert . {\tag{5.4}} \end{equation}$$

In our case we may take $\Lambda ^{p+i\,}{\mathbin{\mathbb{L}}}_A[k-i]= \Lambda ^{p+i}\Omega ^1_A[k-i]$. Then $\prod _{i\geqslant 0}\Lambda ^{p+i\,}\Omega ^1_A[k-i]$ means the complex which as a graded vector space is the product over all $i\geqslant 0$ of the graded vector spaces $\Lambda ^{p+i\,}\Omega ^1_A[k-i]$, with differential ${\mathrm{d}}+{\mathrm{d}_{dR}}$.

The difference between $\prod _{i\geqslant 0}\Lambda ^{p+i}{\mathbin{\mathbb{L}}}_A[k-i]$ and $\bigoplus _{i\geqslant 0}\Lambda ^{p+i}{\mathbin{\mathbb{L}}}_A[k-i]$ is that elements $(\omega ^i)_{i\geqslant 0}\in \bigoplus _{i\geqslant 0}\Lambda ^{p+i}{\mathbin{\mathbb{L}}}_A[k-i]$ have $\omega ^i\ne 0$ for only finitely many $i$, but elements $(\omega ^i)_{i\geqslant 0}\in \prod _{i\geqslant 0}\Lambda ^{p+i}{\mathbin{\mathbb{L}}}_A[k-i]$ can have $\omega ^i\ne 0$ for infinitely many $i$.

Thus, as for 5.1–5.2, equation 5.4 implies that

$$\begin{align*} \pi _0\bigl ({\mathbin{\mathcal{A}}}^{p,\operatorname {cl}}_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)\bigr )&\cong H^0\bigl ( \textstyle \prod _{i\geqslant 0}\Lambda ^{p+i\,}\Omega ^1_A[k-i],{\mathrm{d}}+{\mathrm{d}_{dR}}\bigr )\\ &=H^k\bigl (\textstyle \prod _{i\geqslant 0}\Lambda ^{p+i}\Omega ^1_A[-i],{\mathrm{d}}+{\mathrm{d}_{dR}}\bigr ). \end{align*}$$

As for Definition 5.1, we define:

Definition 5.2.

In the situation above with ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$, a closed $p$-form of degree $k$ on ${\boldsymbol{X}}$ for $p\geqslant 0$ and $k\leqslant 0$ is $\omega =(\omega ^0,\omega ^1,\omega ^2,\ldots )$ with $\omega ^i\in (\Lambda ^{p+i} \Omega ^1_A)^{k-i}$ for $i=0,1,2,\ldots ,$ satisfying the equations

$$\begin{equation} \begin{aligned} {\mathrm{d}}\omega ^0&=0 &&\text{in $(\Lambda ^p \Omega ^1_A)^{k+1}$, and}\\ {\mathrm{d}_{dR}}\omega ^i+{\mathrm{d}}\omega ^{i+1}&=0 &&\text{in $(\Lambda ^{p+i+1}\Omega ^1_A)^{k-i}$, for all $i\geqslant 0$.} \end{aligned} {\tag{5.5}} \end{equation}$$

We call two closed $p$-forms $\omega =(\omega ^0,\ldots )$, $\tilde{\omega }=(\tilde{\omega }^0,\ldots )$ of degree $k$ equivalent, written $\omega \sim \tilde{\omega }$, if there exists $\alpha =(\alpha ^0,\alpha ^1,\ldots )$ with $\alpha ^i\in (\Lambda ^{p+i}\Omega ^1_A)^{k-i-1}$ for $i=0,1,\ldots ,$ satisfying

$$\begin{align*} \omega ^0-\tilde{\omega }^0&={\mathrm{d}}\alpha ^0 &&\text{in $(\Lambda ^p\Omega ^1_A)^k$, and}\\ \omega ^{i+1}-\tilde{\omega }^{i+1}&={\mathrm{d}_{dR}}\alpha ^i+{\mathrm{d}}\alpha ^{i+1} &&\text{in $(\Lambda ^{p+i+1}\Omega ^1_A)^{k-i-1}$, for all $i\geqslant 0$.} \end{align*}$$

The morphism $\pi :{\mathbin{\mathcal{A}}}^{p,\operatorname {cl}}_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)\rightarrow {\mathbin{\mathcal{A}}}^p_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)$ from closed $p$-forms of degree $k$ to $p$-forms of degree $k$ in §3.4 corresponds to $\pi :\omega =(\omega ^0,\ldots )\mapsto \omega ^0$.

A closed 2-form $\omega =(\omega ^0,\omega ^1,\ldots )$ of degree $k$ on ${\boldsymbol{X}}$ is called a $k$-shifted symplectic form if $\omega ^0=\pi (\omega )$ is a nondegenerate 2-form of degree $k$.

Then equivalence classes $[\omega ]$ of closed $p$-forms $\omega$ of degree $k$ on ${\boldsymbol{X}}$ in the sense of Definition 5.1 correspond to connected components of the simplicial set ${\mathbin{\mathcal{A}}}^{p,\operatorname {cl}}_{\mathbin{\mathbb{K}}}({\boldsymbol{X}},k)$ of closed $p$-forms of degree $k$ on ${\boldsymbol{X}}$ in the sense of Pantev et al. 24.

5.2. Closed forms and cyclic homology of mixed complexes

In order to work effectively with such symplectic forms, it is very useful to interpret them in the context of cyclic homology of mixed complexes. The following definitions are essentially as in Loday 21, §2.5.13, except that we use cohomological grading and take into account an extra weight grading in our mixed complexes, as in Pantev et al. 24, §1.1.

Definition 5.3.

A mixed complex $E$ is a complex over ${\mathbin{\mathbb{K}}}$ with a differential $b$ of degree $1$ together with an additional square-zero operator $B$ of degree $-1$ anti-commuting with $b$. A graded mixed complex has in addition a weight grading giving by a decomposition $E=\bigoplus _p E(p)$, where $b$ has degree $0$ and $B$ has degree $1$ with respect to the weight grading. A morphism between graded mixed complexes $E=\bigoplus _p E(p)$ and $F=\bigoplus _p F(p)$ is a ${\mathbin{\mathbb{K}}}$-linear map $\varphi : E \rightarrow F$ of degree $0$ with respect to both the cohomological and weight grading that commutes with $b$ and $B$. The morphism $\varphi : E \rightarrow F$ is a weak equivalence if it is a quasi-isomorphism for the cohomology taken with respect to the differential $b$. For simplicity and since it is sufficient for our applications, we shall consider only graded mixed complexes that are bounded above at $0$ with respect to the cohomological grading and bounded below at $0$ with respect to the weight grading.

Example 5.4.

For us, the main example of a graded mixed complex is $E=\operatorname {DR}(A)=\bigoplus _p\Lambda ^p\Omega ^1_A[p]$ with $b={\mathrm{d}}$ and $B={\mathrm{d}_{dR}}$, for $A$ a standard form cdga.

Definition 5.5.

Given a graded mixed complex $E$, for each $p$ we define three complexes, the negative cyclic complex of weight $p$, denoted $\operatorname {NC}(E)(p)$, the periodic cyclic complex of weight $p$, denoted $\operatorname {PC}(E)(p)$, and the cyclic complex of weight $p$, denoted $\operatorname {CC}(E)(p)$. The degree $k$ terms of these complexes are:

$$\begin{gather*} \operatorname {NC}^k(E)(p) = \prod _{i \geqslant 0} E^{k-2i}(p+i), \qquad \operatorname {PC}^k(E)(p) = \prod _{i} E^{k-2i}(p+i), \\ \operatorname {CC}^k(E)(p) = \prod _{i \leqslant 0} E^{k-2i}(p+i). \end{gather*}$$

In each case, the differential is simply $b+B$, and the complexes can be constructed as $\operatorname {Tot}^{\Pi }$ of appropriate double complexes. The $k$th cohomology of the complexes $\operatorname {NC}(E)(p)$, $\operatorname {PC}(E)(p)$, and $\operatorname {CC}(E)(p)$ are denoted $\operatorname {HN}^k(E)(p)$, $\operatorname {HP}^k(E)(p)$, and $\operatorname {HC}^k(E)(p)$, respectively.

There is an evident short exact sequence of complexes

$$\begin{equation*} 0 \longrightarrow \operatorname {NC}(E)(p) \longrightarrow \operatorname {PC}(E)(p) \longrightarrow \operatorname {CC}(E)(p-1)[2] \longrightarrow 0, \end{equation*}$$

and an induced long exact sequence of cohomology groups

$$\begin{equation} \begin{gathered} \vcenter{\img[][266pt][32pt][{$\xymatrix@C=25pt@R=1pt{ \cdots\ar[r] &amp; \HC^{k+1}(E)(p-1) \ar[r] &amp; \HN^k(E)(p) \ar[r] &amp; \HP^k(E)(p) \\ {\phantom{\cdots}} \ar[r] &amp; \HC^{k+2}(E)(p-1) \ar[r] &amp; {}\quad\cdots.\qquad{} }$}]{Images/img563e294dc45ede455a89e407c05e752c.svg}} \end{gathered} {\tag{5.6}} \end{equation}$$

When $E=\operatorname {DR}(A)$ for $A$ a standard form cdga, we denote the corresponding cochain groups by $\operatorname {NC}^k(A)(p),\operatorname {PC}^k(A)(p), \operatorname {CC}^k(A)(p)$, and the cohomology groups by $\operatorname {HN}^k(A)(p),\operatorname {HP}^k(A)(p), \operatorname {HC}^k(A)(p)$. As in Loday 21, Ch. 5, these groups are known to be compatible with other definitions that the reader may have seen. The connection of all this with the material of §5.1 is that closed $p$-forms $\omega =(\omega ^0,\omega ^1,\omega ^2,\ldots )$ of degree $k$ in Definition 5.2 are cocycles in $\operatorname {NC}^{k-p}(A)(p)$, and equivalence classes $[\omega ]$ of closed $p$-forms $\omega =(\omega ^0,\omega ^1,\ldots )$ of degree $k$ on ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ are elements of $\operatorname {HN}^{k-p}(A)(p)$, by the Hochschild–Kostant–Rosenberg theorem.

Here is a useful vanishing result. (Note that the group $\operatorname {HN}^{k-2}(A)(2)$ in 5.7 classifies closed $2$-forms $\omega$ of degree $k$ on ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ up to equivalence.)

Proposition 5.6.

(a)

Let $A$ be a connective cdga over ${\mathbin{\mathbb{K}}},$ with $H^{0}(A)$ of finite type over ${\mathbin{\mathbb{K}}}$. If $p+k \leqslant 2$, then in the sequence 5.6 the map $\operatorname {HP}^{k-3}(A)(p) \rightarrow \operatorname {HC}^{k-1}(A)(p-1)$ is an injection, hence the map $\operatorname {HN}^{k-3}(A)(p) \rightarrow \operatorname {HP}^{k-3}(A)(p)$ is zero and the map $\operatorname {HC}^{k-2}(A)(p-1) \rightarrow \operatorname {HN}^{k-3}(A)(p)$ is a surjection.

In particular, for $k<0,$ we have a short exact sequence$$\begin{equation} 0 \longrightarrow \operatorname {HP}^{k-3}(A)(2) \longrightarrow \operatorname {HC}^{k-1}(A)(1) \longrightarrow \operatorname {HN}^{k-2}(A)(2) \longrightarrow 0. {\tag{5.7}} \end{equation}$$

(b)

For $k=-1,$ the left-hand group $\operatorname {HP}^{-4}(A)(2)$ in 5.7 has $\operatorname {HP}^{-4}(A)(2)\cong \mathrm{H}^0_{\inf }(\operatorname {Spec}H^0(A)),$ where $\mathrm{H}^*_{\inf }(X)$ is the algebraic de Rham cohomology of a ${\mathbin{\mathbb{K}}}$-scheme $X$. Thus, if $\operatorname {Spec}H^0(A)$ is connected, then $\operatorname {HP}^{-4}(A)(2)={\mathbin{\mathbb{K}}}$.

(c)

For $k \leqslant -2,$ we have $\operatorname {HP}^{k-3}(A)(2)=0$.

Proof.

When $A$ is a cdga concentrated in degree $0$, the fact that $\operatorname {HP}^{k-3}(A)(p) \rightarrow \operatorname {HC}^{k-1}(A)(p-1)$ is an injection is essentially Emmanouil 10, Prop. 2.6, noting that $\operatorname {HP}^{k-3}(A)(p) \cong \mathrm{H}^{2p+k-3}_{\inf }(A)$ by 10, Th. 2.2. We therefore have a short exact sequence as claimed. Here $\mathrm{H}^{2p+k-3}_{\inf }(A)$ is infinitesimal or algebraic de Rham cohomology. In particular, for $k=-1$, $\operatorname {HP}^{-4}(A)(2)\cong \mathrm{H}^0_{\inf }(A) \cong {\mathbin{\mathbb{K}}}$ when $\operatorname {Spec}A$ is connected, and for $k\leqslant -2$, $\operatorname {HP}^{k-3}(A)(2)\cong \mathrm{H}^{k+1}_{\inf }(A)=0$.

Now let $A$ be a connective cdga with $H^{0}(A)$ of finite type over ${\mathbin{\mathbb{K}}}$ and consider the natural map $A\rightarrow H^0(A)$. By Goodwillie 13, Th. IV.2.1, the induced map $\operatorname {HP}^{k-3}(A)(p) \rightarrow \operatorname {HP}^{k-3}(H^0(A))(p)$ is an isomorphism, so that (b) and (c) follow from the above. The injectivity of the map $\operatorname {HP}^{k-3}(A)(p) \rightarrow \operatorname {HC}^{k-1}(A)(p-1)$ follows from the injectivity of the map $\operatorname {HP}^{k-3}(H^0(A))(p) \rightarrow \operatorname {HC}^{k-1}(H^0(A))(p-1)$ by functoriality of the long exact sequence 5.6, and this proves (a).

■

Using Proposition 5.6 we show that if $\omega =(\omega ^0,\omega ^1,\ldots )$ is a $k$-shifted symplectic structure on ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ for $k<0$, then up to equivalence we can take $\omega ^0$ to be exact and $\omega ^i=0$ for $i=1,2,\ldots ,$ which is a considerable simplification. Also we parametrise how to write $\omega$ this way in its equivalence class.

Proposition 5.7.

(a)

Let $\omega =(\omega ^0,\omega ^1, \omega ^2,\ldots )$ be a closed $2$-form of degree $k<0$ on ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A,$ for $A$ a standard form cdga over ${\mathbin{\mathbb{K}}}$. Then there exist $\Phi \in A^{k+1}$ and $\phi \in (\Omega ^1_A)^k$ such that ${\mathrm{d}}\Phi =0$ in $A^{k+2}$ and ${\mathrm{d}_{dR}}\Phi +{\mathrm{d}}\phi =0$ in $(\Omega ^1_A)^{k+1}$ and $\omega \sim ({\mathrm{d}_{dR}}\phi ,0,0,\ldots ),$ in the sense of Definition 5.2.

(b)

In the case $k=-1$ in (a), we have $\Phi \in A^0=A(0),$ so we can consider the restriction $\Phi \vert _{X^{\mathrm{red}}}$ of $\Phi$ to the reduced ${\mathbin{\mathbb{K}}}$-subscheme $X^{\mathrm{red}}$ of $X=t_0({\boldsymbol{X}})=\operatorname {Spec}H^0(A)$. Then $\Phi \vert _{X^{\mathrm{red}}}$ is locally constant on $X^{\mathrm{red}},$ and we may choose $(\Phi ,\phi )$ in (a) such that $\Phi \vert _{X^{\mathrm{red}}}=0$.

(c)

Suppose $(\Phi ,\phi )$ and $(\Phi ',\phi ')$ are alternative choices in (a) for fixed $\omega ,k,{\boldsymbol{X}},A,$ where if $k=-1$ we suppose $\Phi \vert _{X^{\mathrm{red}}}=0= \Phi '\vert _{X^{\mathrm{red}}}$ as in (b). Then there exist $\Psi \in A^k$ and $\psi \in (\Omega ^1_A)^{k-1}$ with $\Phi -\Phi '={\mathrm{d}}\Psi$ and $\phi -\phi '={\mathrm{d}_{dR}}\Psi +{\mathrm{d}}\psi$.

Proof.

As in Definition 5.5, the $\sim$-equivalence class $[\omega ]$ of $\omega$ lies in $\operatorname {HN}^{k-2}(A)(2)$. Thus by 5.7 in Proposition 5.6, $[\omega ]$ lies in the image of the map $\operatorname {HC}^{k-1}(A)(1) \rightarrow \operatorname {HN}^{k-2}(A)(2)$. A class in $\operatorname {HC}^{k-1}(A)(1)$ is represented by $(\Phi ,\phi )\!\in \! \operatorname {CC}^{k-1}(A)(1)\!=\!A^{k+1}\!\times \!(\Omega ^1_A)^k$ with ${\mathrm{d}}\Phi \!=\!0\!=\!{\mathrm{d}_{dR}}\Phi \!+\!{\mathrm{d}}\phi$, and the map $\operatorname {CC}^{k-1}(A)(1)\rightarrow \operatorname {NC}^{k-2}(A)(2)$ takes $(\Phi ,\phi )\mapsto ({\mathrm{d}_{dR}}\phi ,0,0,\ldots )$. Note that

$$\begin{equation*} {\mathrm{d}}({\mathrm{d}_{dR}}\phi )=-{\mathrm{d}_{dR}}\circ {\mathrm{d}}\phi =-{\mathrm{d}_{dR}}\circ {\mathrm{d}_{dR}}\Phi -{\mathrm{d}_{dR}}\circ {\mathrm{d}}\phi =-{\mathrm{d}_{dR}}\bigl ({\mathrm{d}_{dR}}\Phi +{\mathrm{d}}\phi \bigr )=0, \end{equation*}$$

so that $({\mathrm{d}_{dR}}\phi ,0,0,\ldots )$ is a closed 2-form of degree $k$ on ${\boldsymbol{X}}$. This proves (a).

For (b), $\Phi \vert _X:X\rightarrow {\mathbin{\mathbb{A}}}^1$ is a regular function on $X$ when $k=-1$ . We have $H^0(T^*X)\cong H^0(\Omega ^1_A)$. As ${\mathrm{d}_{dR}}\Phi +{\mathrm{d}}\phi =0,$ we have $[{\mathrm{d}_{dR}}\Phi ]=0$ in $H^0(\Omega ^1_A)$, so ${\mathrm{d}_{dR}}\bigl (\Phi \vert _X\bigr )=0$ in $H^0(T^*X)$. Therefore $\Phi \vert _{X^{\mathrm{red}}}:X^{\mathrm{red}}\rightarrow {\mathbin{\mathbb{A}}}^1$ is locally constant.

This is related to the isomorphism $\operatorname {HP}^{-4}(A)(2)\cong \mathrm{H}^0_{\inf }(\operatorname {Spec}H^0(A))$ in Proposition 5.6(b). There is a natural isomorphism from $\mathrm{H}^0_{\inf }(\operatorname {Spec}H^0(A))$ to the ${\mathbin{\mathbb{K}}}$-vector space of locally constant maps $\Phi \vert _{X^{\mathrm{red}}}:X^{\mathrm{red}}\rightarrow {\mathbin{\mathbb{A}}}^1$. When we lift $[\omega ]\in \operatorname {HN}^{-3}(A)(2)$ to $[\Phi ,\phi ]\in \operatorname {HC}^{-2}(A)(1)$ in 5.7 for $k=-1$, the possible choice in the lift $[\Phi ,\phi ]$ is $\operatorname {HP}^{-4}(A)(2)\cong \mathrm{H}^0_{\inf }(\operatorname {Spec}H^0(A))$, which corresponds exactly to the space of locally constant maps $\Phi \vert _{X^{\mathrm{red}}}:X^{\mathrm{red}}\rightarrow {\mathbin{\mathbb{A}}}^1$.

So, by adjusting the choice of lift $[\Phi ,\phi ]$ of $[\omega ]$ to $\operatorname {HC}^{-2}(A)(1)$, we can take $\Phi \vert _{X^{\mathrm{red}}}=0$, proving (b). Note that this determines the class $[\Phi ,\phi ]$ in $\operatorname {HC}^{-2}(A)(1)$ lifting $[\omega ]$ uniquely. That is, we have constructed a canonical splitting of the exact sequence 5.7 when $k=-1$.

For (c), note that for all $k<0$ the class $[\Phi ,\phi ]\in \operatorname {HC}^{k-1}(A)(1)$ in (a) lifting $[\omega ]\in \operatorname {HN}^{k-2}(A)(2)$ is uniquely determined, requiring $\Phi \vert _{X^{\mathrm{red}}}=0$ when $k=-1$ as above, and since $\operatorname {HP}^{k-3}(A)(2)=0$ in 5.7 for $k\leqslant -2$ by Proposition 5.6(c). Hence, if $(\Phi ,\phi )$ and $(\Phi ',\phi ')$ are alternative choices in (a), then $[\Phi ,\phi ]=[\Phi ',\phi ']\in \operatorname {HC}^{k-1}(A)(1)$. Thus, there exists $(\Psi ,\psi )\in \operatorname {CC}^{k-2}(A)(1)$ with ${\mathrm{d}}(\Psi ,\psi )=(\Phi ,\phi )-(\Phi ',\phi ')$. From the definitions, this means that $\Psi \in A^k$ and $\psi \in (\Omega ^1_A)^{k-1}$ with $\Phi -\Phi '={\mathrm{d}}\Psi$ and $\phi -\phi '={\mathrm{d}_{dR}}\Psi +{\mathrm{d}}\psi$.

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When we apply Proposition 5.7 in §5.4–§5.7, we will do it with $k\omega$ in place of $\omega$, yielding $\Phi ,\phi$ with ${\mathrm{d}}\Phi =0$, ${\mathrm{d}_{dR}}\Phi +{\mathrm{d}}\phi =0$, and $k\omega \sim ({\mathrm{d}_{dR}}\phi ,0,0,\ldots )$. This will give simpler formulae, eliminating factors of $k,1/k$.

5.3. Darboux forms for $k$-shifted symplectic structures

The next four examples give standard models for $k$-shifted symplectic affine derived ${\mathbin{\mathbb{K}}}$-schemes for $k<0$, which we will call “in Darboux form”. Theorem 5.18 will prove that every $k$-shifted symplectic derived scheme $({\boldsymbol{X}},\omega )$ is Zariski/étale locally equivalent to one in Darboux form. We divide into three cases:

(a)

$k$ is odd, so that $k=-2d-1$ for $d=0,1,2,\ldots ;$

(b)

$k\equiv 0\mod 4$, so that $k=-4d$ for $d=1,2,\ldots ;$ and

(c)

$k\equiv 2\mod 4$, so that $k=-4d-2$ for $d=0,1,\ldots .$

The difference is in the behaviour of 2-forms in the variables of middle degree $k/2$. In case (a) $k/2\notin {\mathbin{\mathbb{Z}}}$, so there is no middle degree, and this is the simplest case, which we handle in Example 5.8. In (b) $k/2$ is even, so 2-forms in the middle degree variables are antisymmetric; we discuss this in Example 5.9. In (c) $k/2$ is odd, so 2-forms in the middle degree variables are symmetric. For this case we give both a strong Darboux form in Example 5.10, to which $k$-shifted symplectic derived ${\mathbin{\mathbb{K}}}$-schemes are equivalent étale locally, and a weak Darboux form in Example 5.12, to which they are equivalent Zariski locally.

For $A$ as in Example 5.8 below, as in Bouaziz and Grojnowski 6 we can regard ${\boldsymbol{X}}=\operatorname {\mathbf{Spec}}A$ as a twisted shifted cotangent bundle $T^*_\sigma [k]{\boldsymbol{Y}}$ for ${\boldsymbol{Y}}=\operatorname {\mathbf{Spec}}B$, where $B\subset A$ is the sub-cdga generated by the variables $x_j^{-i}$. Then in 5.8 the $x_j^{-i}$ are coordinates on the base ${\boldsymbol{Y}}$, and $y_j^{k-i}$ are the dual coordinates on the fibres of $T^*_\sigma [k]{\boldsymbol{Y}}\rightarrow {\boldsymbol{Y}}$.

Example 5.8.

Fix $d=0,1,\ldots .$ We will explain how to define a class of explicit standard form cdgas