Relation to existing work

Throughout, we build on the constructions and results of [Reference Huybrechts and Thomas6], [Reference Illusie7] and [Reference Gillam3], which we generalize to algebraic stacks. Our construction also recovers the G-equivariant Atiyah class considered by Ricolfi [Reference Ricolfi20]. Throughout, we use Olsson’s definition of the quasi-coherent derived category and the cotangent complex for algebraic stacks developed in [Reference Olsson16] and [Reference Laszlo and Olsson10]; see also the excellent discussion in [Reference Hall and Rydh4, §1].

Generalizations of the Atiyah class to the theory of algebraic stacks have appeared before in different contexts: In the setting of derived algebraic geometry of Schürg, Toën and Vezzosi [Reference Schürg, Toën and Vezzosi21], a perfect complex E on a derived geometry stack $\mathcal {Y}$ with a perfect cotangent complex gives rise to a map from $\mathcal {Y}$ to the (derived) moduli stack of perfect complexes. In ([Reference Schürg, Toën and Vezzosi21], Appendix A), they define the Atiyah class as the induced pullback map on cotangent complexes and argue that it recovers Illusie’s definition in the case of schemes. Moreover, Lurie has constructed the Atiyah class in the context of spectral algebraic geometry [Reference Lurie12, §19.2.2]. In the purely classical setting, our definition goes beyond these, as we do not require the perfectness assumptions made in [Reference Schürg, Toën and Vezzosi21], and since the current version of [Reference Lurie12] presently only deals with Deligne–Mumford stacks.Footnote ¹

Finally, the paper [Reference Aranha and Pstragowski1] shows how to construct virtual cycles from perfect obstruction theories on algebraic stacks and thus fits in neatly with the viewpoint taken here, that it is often advantageous to consider fundamental constructions directly on the moduli stack.

Notations and conventions

For a Grothendieck topos T and a ring R in T, we let $\operatorname {\mathrm {Mod}}(R)$ denote the category of R-modules and $D(R)$ its derived category. We identify $\operatorname {\mathrm {Mod}}(R)$ with the subcategory of $D(R)$ generated by complexes concentrated in degree zero.

For any abelian category $\mathcal {A}$ , write $C(\mathcal {A})$ for the category of complexes of objects in $\mathcal {A}$ . Write $C^{\leq 0}(\mathcal {A})$ and $C^{[-1,0]}(\mathcal {A})$ for the full sub-category of complexes bounded in degrees $\leq 0$ and in degrees $-1,0$ , respectively. When $\mathcal {A}= \operatorname {\mathrm {Mod}}(R)$ , we simply write $C(R)$ and $C^{\leq 0}(R)$ , $C^{[-1,0]}(R)$ , respectively.

We will use the following convention regarding shift functors: For any complex E of R-modules, we have a natural isomorphism

$$\begin{align*}E[1]={\mathbb{Z}}[1]\otimes_{{\mathbb{Z}}} E.\end{align*}$$

If $E,F$ are complexes, then by $E\otimes F[1]$ we mean $(E\otimes F)[1]$ rather than $E\otimes (F[1])$ .

All tensor products, pullbacks and duals of objects in a derived category be in the derived sense. For modules and complexes, we will consider the underived tensor products and pullbacks but will point out when these do not necessarily compute the derived operation.

We use the definition of algebraic stacks as in the Stacks project [22, Tag 026N].

Functoriality

Given a map $F\to E$ in $D^-_{qcoh}(\mathcal {X})$ , the induced diagram

commutes.

Pullback

Given another morphism $f':\mathcal {X}'\to \mathcal {Y}'$ together with maps $A:\mathcal {X}'\to \mathcal {X}$ and $B:\mathcal {Y}'\to \mathcal {Y}$ , and a $2$ -isomorphism $B\circ f'\Rightarrow f\circ A$ , the induced diagram

commutes. If E is perfect, then equivalently, the diagram

commutes.

Tensor products

Identify $E\otimes L_{\mathcal {X}/\mathcal {Y}}[1]\otimes F\simeq L_{\mathcal {X}/\mathcal {Y}}[1]\otimes E\otimes F$ using the standard symmetry isomorphism of the derived tensor product. Then, up to this identification, we have an equality

$$\begin{align*}\operatorname{\mathrm{at}}_{E\otimes F} = \operatorname{\mathrm{at}}_E\otimes F +E\otimes {\operatorname{\mathrm{at}}_F}.\end{align*}$$

As a special case of this, if E and F are perfect and $\operatorname {\mathrm {at}}_F$ is trivial (e.g., if F is pulled back from $\mathcal {Y}$ ), then the following diagram commutes:

Here, the left vertical map is induced by the diagonal map $\mathcal {O}_X\to F\otimes F^{\vee }$ and the symmetry isomorphisms of the tensor product.

Determinants

Suppose that E is perfect and consider the natural trace map $\operatorname {tr}:\operatorname {\mathrm {Hom}}(E,L_{\mathcal {X}/\mathcal {Y}}[1]\otimes E)\to \operatorname {\mathrm {Hom}}(\mathcal {O}_{\mathcal {X}}, L_{\mathcal {X}/\mathcal {Y}}[1])$ . Then we have $\operatorname {\mathrm {at}}_{\det (E)} = \operatorname {tr}(\operatorname {\mathrm {at}}_E)\otimes \det (E)$ as morphisms $\det E\to L_{\mathcal {X}/\mathcal {Y}}\otimes \det E[1]$ , at least when E has a global finite length resolution by locally free sheaves. In particular, if the latter condition holds, the following diagram commutes

where the left vertical map is induced by the natural diagonal map $\mathcal {O}_X\to E\otimes E^{\vee }$ .

Pushforward

Consider a cartesian diagram

and suppose that $\mathcal {X}'\to \mathcal {X}$ is concentrated [Reference Hall and Rydh4, Definition 2.4]. Let $E\in D^-_{qcoh}(\mathcal {X}')$ . Then we have a commutative diagram

(1.1)

Suppose that, moreover, the diagram is Cartesian and that the morphisms $\mathcal {X}\to \mathcal {Y}$ and $\mathcal {Y}'\to \mathcal {X}$ are Tor-independent, and that E is a perfect complex. Then the right vertical morphism in (1.1) is an isomorphism, and this gives a natural identification $Rp_*(\operatorname {\mathrm {at}}_E)=\operatorname {\mathrm {at}}_{Rp_*E}$ as morphisms $Rp_*E\to L_{\mathcal {X}/\mathcal {Y}}[1]\otimes Rp_*E$ . If, moreover, $Rp_*E$ is perfect, this can be restated as commutativity of the following diagram:

Obstruction theory on moduli spaces of sheaves

Let $\mathcal {M}$ be a stack over $\operatorname {\mathrm {Spec}} k$ and let $E\in D^-_{qcoh}(X\times \mathcal {M})$ be perfect. Consider the Atiyah class map of E relative to X:

$$\begin{align*}\operatorname{\mathrm{at}}_{E}:E\to L_{X\times \mathcal{M}/X}\otimes E[1]\simeq \pi_{\mathcal{M}}^{*}L_{\mathcal{M}}\otimes E[1].\end{align*}$$

Since E is dualizable, and by the projection formula, this data is equivalent to a map

$$\begin{align*}\mathcal{O}_{\mathcal{M}}\to R\pi_*(\pi_{\mathcal{M}}^{*}L_{\mathcal{M}}\otimes (E\otimes E^{\vee}))[1] \simeq L_{\mathcal{M}}\otimes R\pi_*(E\otimes E^{\vee})[1].\end{align*}$$

Using dualizability again, we obtain a morphism

(1.3) $$ \begin{align} \operatorname{\mathrm{At}}_E:R\pi_{\mathcal{M}*}(E\otimes E^{\vee})^{\vee}[-1]\to L_{\mathcal{M}}. \end{align} $$

This is proven in §§6.3–6.5 for moduli of sheaves. The statement for complexes is addressed in Remark 6.16. Recall that an obstruction theory on an algebraic stack $\mathcal {Y}$ consists of a map $g:\mathbb {E}\to L_{\mathcal {Y}}$ in $D_{qcoh}(\mathcal {Y})$ , so that $h^i(g)$ is an isomorphism for $i\geq 0$ , and so that $h^{-1}(g)$ is surjective [Reference Mochizuki14, Definition 2.4.1].

Obstruction theory on Quot-schemes

Let $\mathcal {Y}$ be an algebraic stack and let E be a $\mathcal {Y}$ -flat coherent sheaf on $X\times \mathcal {Y}$ . Let $f:\mathcal {Q}\to \mathcal {Y}$ be an open substack of the relative Quot-scheme of E over $\mathcal {Y}$ and let

$$\begin{align*}0\to F\to E_{\mathcal{Q}}\to G\to 0\end{align*}$$

be the universal exact sequence on $X\times \mathcal {Q}$ . We consider the associated reduced Atiyah class $\overline {\operatorname {\mathrm {at}}}_{E}:=\overline {\operatorname {\mathrm {at}}}_{E,V\times \mathcal {X}/V\times \mathcal {Y},G}$ as a map $\overline {\operatorname {\mathrm {at}}}_E:F\to \pi _{\mathcal {X}}^{*} L_{\mathcal {X}/\mathcal {Y}}\otimes G$ in the derived category. As before, this data is equivalent to a morphism

$$\begin{align*}\overline{\operatorname{\mathrm{At}}}_E: Rf_{*}(G\otimes F^{\vee})^{\vee}\to L_{\mathcal{Q}/\mathcal{Y}}.\end{align*}$$

Simplicial modules

For a simplicial ring A in T, we denote by $A-\operatorname {\mathrm {Mods}}$ the category of A-modules. When A is an ordinary, we regard it as a constant simplicial ring, so that $A-\operatorname {\mathrm {Mods}}$ denotes the category of simplicial A-modules. In either case, we denote by $D^{\Delta }(A)$ the derived category obtained by localizing $A-\operatorname {\mathrm {Mods}}$ at the class of quasi-isomorphisms.

Dold–Kan correspondence

Let A be an ordinary ring in T. The normalized chain functor induces an equivalence of abelian categories $N:A-\operatorname {\mathrm {Mods}} \to C^{\leq 0}(A)$ ; see [Reference Illusie7, I 1.3]. It sends homotopic maps to homotopic maps and there are natural identifications $\pi _i(M)\simeq h^{-i}(NM)$ for a simplicial A-module M. In particular, N preserves quasi-isomorphisms and induces an equivalence $N:D^{\Delta }(A)\to D^{\leq 0}(A)$ .

Cones and distinguished triangles

As in [Reference Illusie7, I 3.2.1], let $\sigma $ denote the simplicial ${\mathbb {Z}}$ -module satisfying $N\sigma ={\mathbb {Z}}[1]$ and let $\gamma $ be the simplicial ${\mathbb {Z}}$ -module such that $N\gamma $ is the complex ${\mathbb {Z}}\to {\mathbb {Z}}$ concentrated in degrees $[-1,0]$ . Let A be a simplicial ring in the topos T and let E be an A-module. We write $\sigma E:=\sigma \otimes _{\mathbb {Z}} E$ and $\gamma E:=\gamma \otimes _{\mathbb {Z}} E$ . One has canonical isomorphisms $\pi _i(E) = \pi _{i+1}(\sigma E)$ for any $i\geq 0$ . We have a natural exact sequence of A-modules

$$\begin{align*}0\to E\xrightarrow{\iota} \gamma E\xrightarrow{q} \sigma E\to 0. \end{align*}$$

For a map $\alpha :E\to F$ of A-modules, we define

$$\begin{align*}\operatorname{Cone}^{\Delta}(\alpha):= \operatorname{Coker}(E\xrightarrow{(\iota,\alpha)} \gamma E\oplus F).\end{align*}$$

We have the sequence of natural maps

(2.4) $$ \begin{align} E\xrightarrow{\alpha} F\to \operatorname{Cone}^{\Delta}(\alpha)\to \sigma E, \end{align} $$

and the induced maps on homotopy groups fit into a long exact sequence. One declares a sequence $E\to F\to G\to \sigma E$ in $D^{\Delta }(A)$ to be a distinguished triangle if it is isomorphic in $D^{\Delta }(A)$ to a sequence of the form (2.4); see [Reference Illusie7, I 3.2.2]. If A is an ordinary ring in T, and E a simplicial A-module, then in $D^{\leq 0}(A)$ , we have natural isomorphisms $N\sigma E\simeq (NE)[1]$ , and the Dold–Kan correspondence preserves the notions of distinguished triangle.

Derived tensor product

Let A be a simplicial ring in T. The derived tensor product defines a functor $D^{\Delta }(A)\times D^{\Delta }(A)\to D^{\Delta }(A), (E,F)\mapsto E\otimes _A F$ , which can be computed as follows: For any quasi-isomorphism $L\to E$ where L is a flat (i.e., degreewise flat) A-module, the derived tensor product is computed by the (usual) tensor product $L\otimes _A F$ of A-modules (taken degree-wise). The analogous statement holds with a flat replacement of F. For fixed E, the functor $E\otimes -:D^{\Delta }(A)\to D^{\Delta }(A)$ is naturally triangulated, and similarly for $-\otimes E$ . If A is an ordinary ring and E, F are simplicial A-modules, then we have canonical natural isomorphisms $N(E\otimes ^{\ell } F) \simeq NE\otimes ^{L} NF$ in $D^{\leq 0}(A)$ , which are compatible with the symmetry isomorphism of the tensor product.

Now let $P\to B$ be a morphism of A-algebras. Then, the derived tensor product $B\otimes _{P} -$ induces a triangulated functor $D^{\Delta }(P)\to D^{\Delta }(B)$ , which is left-adjoint to the functor $D^{\Delta }(B)\to D^{\Delta }(P), N\mapsto N_P$ given by restriction of scalars.

We have the following:

Simplicial resolutions

Let $A\to B$ be a map of ordinary rings in a topos T. We denote by

$$\begin{align*}P_A(B)\end{align*}$$

the standard simplicial resolution of B over A [Reference Illusie7, I 1.5]. It is a simplicial A-algebra and flat over A in each degree. There is a natural quasi-isomorphism $P_A(B)\to B$ , where we regard B as a constant simplicial A-algebra in T.

We will use the following result.

Lemma 2.3. Let

be a commutative diagram of locally ringed topoi with enough points. Assume that a and b are flat. Then the natural map $a^{-1}P_{g^{-1}\mathcal {O}_{Y_1}}(\mathcal {O}_{W_1})\to P_{h^{-1}\mathcal {O}_{Y_2}}(\mathcal {O}_{W_2})$ of simplicial sheaves of rings on $W_2$ is flat in each degree.

Proof. This can be checked on stalks of $W_2$ . Since taking the standard simplicial resolution commutes with pullback of topoi and with filtered direct limits, we are reduced to the following setting: We have a diagram in the category of local rings

with $B_2$ flat over $B_1$ and $A_2$ flat over $A_1$ , and we need to show that the natural map $P_{A_1}(B_1)\to P_{A_2}(B_2)$ is degreewise flat. Denote this map by $F:P\to R$ with n-th part $F_n:P_n\to R_n$ for $n\geq 0$ . We also define $F_{-1}: B_1\to B_2$ . We show by induction on $n\geq -1$ that $F_n$ is flat and injective. The base case follows from the fact that a flat morphism of local rings is faithfully flat and therefore injective. By the construction of the standard simplicial resolution, we have that $P_{n+1}$ is the free polynomial algebra $A_1[P_n]$ over the set of elements of $P_n$ . The analogue is true for $R_{n+1}$ , and the map $F_{n+1}$ is the map $P_{n+1}=A_1[P_n]\to A_2[R_n]=R_{n+1}$ obtained by functoriality of this construction. The induction step then follows from Lemma 2.4 (note that $A_1\to A_2$ is injective since it is a flat map of local rings).

Lemma 2.4. Consider a commutative diagram of rings

where the map $A_1\to A_2$ is flat and injective and $B_1\to B_2$ is injective. Then the induced map between polynomial algebras $A_1[B_1]\to A_2[B_2]$ is flat and injective.

Module of principal parts

Let $A\to B$ be a map of rings in T. The (first) module of principal parts for the ring map $A\to B$ is given by $P_{B/A}^1:= (B\otimes _A B)/I_{\Delta }^2$ , where $I_{\Delta }$ is the kernel of the multiplication map $B\otimes _AB\to B$ . (Here, the tensor product is in general not a derived one). The A-module $P_{B/A}^1$ is naturally a $(B,B)$ -bimodule. Recall that $\Omega ^1_{B/A}=I_{\Delta }/I_{\Delta }^2$ , so that we have an exact sequence of $(B,B)$ -bimodules called the exact sequence of principal parts

$$\begin{align*}0\to \Omega^1_{B/A}\to P^1_{B/A}\to B \to 0.\end{align*}$$

Here, for each of the outer terms, the two B-module structures agree. We will denote this sequence by $\underline {P}^1_{B/A}$ . The map $b\mapsto b\otimes 1$ gives a splitting of this sequence for the left B-module structures, and $b\to 1\otimes b$ gives a splitting for the right B-module structures. Now let E be a B-module. Then we set $P^1_{B/A}(E):=P^1_{B/A}\otimes _B E$ , where we take the tensor product with respect to the right B-module structure on $P^1_{B/A}$ . Equivalently, $P_{B/A}^1(E)=B\otimes _A E/(I_{\Delta }^2 B\otimes _A E)$ . Then the sequence of principal parts for E is $\underline {P}^1_{B/A}(E):=\underline {P}^1_{B/A}\otimes _B E$ , or explicitly,

$$ \begin{align*} 0\to \Omega_{B/A}\otimes_B E\to P_{B/A}^1(E)\to E\to 0. \end{align*} $$

We usually regard this as a sequence of B-modules with respect to the left B-module structure. Note that it is in general not split.

Illusie’s Atiyah class

With these ingredients, we recall Illusie’s definition of Atiyah class: Let $f:X\to Y$ be a morphism of ringed topoi and let $E\in D^{\leq 0}(X)$ . Let $P:= P_{f^{-1}\mathcal {O}_Y}(\mathcal {O}_X)$ be the standard simplicial resolution. By the Dold–Kan correspondence, we may regard E as an object of $D^{\Delta }(\mathcal {O}_X)$ and thus of $D^{\Delta }(P)$ by restriction of scalars. We have the exact sequence of principal parts associated to $E_P$ , which is an exact sequence of P-modules:

$$\begin{align*}\underline{P}^1_{P/f^{-1}\mathcal{O}_Y}(E):\quad 0\to \Omega^1_{P/f^{-1}\mathcal{O}_Y}\otimes_P E_P\to P^1_{P/f^{-1}\mathcal{O}_Y}(E_P)\to E_P\to 0.\end{align*}$$

Note that the leftmost term here computes the derived tensor product since $\Omega _{P/f^{-1}\mathcal {O}_Y}$ is flat over P. Moreover, it is canonically quasi-isomorphic to the restriction of scalars of $L_{X/Y}\otimes _{\mathcal {O}_X} E$ . From the sequence $\underline {P}_{P/f^{-1}\mathcal {O}_Y}(E_P)$ , we obtain a morphism $E_P\to \sigma (L_{X/Y}\otimes _{\mathcal {O}_X} E)_P$ in $D^{\Delta }(P)$ . Extending scalars to $\mathcal {O}_X$ , this defines a canonical morphism $E\to \sigma L_{X/Y}\otimes _{\mathcal {O}_X} E$ in $D^{\Delta }(\mathcal {O}_X)$ . The Atiyah class is the corresponding morphism

$$\begin{align*}E\to L_{X/Y}[1]\otimes E\end{align*}$$

in $D^{\leq 0}(\mathcal {O}_X)$ obtained via the Dold–Kan correspondence.

Application to algebraic stacks

We apply the preceding discussion to the derived categories of algebraic stacks and the cotangent complex. This lays the groundwork for our definition of the Atiyah class on a stack using the modules of principal parts.

Situation 2.9. Consider the diagram (2.3) with the associated strictly simplicial algebraic spaces $X_{\bullet }$ and $Y_{\bullet }$ . Let $W:=X\times _{\mathcal {X}_Y} X$ , with associated projections $s,t:W\to X$ on the first and second factor, respectively, and let $h:W\to Y$ be the induced map. Let $W_{\bullet }$ be the strictly simplicial algebraic space associated to the covering $W\xrightarrow {s} X\to \mathcal {X}$ . One has canonical isomorphisms $W_n\simeq X_n\times _{\mathcal {X}_{Y_n}}X_n$ and maps $h_n:W_n\to Y_n$ . We denote by ${W}_{\parallel }$ the topos associated to the diagram

$$\begin{align*}W_{\bullet} \overset{s_{\bullet}}{\underset{t_{\bullet}}{\rightrightarrows}} X_{\bullet}.\end{align*}$$

It has a natural structure of ringed topos with flat pullback maps $s_{\bullet }^{\sharp }, t_{\bullet }^{\sharp }$ , and we write $\mathcal {O}_{{W}_{\parallel }}$ for the structure sheaf.

We denote by $({Y}_{\parallel },\mathcal {O}_{{Y}_{\parallel }})$ the ringed topos associated to the diagram

$$\begin{align*}Y_{\bullet}\rightrightarrows Y_{\bullet}\end{align*}$$

with both arrows the identity. There is a natural map of topoi ${h}_{\parallel }:{W}_{\parallel }\to {Y}_{\parallel }$ .

For E a complex of $\mathcal {O}_{\mathcal {X}}$ -modules, we write $E_{X_{\bullet }}:=\eta _X^{*}E$ and $E_{W_{\bullet }}:=\eta _W^{*}E$ . We have natural morphisms $s_{\bullet }^{*}E_{X_{\bullet }}\to E_{W_{\bullet }}$ and $t_{\bullet }^{*}E_{X_{\bullet }}\to E_{W_{\bullet }}$ . Thus, we get naturally a complex of $\mathcal {O}_{{W}_{\parallel }}$ -modules, which we denote $E_{{W}_{\parallel }}$ .

In Situation 2.9, we have the following properties:

Lemma 2.11. In $D_{qcoh}(W_{\bullet })$ , the object $\eta _{W}^{*}L_{\mathcal {X}/\mathcal {Y}}$ is naturally isomorphic to

$$\begin{align*}\operatorname{\mathrm{Cone}}_{\mathcal{O}_{{W}_{\parallel}}}(L_{{W}_{\parallel}/{Y}_{\parallel}})[-1]=\operatorname{Cone} \left(s_{\bullet}^{*}L_{X_{\bullet}/Y_{\bullet}}\oplus t_{\bullet}^{*}L_{X_{\bullet}/Y_{\bullet}}\xrightarrow{-s^{*}_{\bullet}+t_{\bullet}^{*}} L_{W_{\bullet}/Y_{\bullet}}\right)[-1].\end{align*}$$

This isomorphism is functorial in the diagram (2.3).

Proof. The displayed equality is immediate from the definition of ${W}_{\parallel }$ and the definition of cotangent complex for a morphism of topoi. By the construction of the cotangent complex and Remark 2.1, we have a canonical isomorphism

(2.5) $$ \begin{align} \eta_X^{*}L_{\mathcal{X}/\mathcal{Y}}\simeq L_{\mathcal{X}/\mathcal{Y},X/Y} = \operatorname{Cone}\left(L_{X_{\bullet}/Y_{\bullet}}\xrightarrow{-} \Omega_{X_{\bullet}/\mathcal{X}_{Y_{\bullet}}}\right)[-1]. \end{align} $$

We pull this isomorphism back via $s_{\bullet }$ . Due to the diagram

we have natural quasi-isomorphisms of complexes

$$\begin{align*}s_{\bullet}^{*}\Omega_{X_{\bullet}/\mathcal{X}_{Y_{\bullet}}} \simeq \Omega_{W_{\bullet}/X_{\bullet},t_{\bullet}}\leftarrow \operatorname{Cone}(t_{\bullet}^{*}L_{X_{\bullet}/Y_{\bullet}} \xrightarrow{t^{*}}L_{W_{\bullet}/Y_{\bullet}}).\end{align*}$$

We claim that the following diagram commutes:

where the upper horizontal map is given by the pullback $s^{*}:s_{\bullet }^{*}L_{X_{\bullet }/Y_{\bullet }}\to L_{W_{\bullet }/Y_{\bullet }}$ followed by the inclusion of $L_{W_{\bullet }/Y_{\bullet }}$ into the cone. Indeed, this follows from the observation that the map $s^{*}_{\bullet }L_{X_{\bullet }/Y_{\bullet }}\to \Omega _{W_{\bullet }/X_{\bullet },t_{\bullet }}$ factors through $L_{W_{\bullet }/Y_{\bullet }}$ . It follows from this that, after pulling back the right-hand side of (2.5) along $s_{\bullet }$ , the result is naturally quasi-isomorphic to

$$\begin{align*}\operatorname{Cone}\left(s_{\bullet}^{*}L_{X_{\bullet}/Y_{\bullet}}\xrightarrow{-a} \operatorname{Cone}\left(t_{\bullet}^{*}L_{X_{\bullet}/Y_{\bullet}}\xrightarrow{t^{*}} L_{W_{\bullet}/Y_{\bullet}}\right)\right)[-1].\end{align*}$$

An easy calculation shows that this iterated cone is identical to

$$\begin{align*}\operatorname{Cone} \left(s_{\bullet}^{*}L_{X_{\bullet}/Y_{\bullet}}\oplus t_{\bullet}^{*}L_{X_{\bullet}/Y_{\bullet}}\xrightarrow{-s^{*}_{\bullet}+t_{\bullet}^{*}} L_{W_{\bullet}/Y_{\bullet}}\right)[-1],\end{align*}$$

as desired. The compatibility with pullback follows from the compatibility of (2.5) and the usual pullback compatibilities of the cotangent complex for algebraic spaces.

Remark 2.12. In Situation 2.9, suppose that R is a simplicial ring on ${W}_{\parallel }$ with a given map $R\to \mathcal {O}_{{W}_{\parallel }}$ . Then we have the following diagram of functors, which commutes up to canonical natural isomorphisms:

(2.6)

For the following lemma, let $W_{\wedge }$ be the ringed topos associated to the diagram $X_{\bullet }\xleftarrow {s_{\bullet }} W_{\bullet }\xrightarrow {t_{\bullet }} X_{\bullet }$ . Recall that we have a natural restriction functor $D({W}_{\parallel })\to D(W_{\wedge })$ . The following will be used in the proof of Lemma 4.4.

Proof. Let $W_{\wedge ,lis-et}$ be the ringed topos associated to the diagram

$$\begin{align*}X_{\bullet, lis-et}\xleftarrow{s_{\bullet}}W_{\bullet, lis-et}\xrightarrow{t_{\bullet}} X_{\bullet, lis-et}.\end{align*}$$

From the diagram of ringed topoi

we see that there is an induced morphism $\epsilon _{\wedge }: {W}_{\wedge ,lis-et} \to W_{\wedge }$ . Let $F_{\wedge ,lis-et}=(\pi _X^{*}F,\pi _W^{*}F, \pi _X^{*}F)$ , with pullback maps induced by the identifications $\pi _W^{*}=s^{*}\pi _X^{*}$ and $\pi _W^{*}=t^{*}\pi _X^{*}$ , respectively. Since the functors $\epsilon _*$ are exact, we have $F_{\wedge }={\epsilon }_{\wedge *}F_{\wedge , lis-et}$ . In particular, since $\epsilon _*\epsilon ^{*}$ is naturally isomorphic to the identity, it is enough to show that there is a natural quasi-isomorphism $\epsilon _{\wedge }^{*}E_{W_{\wedge }}\to F_{\wedge ,lis-et}$ .

We are reduced to the following situation: Let $G=(G_{X},G_W,G_{X},s^{*},t^{*})$ be a complex of $\mathcal {O}_{W_{\wedge , lis-et}}$ -modules such that $G_X$ has quasicoherent cohomology sheaves, and such that $s^{*}$ , $t^{*}$ are quasi-isomorphisms, and suppose that $F=R(\pi _W)_*G_W$ . Then we need to show that G is naturally isomorphic to $F_{\wedge ,lis-et}$ . Without loss of generality, we may assume that G is K-injective. Then $G_W$ and $G_X$ are themselves K-injective. We can therefore assume that $F=\pi _{W*}G_W$ . Let also $F':=\pi _{X*}G_X$ . The morphism $s^{*}:s^{*}G_X\to G_W$ corresponds to a morphism $G_X\to s_*G_W$ , and by applying $\pi _{W*}$ we obtain a map $\sigma :F'\to F$ . Similarly, we obtain $\tau :F'\to F$ from $t^{*}:t^{*}G_X\to G_W$ . The maps $\sigma $ and $\tau $ are quasi-isomorphisms, as one can check after applying $\pi _W^{*}$ , since $F'$ and F have quasi-coherent comohology.

Then we have the following commutative diagram of complexes of $\mathcal {O}_{W_{\bullet ,lis-et}}$ -modules:

The horizontal maps going to the right are the quasi-isomorphisms coming from the push-pull adjunctions associated to $\pi _X$ and $\pi _W$ , respectively. Setting $\widetilde {F}_{\wedge , lis-et}= (\pi _X^{*}F', \pi _W^{*}F, \pi _X^{*}F', s^{*}\pi _X^{*}\sigma , t^{*}\pi _X^{*}\tau )$ , it follows that we have quasi-isomorphisms

$$\begin{align*}F_{\wedge, lis-et} \leftarrow\widetilde{F}_{\wedge,lis-et}\to G,\end{align*}$$

as desired.

Traces in a closed symmetric monoidal category

Let $\mathcal {C}$ be a symmetric closed monoidal category with product $-\otimes -$ , unit $\mathcal {O}$ , and internal Hom-functor $\mathcal {H}om(-,-)$ . Let $\tau $ denote the symmetry morphism of the tensor product. We let $E,F,G$ denote arbitrary elements of $\mathcal {C}$ and write $E^{\vee }:=\mathcal {H}om(E,\mathcal {O})$ . Recall that we have an adjunction between $-\otimes E$ and $\mathcal {H}om(E,-)$ . We have various natural maps in $\mathcal {C}$ :

1. (1) Evaluation. We have a natural map $ev: \mathcal {H}om(E,F)\otimes E\to F$ , corresponding to the identity on $\mathcal {H}om(E,F)$ under adjunction.

2. (2) As a special case, we get the evaluation map $ev:E^{\vee }\otimes E\to \mathcal {O}$ .

3. (3) Composition. We have a natural map $comp: \mathcal {H}om(F,G)\otimes \mathcal {H}om(E,F)\to \mathcal {H}om(E,G)$ . This map is adjoint to the map $\mathcal {H}om(F,G)\otimes \mathcal {H}om(E,F)\otimes E\to G$ which is obtained from composing two evaluation maps.

4. (4) As a special case, we get the composition map $comp:F\otimes E^{\vee }\to \mathcal {H}om(E,F)$ , where we use the canonical isomorphism $F\simeq \mathcal {H}om(\mathcal {O},F)$ .

5. (5) Diagonal. We have a diagonal map $s:\mathcal {O}\to \mathcal {H}om(E,E)$ which corresponds to $\operatorname {id}_{E}$ under adjunction.

We recall the notion of dualizable object:

For dualizable E and arbitrary F, the canonical map $comp: F\otimes E^{\vee }\to \mathcal {H}om(E,F)$ is an isomorphism, and we will often use this isomorphism to identify the source and target.

Let E be a dualizable object of $\mathcal {C}$ . Then we have further natural maps

1. (1) Trace I. We define the trace map $\operatorname {tr}:\mathcal {H}om({E},{E})\to \mathcal {O}$ as the composition $\mathcal {H}om(E,E)\simeq {E}\otimes {E}^{\vee }\xrightarrow {\tau } E^{\vee }\otimes {E}\xrightarrow {ev} \mathcal {O}$ .

2. (2) Trace II. We have a map $\operatorname {tr}:\mathcal {H}om({E}, {F} \otimes {E})\to {F}$ given by the composition $\mathcal {H}om({E}, {F}\otimes {E})\simeq {F}\otimes {E}\otimes {E}^{\vee }\xrightarrow {{F}\otimes tr}{F}$ . By abuse of language, we call this also the trace map.

3. (3) Diagonal II. We have the composition $\eta :=comp^{-1}\circ s:\mathcal {O}\to {E}\otimes {E}^{\vee }$ , which we will also call diagonal.

A morphism $f:{E}\to {F}\otimes {E}$ corresponds by adjunction to a map $\mathcal {O}\to \mathcal {H}om({E},{F}\otimes {E})$ . When E is dualizable, we define $\operatorname {tr}_{f}$ as the composition $\mathcal {O}\to \mathcal {H}om({E},{F}\otimes {E})\xrightarrow {\operatorname {tr}} {F}.$

As a consequence of the definition and the naturality of adjunction, we have the following:

Lemma 2.15. For $f:{E}\to {F}\otimes {E}$ , the map $tr_{f}$ is equal to the composition

$$\begin{align*}\mathcal{O}\xrightarrow{\eta} {E}\otimes {E}^{\vee}\xrightarrow{f\otimes \operatorname{id}_{{E}^{\vee}}} {F}\otimes {E}\otimes {E}^{\vee} \xrightarrow{\operatorname{id}_{F} \otimes \tau} {F}\otimes {E}^{\vee} \otimes {E}\xrightarrow{\operatorname{id}_{{F} }\otimes ev} {F}, \end{align*}$$

where $\tau $ denotes the symmetry isomorphism of the tensor product.

Proof. The composition of the final two maps is by definition equal to $\operatorname {id}_F\otimes \operatorname {tr}$ . Now consider the following diagram:

The left and right triangles commute by definition, while the square commutes by naturality of the composition map. By naturality of adjunction, the first two arrows in the lower row compose to the map adjoint to f. Thus, by definition, the lower row composes to $\operatorname {tr}_f$ , and the same must be true for the composition along the top.

We will need the following compatibility of traces with tensor products

Lemma 2.16. Let $f:E\to F\otimes E$ be a morphism in $\mathcal {C}$ with E dualizable and let V be another dualizable object of $\mathcal {C}$ . Then

$$\begin{align*}\operatorname{tr}(f\otimes \operatorname{id}_V) = \operatorname{tr}(f)\circ \operatorname{tr}(\operatorname{id}_V).\end{align*}$$

Criterion for additivity of traces

We now assume that, say, $\mathcal {C}=D(R)$ is the derived category of a category of sheaves of modules over a ring R in a topos T. In particular, it is closed monoidal with a compatible triangulated structure. Let ${F}, {E}, {G}$ be dualizable objects of $D(R)$ that are part of an exact triangle

(2.7) $$ \begin{align} F\to E\to G\to F[1]. \end{align} $$

Suppose that we have a commutative diagram for some element $L\in D(R)$ :

where the lower row is obtained from the upper by tensoring with L. We want a criterion that allows us to conclude that $tr(f)=tr(g)+tr(h)$ . We follow the strategy in [Reference May13, §8].

We will make the following

Situation 2.18. Suppose Assumption 2.17 holds. (This is likely unnecessary if one works with K-flat complexes in what follows.) Choose a representation of (2.7) by a short exact sequence $0\to {F}\to {E}\to {G}\to 0$ of bounded above complexes of flat R-modules (or suppose one is given). Further, choose an injective resolution J of R, so that the derived dual of a complex is explicitly realized by $(-)^{\vee }:=\mathcal {H}om_R(-,J)$ . Then we have again an exact sequence $0\to G^{\vee }\to {E}^\vee \to {F}^{\vee }\to 0$ . We define complexes W and $\overline {W}$ via

$$ \begin{align*} W&:= {E}\otimes {E}^{\vee}/{F}\otimes {G}^{\vee}\\ \overline{W}&:= {E}^{\vee}\otimes {E}/{G}^\vee \otimes {F}. \end{align*} $$

The tensor product here is the ordinary tensor product of complexes, which represents the derived tensor product due to the choice of ${F}$ and ${E}$ . We have a natural isomorphism $W\to \overline {W}$ coming from the symmetry isomorphism of the tensor product. Moreover, we have natural inclusions of complexes

$$ \begin{align*} {F}^{\phantom{\vee}}\otimes {F}^{\vee}\hookrightarrow &\, W\hookleftarrow \, {G}^{\phantom{\vee}} \otimes {G}^{\vee}, \end{align*} $$

and

$$ \begin{align*} {F}^{\vee}\otimes {F}^{\phantom{\vee}}\hookrightarrow&\, \overline{W} \hookleftarrow \, {G}^{\vee}\otimes {G}. \end{align*} $$

The inclusions into W are given by

$$\begin{align*}{F}\otimes {F}^{\vee}\simeq {F}\otimes {E}^{\vee}/{F}\otimes {G}^{\vee}\subset W\end{align*}$$

and

$$\begin{align*}{G}\otimes {G}^{\vee}\simeq {E}\otimes {G}^{\vee}/{F}\otimes {G}^{\vee}\subset W,\end{align*}$$

respectively, and analogously for the inclusions into $\overline {W}$ .

Now we have the following result:

Proposition 2.20 [Reference May13, §8].

Suppose there exists a dotted arrow making its two adjacent squares in the following diagram commute. Then $tr(f)=tr(g)+tr(h)$ .

A diagram of exact sequences

Let $\mathcal {A}$ be an abelian category and consider the following diagram in $\mathcal {A}$ in which the solid arrows commute:

(3.1)

Assume that s is a section of e (i.e., $e\circ s =\operatorname {\mathrm {id}}_{E"}$ ). Then, the composition $q\circ s$ induces a morphism $\delta :\operatorname {\mathrm {Ker}}(q")\to \operatorname {\mathrm {Ker}}(f)$ .

Now suppose this diagram extends to a commutative diagram in which all solid rows and columns are exact and where the dotted arrows give a spliting of the middle exact sequence

Then we have the morphism $\delta : F"\to G'$ as described above. From the composition $r\circ j: F\to E'$ , we get an induced map $\operatorname {Coker}(a)\to \operatorname {Coker}(j')$ and thus another map $F"\to G'$ . By a diagram chase, one checks this to be equal to $-\delta $ .

Now assume that, further, $\mathcal {A}$ is the category of modules over a simplicial ring A in a topos. Then, the above morphisms fit into the following diagrams of triangles in $D^{\Delta }(A)$ :

(3.2)

This shows that composing $\delta $ with the connecting map $G'\to \sigma F'$ yields minus the connecting map associated to the exact sequence of the F’s, while composing $\sigma \delta $ with the connecting map $G"\to \sigma F"$ yields the connecting map associated to the exact sequence of the G’s.

The reduced Atiyah class

Now let $f:X\to Y$ be a morphism of ringed topoi. Let $E\in C^{\leq 0}(\mathcal {O}_Y)$ be a complex whose components are Tor-independent to f, so that, in particular, the component-wise pullback $E_X:=f^{*}E$ equals the derived pullback. Let

(3.3) $$ \begin{align} 0\to F\to E_X\to G\to 0 \end{align} $$

be an exact sequence in $C^{\leq 0}(\mathcal {O}_X)$ , which we also view as a sequence of simplicial $\mathcal {O}_X$ -modules via the Dold–Kan correspondence. Let $R:=P_{f^{-1}\mathcal {O}_Y}(\mathcal {O}_X)$ be the standard simplicial resolution and let $E_R:=f^{-1}E\otimes _{f^{-1}\mathcal {O}_Y} R$ . Denote by $\cdot \mid _R$ restriction of scalars from $\mathcal {O}_X$ to R. Since R is flat, the natural morphism $E_R\to E_X|_R$ is a quasi-isomorphism, and it is termwise surjective, since $R\to \mathcal {O}_X$ is. Let $G_R:=G|_R$ and let $F_R$ be the kernel of the induced map $E_R\to G_R$ . We have an induced map of exact sequences of R-modules, in which the vertical arrows are quasi-isomorphisms

We now take the exact sequence of principal parts with respect to the upper row. This gives the following commutative diagram of solid arrows with exact rows and columns:

(3.4)

Here, the arrows denoted by s and r come from a splitting of the middle exact sequence, which is defined as follows: Since $E_R = R\otimes _{f^{-1}\mathcal {O}_Y}f^{-1}E$ , we have by definition

$$ \begin{gather*} P^1_{R/f^{-1}\mathcal{O}_Y}(E_R) = (R\otimes_{f^{-1}\mathcal{O}_Y} R/I_{\Delta}^2)\otimes_R E_R \\ = (R\otimes_{f^{-1}\mathcal{O}_Y} R \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}E)/(I_{\Delta}^{2}\cdot R\otimes R\otimes f^{-1}E). \end{gather*} $$

Then s is given on local sections by $a\otimes m\mapsto a\otimes 1 \otimes m$ , which one checks to be a morphism of left R-modules. We see that the lower right square of (3.4) is of the form (3.1). In particular, we get the induced morphism

(3.5) $$ \begin{align} \delta: F_R\to \Omega_{R/f^{-1}\mathcal{O}_Y}\otimes_R G_R. \end{align} $$

By passing to derived categories, and taking extensions of scalars along $R\to \mathcal {O}_X$ , this corresponds to a morphism

$$\begin{align*}\overline{\operatorname{\mathrm{at}}}_{E,X/Y,G}:F\to L_{X/Y}\otimes G\end{align*}$$

in $D^{\Delta }(\mathcal {O}_X)$ , or equivalently $D^{\leq 0}(\mathcal {O}_X)$ , which we call the reduced Atiyah class of E over Y with respect to the sequence (3.3) on X. We also write $\overline {\operatorname {\mathrm {at}}}_E$ if the remaining data is understood.

Remark 3.1. The morphism $\delta :F_R\to \Omega _{R/f^{-1}\mathcal {O}_Y}\otimes _R G_R$ obtained by using the lower right corner of (3.4) is explicitly given as follows: For a local section $f=\sum r_i\otimes f^{-1}e_i$ , where $r_i$ are sections of R and $e_i$ are sections of E (over Y), we have

$$\begin{align*}\delta:f\mapsto \sum dr_i\otimes \overline{e_i}.\end{align*}$$

Lemma 3.4 (Shift invariance).

Consider the reduced Atiyah class $\overline {\operatorname {\mathrm {at}}}_{E[1]}$ associated to the shift of the sequence (3.3). Then, the following diagram commutes:

where $\tau $ denotes the compatibility morphism making the tensor product $L_{X/Y}\otimes -$ into a triangulated functor.

Proof. On the level of simplicial modules, the shift functor corresponds to taking a tensor product $\sigma \otimes _{\mathbb {Z}}- $ , where $\sigma $ is the simplicial ${\mathbb {Z}}$ -module defined in §2.2. The statement then follows from the fact that for any R-module M, we have canonical isomorphisms of exact sequences, functorial in M, which are induced by the properties of the tensor product:

We extend the definition of the reduced Atiyah class to objects in $D^-(X)$ (cf. Construction 5.6 for the analogue for the usual Atiyah class):

Construction 3.5. Let $f:X\to Y$ be a morphism of ringed topoi, let $E\in D^{-}(Y)$ , with $E_X:=f^{*}E$ and let

(3.6) $$ \begin{align} F\to E_X\to G\xrightarrow{+1} F[1] \end{align} $$

be an exact triangle in $D^-(X)$ . Assume that $\operatorname {\mathrm {Ext}}^{-1}(F,G) = 0$ . Choose $N>0$ so that the $E[N],F[N]$ and $G[N]$ all lie in $D^{\leq 0}(X)$ . By Remark 3.3, we obtain a well-defined map

$$\begin{align*}\overline{\operatorname{\mathrm{at}}}_{E[N]}: F[N]\to L_{X/Y}\otimes (G[N]).\end{align*}$$

We define

$$\begin{align*}\overline{\operatorname{\mathrm{at}}}_{E}:= (\tau^{(N)}\circ \operatorname{\mathrm{at}}_{E[N]}) [-N],\end{align*}$$

where $\tau ^{(N)}: L_{X/Y}\otimes (G[N])\to L_{X/Y}\otimes G [N]$ is the N-fold application of the compatibility map of tensor product with the shift functor. This is independent of choice of N by Lemma 3.4.

Well-definedness and compatibility with pullback

Suppose that we are given a map of diagrams (2.3) – that is, that we have a $2$ -commuting cube

(4.4)

whose front and back faces are as in (2.3). (This means, in particular, that for any two maps $X'\to \mathcal {Y}$ obtained by traveling along the edges of the cube, the two two-isomorphisms relating them by traversing the faces are identical.)

Lemma 4.4. Let $E\in D^{\leq 0}_{qcoh}(\mathcal {X})$ . Let

$$\begin{align*}\operatorname{\mathrm{at}}_E:=\operatorname{\mathrm{at}}_{E,\mathcal{X}/\mathcal{Y},X/Y} \mbox{ and } \operatorname{\mathrm{at}}_{A^{*}E}:=\operatorname{\mathrm{at}}_{A^{*}E,\mathcal{X}'/\mathcal{Y}',X'/Y' }.\end{align*}$$

Then the diagram

commutes.

Proof. By using the setup of Situation 2.9 and Construction 4.1 for the primed objects, we obtain a commutative cube of simplicial algebraic spaces

as well as morphisms of topoi ${a}_{\parallel }:{W}_{\parallel }^{\prime }\to {W}_{\parallel }$ and ${Y}_{\parallel }^{\prime }\to {Y}_{\parallel }$ , which fit in a $2$ -commutative square

By construction, the Atiyah class $\operatorname {\mathrm {at}}_E$ corresponds via $\eta _W^{*}$ to a map $\alpha :E_{W_{\bullet }}\to \eta _W^{*}L_{\mathcal {X}/\mathcal {Y}}[1]\otimes E_{W_{\bullet }}$ , obtained as the shift of the map (4.2). Since the pullback $A^{*}$ can be computed as $\eta _{W'*}a_{\bullet }^{*}\eta _W^{*}$ , we have a natural commutative diagram

Let $E':=\eta _{W'*}(a_{\bullet }^{*}E_W)$ . We apply Construction 4.1 to the primed objects (i.e., with respect to the backside of (4.4) and $E'$ ). We obtain the ring $R'=P_{{h}_{\parallel }^{\prime -1}\mathcal {O}_{{Y}_{\parallel }^{\prime }}}(\mathcal {O}_{{W}_{\parallel }^{\prime }})$ and the map

$$\begin{align*}\alpha':E^{\prime}_{W^{\prime}_{\bullet}}\to \eta_{W'}^{*}L_{\mathcal{X}'/\mathcal{Y}'}[1]\otimes E^{\prime}_{W^{\prime}_{\bullet}},\end{align*}$$

as the shift of the map (4.2). By functoriality of simplicial resolutions, we have a natural map ${a}_{\parallel }^{-1}R\to R'$ of rings on ${W}_{\parallel }^{\prime }$ . Now the statement of the lemma is equivalent to the following:

Claim 4.5. The diagram

in $D(W^{\prime }_{\bullet })$ commutes, where the vertical maps are induced by the natural isomorphism $a_{\bullet }^{*}E_{W_{\bullet }}\xrightarrow {\sim } E^{\prime }_{W'}$ and the pullback map on cotangent complexes.

Proof. We may assume that E is represented by a complex of flat modules concentrated in degrees $\leq 0$ , so that we have an induced representative with flat terms for $E_{{W}_{\parallel }}$ . Then the pullback ${a}_{\parallel }^{*}E_{{W}_{\parallel }}={a}_{\parallel }^{*}E_{{W}_{\parallel }}$ can be computed termwise, and its components are given by $b_{\bullet }^{*}E_X$ and $a_{\bullet }^{*}E_W$ , respectively.

By functoriality of the principal parts construction, we have a morphism of exact sequences of ${a}_{\parallel }^{-1}R$ -modules ${a}_{\parallel }^{-1}\underline {P}^1_{\mathcal {O}_{{W}_{\parallel }}/{h}_{\parallel }^{-1}\mathcal {O}_{{Y}_{\parallel }}}(E_{{W}_{\parallel }})\to \underline {P}^1_{\mathcal {O}_{{W}_{\parallel }^{\prime }}/{h'}_{\parallel }^{-1}\mathcal {O}_{{Y'}_{\parallel }}}({a}_{\parallel }^{*}E_{{W}_{\parallel }}).$ We get the induced diagram of connecting maps in $D^{\Delta }({a}_{\parallel }^{-1}R)$

By adjunction, this corresponds to a diagram in $D^{\Delta }(R')$ , which in turn corresponds to a diagram in $D(\mathcal {O}_{{W}_{\parallel }})$ :

After applying $\operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W'}_{\parallel }}}$ and shifting, the upper line is canonically identified with $a_{\bullet }^{*}\alpha $ (this uses that we have a natural isomorphism $\operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W'}_{\parallel }}}\circ {a}_{\parallel }^{*}\simeq a_{\bullet }^{*}\circ \operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W}_{\parallel }}}$ , which one can check by computing with K-flat complexes of $\mathcal {O}_{{W}_{\parallel }}$ modules). The lower line yields a map

$$\begin{align*}\alpha":a_{\bullet}^{*}E_{W_{\bullet}}\to \eta_{W'}^{*}L_{\mathcal{X}'/\mathcal{Y}'}[1]\otimes a_{\bullet}^{*}E_{W_{\bullet}}.\end{align*}$$

To finish the proof of the claim, we need to show that under the isomorphism $a^{*}E_{W}\to E^{\prime }_{W'}$ , the maps $\alpha "$ and $\alpha $ get identified. This follows from Lemma 2.13 and Remark 4.3 ii).

Proof. Suppose we are given two choices

We need to show that $\operatorname {\mathrm {at}}_{E,\mathcal {X}/\mathcal {Y},X_1/Y_1}$ and $\operatorname {\mathrm {at}}_{E,\mathcal {X}/{\mathcal {Y}},X_2/Y_2}$ agree. By replacing $X_2\to Y_2$ with the fiber product $X_{1}\times _{\mathcal {X}}X_2\to Y_1\times _{\mathcal {Y}}Y_2$ , we may without loss of generality assume that we have a $2$ -commuting diagram

By adding in the identity morphisms on $\mathcal {X}$ and $\mathcal {Y}$ , this gives a $2$ -commuting cube (4.4), and we obtain the desired equality from Lemma 4.4.

Corollary 4.7. The Atiyah class is compatible with pullback. More precisely, given a $2$ -commutative square

the natural diagram

commutes.

Compatibility with Illusie’s definition

Proof. In this case, we may choose $X=\mathcal {X}$ and $Y=\mathcal {Y}$ in diagram (2.3), so that we get $W=X$ . Then $X_{\bullet }$ and $W_{\bullet }$ are constant strictly simplicial algebraic spaces with value X, and $Y_{\bullet }$ is the constant strictly simplicial algebraic space with value Y. In this case, we have the morphism of topoi $\Delta :{W}_{\parallel }\to X_{\bullet }$ , where $\Delta ^{*}M_X = (M_X,M_X,\operatorname {id},\operatorname {id})$ . The morphism $c_E$ of Lemma 2.6 gives a natural transformation of functors $\operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W}_{\parallel }}}\circ \Delta ^{*}\Rightarrow [-1]$ , which induces to a natural isomorphism of functors from $D(\mathcal {O}_{X_{\bullet }})$ to itself. It follows from Construction 4.1 that the morphism $E_{{W}_{\parallel }} \to L_{{W}_{\parallel }/{Y}_{\parallel }}[1]\otimes _{\mathcal {O}_{{W}_{\parallel }}} E_{{W}_{\parallel }}$ obtained from (4.1) by applying the functors $D^{\Delta }(R)\to D^{\Delta }(\mathcal {O}_{{W}_{\parallel }})\to D({W}_{\parallel })$ is naturally identified with the pullback $\Delta ^{*}\operatorname {\mathrm {at}}_{E_{X_{\bullet }}, X_{\bullet }/Y_{\bullet }}$ of the Atiyah class for the map of topoi $X_{\bullet }\to Y_{\bullet }$ . Thus, taking the cone, we see that (4.2) is identified with $\operatorname {\mathrm {at}}_{E_{X_{\bullet }}, X_{\bullet }/Y_{\bullet }}[1]$ . From the commutative diagram of topoi

we see that $\eta _{X*}\operatorname {\mathrm {at}}_{E_{X_{\bullet }}, X_{\bullet }/Y_{\bullet }}$ is naturally identified with the pullback to the lisse-étale site of the usual Atiyah class for the map $X_{\operatorname {et}}\to Y_{\operatorname {et}}$ , from which the result follows.

As a first application of our definition, we compute the Atiyah class of the universal vector bundle on $BGL_n$ over a chosen base field k.

Example 4.9. Let $\mathcal {V}$ denote the universal rank n locally free sheaf on $BGL_n$ , which is the sheaf of sections of the vector bundle associated to the universal principal $GL_n$ -bundle. We have the $2$ -cartesian diagram, where the map x from $\operatorname {\mathrm {Spec}} k$ is the one corresponding to the trivial $GL_n$ -torsor

There is a natural trivialization of the pullback $\mathcal {V}_{\operatorname {\mathrm {Spec}} k}$ . Then for every T-valued point $g=(g_{i,j})$ on $GL_n$ , the $2$ -morphism $\rho $ indicated in the diagram induces a natural pullback map $\rho ^{*}: t^{*}\mathcal {V}_{\operatorname {\mathrm {Spec}} k}|_T\xrightarrow {\sim }s^{*}\mathcal {V}_{\operatorname {\mathrm {Spec}} k}|_T$ . With respect to the given trivialization of $\mathcal {V}$ over $\operatorname {\mathrm {Spec}} k$ , this morphism is just given by $g^{-1}:\mathcal {O}_T^{\oplus n} \to \mathcal {O}_T^{\oplus n}$ .

We now compute the Atiyah class of $\mathcal {V}$ using Construction 4.1, where we chose $X=Y=\operatorname {\mathrm {Spec}} k$ , so that we get $W=GL_n$ . More precisely, we will calculate the restriction of $\operatorname {\mathrm {at}}_{\mathcal {V}}$ to the etale site of W, so that we do not have to consider the associated strictly simplicial algebraic spaces. Thus, here we let ${W}_{\parallel }$ and ${Y}_{\parallel }$ be the topos associated to the diagrams

$$\begin{align*}W\underset{t}{\overset{s}{\rightrightarrows}}X \mbox{ and } {Y}\rightrightarrows Y,\end{align*}$$

respectively, and ${h}_{\parallel }:{W}_{\parallel }\to {Y}_{\parallel }$ the natural map, and we let $\mathcal {V}_{{W}_{\parallel }}$ denote the locally free sheaf on ${W}_{\parallel }$ obtained by pullback. First, to calculate $\operatorname {\mathrm {at}}_{\mathcal {V}_{{W}_{\parallel }},{W}_{\parallel }/{Y}_{\parallel }}$ , we do not need to take a simplicial resolution of $\mathcal {O}_{{W}_{\parallel }}$ over ${h}_{\parallel }^{-1}\mathcal {O}_{{Y}_{\parallel }}$ since the map ${W}_{\parallel }\to {Y}_{\parallel }$ is composed of smooth maps of algebraic spaces. Moreover, since $\operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W}_{\parallel }}}$ preserves mapping cones of complexes, we may calculate $\operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W}_{\parallel }}}(\operatorname {\mathrm {at}}_{\mathcal {V}_{{W}_{\parallel }},{W}_{\parallel }/{Y}_{\parallel }})$ by first applying $\operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W}_{\parallel }}}$ to the sequence of principal parts $P_{{W}_{\parallel }/{Y}_{\parallel }}(\mathcal {V}_{{W}_{\parallel }})$ and only then taking connecting homomorphisms. Writing out the pullback maps in the sequence of principal parts $P_{{W}_{\parallel }/{Y}_{\parallel }}(\mathcal {V}_{{W}_{\parallel }})$ gives the following diagram on W, where we use that $\Omega _{X/Y}=0$ :

Since $\mathcal {V}_W$ is pulled back from $X=Y=\operatorname {\mathrm {Spec}} k$ via s, we have a natural splitting $P_{W/Y}(\mathcal {V}_W)=\Omega _{W/Y}\otimes s^{*}\mathcal {V}_{X}\oplus s^{*}\mathcal {V}_X$ . With respect to this splitting, one calculates that the induced pullback map along t is given by

$$\begin{align*}t^{*}\mathcal{V}_{X} \xrightarrow{(d\rho, \rho)} \Omega_{W/Y}\otimes s^{*}\mathcal{V}_{X}\oplus s^{*}\mathcal{V}_X. \end{align*}$$

It follows, that after applying $\operatorname {\mathrm {Cone}}_{\mathcal {O}_{{W}_{\parallel }}}$ , the resulting exact sequence has the following exact subsequence, and the termwise inclusions give quasi-isomorphisms:

$$\begin{align*}0\to \Omega_{W/Y}\otimes s^{*}\mathcal{V}_X\to \operatorname{Cone}(t^{*}\mathcal{V}_X\xrightarrow{d\rho}\Omega_{W/Y}\otimes s^{*}\mathcal{V}_X) \to t^{*}\mathcal{V}_X[1]\to 0. \end{align*}$$

It is a general fact that for an exact sequence of this form, the connecting map is canonically quasi-isomorphic to minus the shift of the map the cone is taken over for the middle term – that is, in our case, this gives

$$\begin{align*}d\rho[1]: t^{*}\mathcal{V}_X[1]\to \Omega_{W/Y}\otimes s^{*}\mathcal{V}_X[1]. \end{align*}$$

In particular, we find that there is a natural isomorphism $s^{*}x^{*}L_{BGL_n}[1]\simeq \Omega _{W/Y}$ and that using this identification and the trivialization of $\mathcal {V}_X$ , we have that $s^{*}x^{*}\operatorname {\mathrm {at}}_{\mathcal {V}}$ is given by

(4.5) $$ \begin{align} \mathcal{O}_{GL_n}^{\oplus n}\xrightarrow{ d(T^{-1})\circ T} \Omega_{GL_n}\otimes \mathcal{O}_{GL_n}^{\oplus n}, \end{align} $$

where $T=(T_{ij})$ is the universal matrix on $GL_n$ . In particular, for $n=1$ , we have the following:

Example 4.10. By taking the trace in (4.5), we obtain the section

$$\begin{align*}\mathcal{O}_{GL_n}\to \Omega_{GL_n}\end{align*}$$

given by $\operatorname {tr}(d(T^{-1})T)$ . This is equal to

$$\begin{align*}d(\det T^{-1}) \det T.\end{align*}$$

In particular, for $n=1$ , we have $T=(t)$ as a $1\times 1$ matrix, and the pullback of $\operatorname {\mathrm {at}}_{\mathcal {L}}\otimes \mathcal {L}^{-1}$ – with $\mathcal {L}$ the universal rank $1$ sheaf – to $\mathbb {G}_m=GL_1$ is given by

$$\begin{align*}\mathcal{O}_{\mathbb{G}_m}\to \Omega_{\mathbb{G}_m},\end{align*}$$

sending $1$ to $d(1/t) t=-dt/t$ . We have the natural map $DET:BGL_n\to B\mathbb {G}_m$ given by associating to a locally free sheaf its determinant line bundle. By taking fiber products with a point, this induces a map $GL_n\to \mathbb {G}_m$ , (which is again just the determinant) whose pullback map on differentials is given by $dt\mapsto d(\det T)$ . In particular, we have $td(1/t)\mapsto \det T d(\det T^{-1})$ . This shows that we have a natural identification $\operatorname {tr}(s^{*}x^{*}\operatorname {\mathrm {at}}_{\mathcal {V}})= s^{*}x^{*}(\operatorname {\mathrm {at}}_{\det \mathcal {V}}\otimes \det \mathcal {V}^{-1})$ . One can show that this isomorphism is part of a descent datum, and thus we have

$$\begin{align*}\operatorname{tr}(\operatorname{\mathrm{at}}_{\mathcal{V}})= \operatorname{\mathrm{at}}_{\det \mathcal{V}}\otimes \det(\mathcal{V})^{-1}.\end{align*}$$