2.1 Mixed Hochschild complexes

For a CDG category ${\mathcal {A}}$ , we use the following notation:

• ○ $C ({\mathcal {A}}) \ (\mathrm {MC} ({\mathcal {A}}))$ : (mixed) Hochschild complex.

• ○ $\overline {C} ({\mathcal {A}})\ (\overline {\mathrm {MC}} ({\mathcal {A}}))$ : (mixed) normalized Hochschild complex.

• ○ $C ^{II} ({\mathcal {A}}) \ (\mathrm {MC}^{II} ({\mathcal {A}}))$ : (mixed) Hochschild complex of the second kind.

• ○ $\overline {C} ^{II} ({\mathcal {A}})\ (\overline {\mathrm {MC}}^{II} ({\mathcal {A}}))$ : (mixed) normalized Hochschild complex of the second kind.

For notational convenience, we let $C'$ denote either $C, \overline {C}, C^{II}$ or $\overline {C} ^{II}$ and $\mathrm {MC} ' $ denote either $\mathrm {MC}, \overline {\mathrm {MC}}$ , $\mathrm {MC} ^{II}$ , or $\overline {\mathrm {MC}} ^{II}$ . The normalized negative cyclic complex is denoted by $\overline {C} ({\mathcal {A}}) [[u]]$ , where u is a formal variable of degree $2$ . For a mixed complex $(C, b, B)$ , we simply write $C[[u]]$ for the complex $(C [[u]], b + uB)$ .

2.2 Foundational facts frequently in use

2.2.3 Localization in cyclic homology

If ${\mathcal {A}} \to \mathcal {B} \to {\mathcal {C}}$ be an exact sequence of exact dg categories, then it induces an exact triangle of mixed complexes

$$\begin{align*}MC ({\mathcal{A}}) \to \mathrm{MC} (\mathcal{B}) \to \mathrm{MC} ({\mathcal{C}}) \to \mathrm{MC} ({\mathcal{A}} ) [1]; \end{align*}$$

see [Reference Keller22, § 5.6].

2.3 Categorical Chern characters

For $P \in {\mathcal {A}}$ , the cycle class represented by the identity morphism $1_P$ in the normalized Hochschild complex $\overline {C} ({\mathcal {A}})$ (resp. the normalized negative cyclic complex $\overline {C} ({\mathcal {A}}) [[u ]]$ ) of ${\mathcal {A}}$ is denoted by $\mathrm {Ch} _{HH} (P)$ (resp. $\mathrm {Ch} _{HN}(P)$ ).

3.1 The central embedding

The inertia stack has a decomposition:

$$\begin{align*}I{\mathcal{X}} = \sqcup _{r=1}^{\infty} I _{\mu_r} {\mathcal{X}}. \end{align*}$$

An object of $I_{\mu _r}{\mathcal {X}}$ over a k-scheme T is a pair $(\xi , {\alpha })$ of an object $\xi \in {\mathcal {X}} (T)$ and an injective morphism of group-schemes ${\alpha } : \mu _r \times T \to {{\mathcal {A}} ut} _T (\xi )$ ; see [Reference Abramovich, Graber and Vistoli1, § 3]. Note that, for all but finitely many r, $I_{\mu _r} {\mathcal {X}}$ is empty. An automorphism of the pair $(\xi , {\alpha })$ in $I_{\mu _r}{\mathcal {X}}$ is by definition an automorphism $f \in {{\mathcal {A}} ut}_T (\xi )$ such that

$$\begin{align*}f \circ {\alpha} \circ f^{-1} = {\alpha}. \end{align*}$$

In other words, the automorphism group-scheme ${{\mathcal {A}} ut}_T (\xi , {\alpha } )$ of $(\xi , {\alpha })$ over T is the centralizer of ${\alpha }$ in ${{\mathcal {A}} ut}_T (\xi )$ . We have a canonical central embedding $\mathfrak {c}: \mu _r \times T \to {{\mathcal {A}} ut}_T(\xi , {\alpha })$ . This gives a natural morphism

$$ \begin{align*} \mu _r \times T \to T\times _{I_{\mu_r}{\mathcal{X}}} T; (\xi, t) \mapsto (t, t, \mathfrak{c} (\xi , t)). \end{align*} $$

3.2 The central automorphism

Let $T \to I_{\mu _r}{\mathcal {X}}$ be an étale surjection, and let $pr _i : T\times _{I_{\mu _r}{\mathcal {X}}} T \to T$ be the i-th projection. A vector bundle E on $I_{\mu _r}{\mathcal {X}}$ amounts to a vector bundle F on T with an isomorphism $\phi ^F \in \mathrm {Isom}_T (pr _1^* F , pr_2^* F)$ satisfying the cocycle condition. By pulling back the isomorphism $\phi ^F$ to $\mu _r \times T$ , we obtain a morphism of group-schemes $\mu _r\times T \to \mathrm {Aut}_{T} (F)$ . Here, $\mathrm {Aut}_{T} (F)$ denotes the group of automorphisms of F fixing T. Since $\mathfrak {c}$ is central, the homomorphism descends to a homomorphism $\mu _r \to \mathrm {Aut}_{I_{\mu _r}{\mathcal {X}}} (E)$ . Denote by

$$\begin{align*}\mathfrak{can} _E \in \mathrm{Aut}_{I_{\mu_r}{\mathcal{X}}} (E) \end{align*}$$

the image of the chosen r-th root $e^{2\pi i/r}$ of unity.

According to the action $\mu _r$ upon E, the bundle E is decomposable into eigenbundles

$$\begin{align*}\bigoplus _{\chi \in \widehat{\mu _r}} E_{\chi} ,\end{align*}$$

where $\widehat {\mu _r}$ is the character group of $\mu _r$ . Then we have

$$\begin{align*}\mathfrak{can} _E = \bigoplus_{\chi \in \widehat{\mu _r} } \chi (e^{2\pi i /r } ) \mathrm{id}_{E_{\chi}} \in {\mathrm{Hom}} _{I_{\mu_r}{\mathcal{X}}} (E, E) .\end{align*}$$

3.3 The local description

Locally the central automorphisms $\mathfrak {can}$ can be described as follows. Suppose that ${\mathcal {X}} = [X/G]$ , where G is a finite group and X is a scheme. Let $g\in G$ with order r, and write $\mathrm {C} _{G}(g)$ for the centralizer of g in G. For the component $[X^g / \mathrm {C} _{G}(g) ]$ of $I_{\mu _r}{\mathcal {X}}$ and a $\mathrm {C} _{G}(g)$ -equivariant sheaf E on $X^g$ , we have an isomorphism

$$\begin{align*}(g^{-1})^* E \xrightarrow{\varphi _g^E} E \end{align*}$$

from the equivariant structure of E. Since g acts trivially on $X^g$ , $(g^{-1})^* E = E$ . And hence, $\varphi _g^E $ is an automorphism of E, which is the automorphism $\mathfrak {can} _E$ . Since any element of $\mathrm {C} _{G}(g)$ commutes with g, the homomorphism $\varphi ^E_g$ is $\mathrm {C} _{G}(g)$ -equivariant. Thus, $\varphi _g^E \in \mathrm {Hom}_{I_{\mu _r}{\mathcal {X}}} (E, E)$ . For any $\mathrm {C} _{G}(g)$ -equivariant sheaf $E'$ on $X^g$ and any $\mathrm {C} _{G}(g)$ -equivariant ${\mathcal {O}}_{X^g}$ -module homomorphism $a: E\to E'$ , note that $\varphi _g^{E'} \circ a = a \circ \varphi _g^E $ .

3.4 $T_{I{\mathcal {X}}} \cong IT_{{\mathcal {X}}}$

In this subsection, let ${\mathcal {X}}$ be smooth over k. We prove that there is a natural isomorphism $T_{I{\mathcal {X}}} \cong IT_{{\mathcal {X}}}$ .

Consider a commuting diagram of natural morphisms

where the square is a fiber square.

Lemma 3.1. The morphism $\phi $ induces an isomorphism $IT_{{\mathcal {X}}} \cong T_{I{\mathcal {X}}}$ .

Proof. First, note that it is enough to check the isomorphism over the étale site of the coarse moduli space of ${\mathcal {X}}$ . Since ${\mathcal {X}}$ is separated, then ${\mathcal {X}}$ is étale locally a quotient of a nonsingular variety Y by a finite group G action. Hence, we may assume that ${\mathcal {X}} = [Y/G]$ . Since

$$\begin{align*}T_{[Y/G]} \cong [T_Y / G] \text{ and } I[Y/G] \cong \bigsqcup _{g\in G/G} [Y^g / \mathrm{C} _{G}(g) ], \end{align*}$$

we have

$$\begin{align*}IT_{[Y/G]} \cong \bigsqcup _{g \in G/G} [ (T_Y)^g / \mathrm{C} _{G}(g) ] \text{ and } T_{I[Y/G]} \cong \bigsqcup _{g\in G/G} [T_{Y^g} /\mathrm{C} _{G}(g) ]. \end{align*}$$

Since $(T_Y)^g \cong T_{Y^g}$ , we conclude the proof.

Note first that there is a natural short exact sequence of vector bundles

$$\begin{align*}0 \to T_{I{\mathcal{X}}} \to \rho ^* _{{\mathcal{X}}} T_{{\mathcal{X}}} \to N_{\rho_{{\mathcal{X}}}} \to 0. \end{align*}$$

In fact, this sequence splits. The reason is that according to the canonical automorphism of $\rho ^* _{{\mathcal {X}}} T_{{\mathcal {X}}}$ there is a decomposition of $\rho _{{\mathcal {X}}} ^* T_{{\mathcal {X}}} $ into the fixed part and the moving part, which are naturally isomorphic to $T_{I{\mathcal {X}}} $ and $N_{\rho _{{\mathcal {X}}}}$ , respectively. $N_{\rho _{{\mathcal {X}}}}$ is in particular a vector bundle. For a detailed discussion on local embeddings, see [Reference Vistoli45, §1.20]).

Remark 3.2. When ${\mathcal {X}}$ is the global quotient $[Y/G]$ by a finite group G, $N_{\rho _{{\mathcal {X}}}} \cong [N_{Y^g/ Y} / \mathrm {C} _{G}(g) ]$ .

4.1 Matrix factorizations and their derived categories

Whenever an LG model $({\mathcal {X}} , w)$ is given, we consider a sheaf of curved differential graded (CDG for short) algebra $({\mathcal {O}}_{{\mathcal {X}}} , -w)$ over ${\mathcal {X}}$ . It is concentrated in degree $0$ with zero differential and a curvature $-w$ .

We denote the category of QDG modules over $({\mathcal {O}}_{\mathcal {X}}, -w)$ by $q\mathrm {Mod}({\mathcal {X}}, w)$ . It is a CDG category whose morphisms and differentials are

$$ \begin{align*} {\mathrm{Hom}}_{q\mathrm{Mod}}&((P,\delta_P), (Q, \delta_Q))=\left( {\mathrm{Hom}}_{{\mathcal{O}}_{\mathcal{X}}}(P, Q) , \delta \right),\\ \delta(f)&= \delta _Q \circ f - (-1)^{|f|} f \circ \delta _P. \end{align*} $$

The curvature element $h_{(p, \delta _P)}$ of $(P, \delta _P)$ is defined as $\delta _P^2+\rho _{-w} \in \mathrm {End}(P)$ , where $\rho _{-w}$ is the multiplication map by $-w$ .

By definition, factorizations form a dg subcategory inside $q\mathrm {Mod}({\mathcal {X}}, w)$ denoted by $\mathrm {Mod}({\mathcal {X}}, w)$ . We denote a full dg subcategory of (quasi-) coherent and matrix factorizations by $\mathrm {QCoh}({\mathcal {X}}, w)$ , $\mathrm {Coh}({\mathcal {X}}, w)$ and $\mathrm {MF}({\mathcal {X}}, w)$ , respectively.

We recall constructions of the derived category of factorizations following [Reference Positselski37]. Let $[\mathrm {QCoh}({\mathcal {X}}, w)]$ be the homotopy category of $\mathrm {QCoh}({\mathcal {X}}, w)$ . Denote by $\mathrm {AbsAcyc}({\mathcal {X}}, w)$ the smallest triangulated subcategory containing the totalizations of all short exact sequences in $Z^0\mathrm {QCoh}({\mathcal {X}}, w)$ . Its object is called absolutely acyclic factorizations. Also, denote by $\mathrm {CoAcyc}({\mathcal {X}}, w)$ the smallest triangulated subcategory containing the totalizations of all acyclic factorizations which is closed under infinite direct sum; its object are called a coacyclic factorizations.

Definition 4.5. The absolute derived category of $\mathrm {QCoh}({\mathcal {X}}, w)$ is the Verdier quotient

$$\begin{align*}\mathrm{D}^{\mathrm{abs}}(\mathrm{QCoh}({\mathcal{X}}, w)):=[\mathrm{QCoh}({\mathcal{X}}, w)]/\mathrm{AbsAcyc}({\mathcal{X}}, w).\end{align*}$$

The coderived category of $\mathrm {QCoh}({\mathcal {X}}, w)$ is the Verdier quotient

$$\begin{align*}\mathrm{D}^{\mathrm{co}}(\mathrm{QCoh}({\mathcal{X}}, w)):=[\mathrm{QCoh}({\mathcal{X}}, w)]/\mathrm{CoAcyc}({\mathcal{X}}, w). \end{align*}$$

We define an absolute/coderived category of $\mathrm {Coh}({\mathcal {X}}, w)$ and $\mathrm {MF}({\mathcal {X}}, w)$ .

4.2 Čech model

In this subsection, we recall the Čech type dg enhancement of $\mathrm {D}\mathrm {MF}(X, w)$ as in [Reference Chung, Kim and Kim10].

Fix an affine étale surjective morphism $\mathfrak {p} : \mathfrak {U} \to {\mathcal {X}}$ from a k-scheme $\mathfrak {U}$ . Let $\mathfrak {U} ^r$ denote the r-th fold product of $\mathfrak {U}$ over ${\mathcal {X}}$ , and let $\mathfrak {p}_r : \mathfrak {U} ^r \to {\mathcal {X}}$ denote the projection. For a vector bundle E on ${\mathcal {X}}$ , let $\check{\text{C}}^{r} (E)= \mathfrak {p}_{r*} \mathfrak {p}_{r}^{*} E$ and

$$\begin{align*}\check{\text{C}} (E) := \left( \bigoplus _{r \ge 1} \check{\text{C}} ^r (E), d_{\scriptscriptstyle \check{\text{C}}\text{ech}} \right) = \left[ 0 \to p_{1*}p_1^*E \to p_{2*}p_2^*E \to \cdots \right]\end{align*}$$

a Čech complex.

Now, let $(E, \delta _E)$ be a matrix quasi-factorization over $({\mathcal {O}}_{\mathcal {X}}, -w)$ . Observe that $\check{\text{C}}({\mathcal {O}}_{{\mathcal {X}}})$ can be viewed as a sheaf of ${\mathcal {O}}_{\mathcal {X}}$ -algebras equipped with an Alexander–Whitney product. By projection formula $ \check{\text{C}}(E) = E \otimes _{{\mathcal {O}}_{{\mathcal {X}}}}\check{\text{C}}({\mathcal {O}}_{{\mathcal {X}}})$ and $\check{\text{C}}(E)$ carries a natural $\check{\text{C}}({\mathcal {O}}_{\mathcal {X}})$ -module structure. We equip $\check{\text{C}} (E)$ with a curved differential

$$\begin{align*}\delta _{\check{\text{C}}(E)}:= \delta _P \otimes 1 + 1\otimes d_{\scriptscriptstyle \check{\text{C}}\text{ech}}. \end{align*}$$

We regard $(\check{\text{C}}(E), \delta _{\check{\text{C}}(E)})$ as a QDG-module over $(\check{\text{C}}({\mathcal {O}}_{\mathcal {X}}), w)$ . Notice that the curvature of $(\check{\text{C}} (E), \delta _{\check{\text{C}}(E})$ coincides with the curvature of E under the canonical map $\mathrm {End} _{{\mathcal {O}} _{{\mathcal {X}}}} (E) \to \mathrm {End} _{\check{\text{C}} ({\mathcal {O}} _{{\mathcal {X}}})} (\check{\text{C}} (E))$ .

Definition 4.8. For a fixed affine étale open cover $\mathfrak {U}$ , a Čech model CDG category $q\mathrm {MF}_{dg}({\mathcal {X}}, w)$ is a CDG category whose objects are matrix quasi-factorizations and its ${\mathbb G}$ -graded Hom spaces are defined as usual

(4.1) $$ \begin{align} {\mathrm{Hom}}_{q\mathrm{MF}_{dg}}(P, Q)&:=\left( {\mathrm{Hom}}_{{\mathcal{O}}_{{\mathcal{X}}}}(\check{\text{C}}(P), \check{\text{C}}(Q)) , \delta \right) \end{align} $$

(4.2) $$ \begin{align} \delta &= \delta _{\check{\text{C}} (P')} \circ f - (-1)^{|f|} f \circ \delta _{\check{\text{C}}(P)}. \end{align} $$

Similarly, we define a Čech model dg category $ \mathrm {MF}_{dg} ({\mathcal {X}}, w) $ of matrix factorizations for $({\mathcal {X}}, w)$ as a full dg subcategory of $q\mathrm {MF}_{dg} ({\mathcal {X}}, w) $ consisting of matrix factorizations for $({\mathcal {X}}, w)$ . When it is necessary to specify the covering $\mathfrak {U}$ , we write $ \mathrm {MF}_{dg} ({\mathcal {X}}, w; \mathfrak {U}) $ for $\mathrm {MF}_{dg} ({\mathcal {X}}, w)$ .

In the following lemma, we show that $\mathrm {MF}_{dg}({\mathcal {X}}, w)$ is equivalent to the dg-quotient dg enhancement of $\mathrm {D}\mathrm {MF}({\mathcal {X}}, w)$ .

Lemma 4.9. The natural functor

$$ \begin{align*}\epsilon: [\mathrm{MF}_{dg}({\mathcal{X}}, w)] \to \mathrm{D}^{\mathrm{co}}(\mathrm{QCoh}({\mathcal{X}}, w))\end{align*} $$

$$ \begin{align*}\hspace{0.2cm} P\mapsto \check{\text{C}}(P)\end{align*} $$

is fully faithful, and its essential image is equal to $\mathrm {DMF}({\mathcal {X}}, w)$ .

Proof. To show that the essential image of $\epsilon $ is as claimed, one can check $\mathrm {Cone}(P \to \check{\text{C}}(P))$ is coacylic so that P is isomorphic to $\check{\text{C}}(P)$ in $\mathrm {DMF}({\mathcal {X}}, w)$ (see [Reference Ciocan-Fontanine, Favero, Guéré, Kim and Shoemaker9, equation 2.1]).

To prove fully faithfulness, we need to show ${\mathrm {Hom}}_{\mathrm {QCoh}({\mathcal {X}}, w)}(\check{\text{C}}(P), \check{\text{C}}(Q)) \simeq {\mathrm {Hom}}_{\mathrm {DMF}({\mathcal {X}}, w)}(\check{\text{C}}(P), \check{\text{C}}(Q))$ . Using Cech filtration on the second argument, it is enough to show that

$$\begin{align*}{\mathrm{Hom}}_{\mathrm{QCoh}({\mathcal{X}}, w)}(\check{\text{C}}(P),j_*j^*Q) \simeq {\mathrm{Hom}}_{\mathrm{D}^{\mathrm{co}}\mathrm{QCoh}({\mathcal{X}}, w)}(\check{\text{C}}(P), j_*j^*Q)\end{align*}$$

for any open étale affine open $j:V \to {\mathcal {X}}$ . Consider the following diagram:

(4.3)

The right vertical $p_2$ is an isomorphism because $P \to \check{\text{C}}(P)$ is an isomoprhism in $\mathrm {D}^{\mathrm {co}}\mathrm {QCoh}$ . The bottom horizontal q becomes

$$\begin{align*}{\mathrm{Hom}}_{\mathrm{MF}(V, w\vert_V)}(j^*P,j^*Q) \to {\mathrm{Hom}}_{\mathrm{D}^{\mathrm{co}}\mathrm{QCoh}(V, w\vert_V)}(j^*P,j^*Q).\end{align*}$$

This is known to be an isomorphism for all affine V. To show that the left vertical $p_1$ is an isomorphism, it is enough to show that

$$\begin{align*}\left( {\mathrm{Hom}}_{\mathrm{QCoh}(V)}(\check{\text{C}}^{\bullet} (j^* P^m), j^*Q^n), \delta_{Cech} \right) \to {\mathrm{Hom}}_{\mathrm{QCoh}(V)}(j^* P^m, j^*Q^n) \end{align*}$$

for all $m, n \in \mathbb Z/2.$ This is nothing but an (unordered) Cech resolution of ${\mathrm {Hom}}_{\mathrm {QCoh}(V)}(j^* P^m, j^*Q^n)$ associated to the open cover $\mathfrak {U} \times _{\mathcal {X}} V$ on V, which is an isomorphism because V is affine. The claim follows.

Lemma 4.9 is an analogue of the Čech enhancement of [Reference Lunts and Schnürer30, § 4.1] for Deligne–Mumford stacks. It follows from the proof of the lemma that the dg-quotient enhancement and the Čech model dg-enhancement of Definition 4.8 coincide (see [Reference Lunts and Orlov29, § 2]).

Remark 4.10. We view ${\mathrm {Hom}}_{\mathrm {MF}_{dg}}(P, Q)$ as a $\mathbb Z \times \mathbb Z/2$ -graded bicomplex. This complex is not bounded above in $\mathbb Z$ -direction because a Čech cover $\mathfrak {U}$ of a stack is genuinely unordered. One can go around the subtleties by taking a suitable truncation. Suppose $(E, \delta _E)$ is a matrix factorization. Define

$$\begin{align*}\tau\check{\text{C}}(E) := \tau _{ \le \dim {\mathcal{X}}} \left( \bigoplus _{r \ge 1} \check{\text{C}} ^r (E), d_{\scriptscriptstyle \check{\text{C}}\text{ech}} \right),\end{align*}$$

where $\tau _{r \le \dim {\mathcal {X}}}$ denotes the truncation. Note that under the assumptions in this text all stacks have finite cohomological dimension, hence the truncated one does compute the sheaf cohomology of the quasi-coherent sheaf E since the map to the coarse moduli ${\mathcal {X}} \to \underline {{\mathcal {X}}}$ is cohomologically affine. Therefore, the induced map between the spectral sequences associated to Čech filtrations

$$\begin{align*}{\mathrm{Hom}}(\check{\text{C}}(P), \check{\text{C}}(Q)) \to {\mathrm{Hom}}(\tau \check{\text{C}}(P), \check{\text{C}}(Q)) \leftarrow {\mathrm{Hom}}(\tau\check{\text{C}}(P), \tau\check{\text{C}}(Q))\end{align*}$$

are isomorphisms on the first page and, hence, are quasi-isomorphisms themselves. Also, notice that $\tau \check{\text{C}}(E) \to \check{\text{C}}(E)$ is a quasi-isomorphism in $\mathrm {D}\mathrm {MF}({\mathcal {X}}, w)$ .

Remark 4.12. Consider another affine covering $\mathfrak {U}' \to {\mathcal {X}}$ . Let $\mathfrak {U} " = \mathfrak {U} ' \times _{{\mathcal {X}}} \mathfrak {U}$ . Then there is a natural dg functor $\mathrm {MF}_{dg} ({\mathcal {X}}, w, \mathfrak {U}') \to \mathrm {MF}_{dg} ({\mathcal {X}}, w, \mathfrak {U} ")$ , which is a quasi-equivalence. The induced chain map $\mathrm {MC} ' (\mathrm {MF}_{dg} ({\mathcal {X}}, w, \mathfrak {U})) \to \mathrm {MC} '( \mathrm {MF}_{dg} ({\mathcal {X}}, w, \mathfrak {U} "))$ between mixed complexes are quasi-isomorphism. For the mixed complex of the first kind, it follows from Morita invariance. For that of either kind, consider the Hochschild complex $C ' (\mathrm {MF}_{dg} ({\mathcal {X}}, w))$ filtered by Čech degree. Since the filtration is bounded (see 4.10), we may apply the Eilenberg–Moore comparison theorem ([Reference Weibel46, Theorem 5.5.11]) to check that the induced chain map is a quasi-isomorphism. A similar argument shows that the natural chain map

$$\begin{align*}\mathrm{MC} ' (q\mathrm{MF}_{dg} ({\mathcal{X}}, w, \mathfrak{U})) \to \mathrm{MC} '( q\mathrm{MF}_{dg} ({\mathcal{X}}, w, \mathfrak{U} ")) \end{align*}$$

is also a quasi-isomorphism.

4.3 Twisting

For a morphism from a scheme U to ${\mathcal {X}}$ , write $IU$ for the fiber product $U \times _{{\mathcal {X}}} I{\mathcal {X}}$ . Following Toën, Halpern–Leistner and Pomerleano [Reference Halpern-Leistner and Pomerleano18], we consider the assignments

$$\begin{align*}U \mapsto C ' ( q\mathrm{MF}_{dg} (IU, w|_{IU}, \mathfrak{U} \times _{{\mathcal{X}}} IU )) \end{align*}$$

for all étale morphisms $U\to {\mathcal {X}}$ . They form a presheaf of mixed complexes on the small étale site ${\mathcal {X}}$ , which we denote by

$$ \begin{align*} \mathrm{MC} ' (q\mathrm{MF}_{I{\mathcal{X}}, w} (-)). \end{align*} $$

Denote the associated cochain complex by $C ' (q\mathrm {MF}_{I{\mathcal {X}}, w} (-))$ . There is a natural morphism of mixed complexes

$$\begin{align*}nat: \mathrm{MC} ' (q\mathrm{MF}_{dg} (I{\mathcal{X}}, w)) \to (\Gamma ({\mathcal{X}}, \mathrm{MC} ' (q\mathrm{MF}_{I{\mathcal{X}}, w} (-))), \end{align*}$$

where the Čech model $q\mathrm {MF}_{dg} (I{\mathcal {X}}, w)$ uses the affine cover $\mathfrak {U} \times _{{\mathcal {X}}} I{\mathcal {X}} \to I{\mathcal {X}}$ .

In Section 3.2, we defined a canonical twist for every coherent sheaf on $I{\mathcal {X}}$ . In turn, this gives an automorphism $\mathfrak {can} _P$ of P for every object P of $\mathrm {MF}_{dg} (I{\mathcal {X}}, w)$ . It is a morphism in the category $\mathrm {MF}_{dg} (I{\mathcal {X}}, w)$ .

Note that

(4.4) $$ \begin{align} \mathfrak{can} _{P'} \circ a = a \circ \mathfrak{can} _{P} \quad \forall a \in {\mathrm{Hom}} _{I{\mathcal{X}}}(P, P'). \end{align} $$

Hence, the assignments $P\mapsto \mathfrak {can} _P$ yield a natural transformation $\mathfrak {can}$ between the identity functor $\mathrm {id} : \mathrm {MF}_{dg} (I{\mathcal {X}}, w) \to \mathrm {MF}_{dg} (I{\mathcal {X}}, w)$ . We will often drop the subscript P in $\mathfrak {can} _P$ for simplicity.

Define a k-linear map $\mathfrak {tw}: C ' (q\mathrm {MF}_{dg} (I{\mathcal {X}}, w)) \to C ' (q\mathrm {MF}_{dg} (I{\mathcal {X}}, w))$ associated to $\mathfrak {can}$ by

$$\begin{align*}a_0 [a_1 | \cdots | a_n ] \mapsto a_0 [ a_1 | \cdots | \mathfrak{can} \circ a_n ]. \end{align*}$$

Note that by equation (4.4) $b \circ \mathfrak {tw} = \mathfrak {tw} \circ b$ , that is, $\mathfrak {tw}$ is a chain automorphism of the Hochschild complex $C' (q\mathrm {MF}_{dg} (I{\mathcal {X}}, w))$ .

Consider the composition $\tau ' $ of a sequence of chain maps

(4.5) $$ \begin{align} C' (q\mathrm{MF}_{dg} ({\mathcal{X}}, w)) \xrightarrow{pullback} C' &(q\mathrm{MF}_{dg} (I{\mathcal{X}}, w)) \nonumber \\ &\qquad \xrightarrow{\mathfrak{tw}} C' (q\mathrm{MF}_{dg} (I{\mathcal{X}}, w)) \xrightarrow{nat} \Gamma ({\mathcal{X}}, C' (q\mathrm{MF}_{I{\mathcal{X}}, w} (-) )). \end{align} $$

A proof of the above proposition will be given § 4.4 and § 4.5. For simplicity, we will often write $\tau $ for $\tau '$ when there is no risk of confusion.

4.4 Local case

Let ${\mathcal {X}} = [{\mathrm {Spec}} A / G]$ , and let $w \in A^G$ a G-invariant element of A as in Proposition 4.13. Let $\mathrm {MF}_{dg} ^G (A, w)$ denote the dg category of G-equivariant factorizations P for $(A, w)$ which are projective as A-modules. The Hom space from P to Q is the G-invariant part of ${\mathrm {Hom}} _A (P, Q)$ of ${\mathbb G}$ -graded A-module homomorphisms. Likewise, we have the CDG category $q\mathrm {MF}_{dg} ^G (A, w)$ of G-equivariant quasi-modules for $(A, w)$ which are projective as A-modules. In fact, these coincide with the Čech models $\mathrm {MF}_{dg} ({\mathcal {X}}, w)$ and $q\mathrm {MF}_{dg} ({\mathcal {X}}, w)$ with respect to the natural choice of an affine cover: ${\mathrm {Spec}} A \to {\mathcal {X}}$ .

Let $I_g$ be the ideal of A generated by $a - g a $ for all $a \in A$ . Denote $A_g := A / I_g$ and $w_g := w |_{A_g} \in A_g$ . We regard the pair $(A, w)$ (resp. $(A_g, w_g)$ ) as a CDG algebra A (resp. $A_g$ ) with zero differential and curvature w (resp. $w_g$ ).

The algebra A has the induced left G-action. Note that for $a, b \in A$ and $g, h \in G$ ,

$$\begin{align*}g (a \ h (b)) = g (a) \ gh (b) .\end{align*}$$

The cross product algebra $A\rtimes G := A \otimes k[G]$ has the multiplication defined by $(a\otimes g) \cdot (b \otimes h) = a g (b) \otimes g h$ . We also view $A\rtimes G$ as a right A-module with a left G-action by

$$\begin{align*}(a \otimes g ) \cdot b = a g (b) \otimes g \text{ and } h\cdot (a \otimes g) = a \otimes g h^{-1}. \end{align*}$$

Equivalently, $A\rtimes G$ is a right $A \rtimes G$ -module by the multiplication

$$\begin{align*}(a \otimes g) \cdot (b \otimes h') = a g(b) \otimes g h'. \end{align*}$$

Note that the curvature of $A\rtimes G$ with zero differential as a right quasi-module over $(A \rtimes G, w \otimes 1)$ is $-w\otimes 1$ .

Denote by $\{ (A\rtimes G, -w\otimes 1) \}$ the full subcategory of $q\mathrm {MF}_{dg} (A\rtimes G, w\otimes 1)$ consisting of the indicated object $A\rtimes G$ with zero differential and curvature $-w\otimes 1$ . The embedding $\{ (A\rtimes G, -w) \} \hookrightarrow q\mathrm {MF}_{dg} (A\rtimes G, w\otimes 1) $ is called a quasi-Yoneda embedding. It is a pseudo-equivalence; see [Reference Polishchuk and Positselski36]. Consider the embedding $ q\mathrm {MF}_{dg} (A\rtimes G, w\otimes 1) \hookrightarrow q\mathrm {MF}_{dg} ^G (A, w)$ , which is also a pseudo-equivalence; see [Reference Chung, Kim and Kim10]. Hence, by the quasi-Morita invariance the induced morphism of mixed complexes

$$\begin{align*}\overline{\mathrm{MC}}^{II} (A\rtimes G, -w) \to \overline{\mathrm{MC}}^{II} ( q\mathrm{MF}_{dg} ^G (A, w) ) \end{align*}$$

is a quasi-isomorphism.

Consider an embedding $ \{ (A_g \rtimes G, -w_g \otimes 1)\} \hookrightarrow q\mathrm {MF}_{dg} (A_g, w_g)$ . This is a pseudo-equivalence since $(A_g, -w_g)$ is a direct summand of $(A_g \rtimes G, -w_g \otimes 1)$ as a right quasi-module over $(A_g, -w_g)$ . Hence, by the quasi-Morita invariance the induced morphism of mixed complexes

$$\begin{align*}\overline{\mathrm{MC}}^{II} (\mathrm{End} _{A_g} (A_g \rtimes G) , -w_g \otimes 1) \to \overline{\mathrm{MC}}^{II} (q\mathrm{MF}_{dg} (A_g, w_g)) \end{align*}$$

is a quasi-isomorphism.

We have a diagram of chain maps

(4.6)

where:

• ○ two vertical chain maps are quasi-isomorphisms as explained already;

• ○ the middle horizontal map $\tau |_{A\rtimes G}$ is induced from the composition $natural \circ \tau $ ;

• ○ $\psi _A$ is the quasi-isomorphic chain map defined by Baranovsky [Reference Baranovsky4, page 799], Segal[Reference Segal40], and Căldăraru and Tu [Reference Căldăraru and Tu8, Proof of 6.3];

• ○ $\mathrm {Tr}$ is the generalized trace map.

Proof. This will be clear since g acts on $A_g$ trivially. Note that the following diagram commutes

(4.7)

where $\overline {a}$ denotes the element in $A_g$ associated to a. Here, the right-bottom corner is meant to have the g-th component

$$\begin{align*}\left\{\begin{array}{cl} |G| \cdot \overline{a}_0 [ g_0 \overline{a}_1 | \cdots | g_0 \cdots g_{n-1} g^{-1} \overline{a}_n ] & \text{ if } g = g_0 \cdots g_n \\ 0 & \text { otherwise. } \end{array} \right. \end{align*}$$

4.5 Proof of Proposition 4.13

Due to Remark 4.12 and the compatibility of the map $\tau $ with Čech differentials, the proof follows from Lemma 4.14.

4.6 The role of $\mathfrak {can}$ at the level of category

As we have already remarked, the insertion of central automorphism $\mathfrak {tw}$ was essential. We would like to sketch how it appears naturally in the computation of Hochschild invariants to clarify its role. The proof of Proposition 4.13 can be interpreted as a two-step process.

The first step is purely categorical. Suppose a k-linear category $\mathcal C$ carries a strict G-action of a finite group G. We still denote corresponding endofunctors by $g:\mathcal C \to \mathcal C, \hskip 0.2cm g\in G$ . We also denote its category of G-equivariant objects by $\mathcal C_G$ . Its object consists of a pair $(E, \phi _g^E)$ , where E is an object of $\mathcal C$ and $\phi _g^E$ is an isomorphism $\phi _g^E : E \simeq gE$ satisfying a cocycle condition. The morphism is defined as usual.

There is a natural functor

$$\begin{align*}\widetilde{\phantom{a}} : \mathcal C \to \mathcal C_G\end{align*}$$

called linearization which is defined on objects as

$$\begin{align*}\widetilde E = \left(\bigoplus_{h\in G}hE, \phi_g^{\widetilde E}\right), \hskip 0.2cm \phi_g^{\widetilde E} : \bigoplus_{h\in G} hE = \bigoplus_{h\in G}g(hE) \simeq \bigoplus_{h \in G} (gh)E.\end{align*}$$

It is not hard to see that the linearization is a both left and right adjoint to the forgetful functor. Its essential image generates $\mathcal C_G$ and

$$\begin{align*}{\mathrm{Hom}}_{\mathcal C_G}(\widetilde E_1, \widetilde E_2) \simeq {\mathrm{Hom}}_{\mathcal C}(\widetilde E_1, \widetilde E_2)^G.\end{align*}$$

This fact leads to the following simple description of Hochschild homology of $\mathcal C_G$ . (See [Reference Perry33].)

(4.8) $$ \begin{align} HH_*(\mathcal C_G) \simeq (\bigoplus_{g\in G} HH_*(\mathcal C, g))^G \simeq \bigoplus_{g \in \mathrm{Conj}(G)}HH_*(\mathcal C_{C(g)}, g). \end{align} $$

Here, $HH_*(\mathcal C , g)$ denotes Hochschild homology with coefficient g, where the endofunctor g is considered as a bimodule.

The second observation is geometric. For simplicity, let $\mathcal C = D(X)$ be a dg category of coherent sheaves on a smooth affine scheme $X={\mathrm {Spec}}(A)$ acted on by a finite group G. One can easily extend the discussion to the case of matrix factorizations. Each component of equation (4.8) has a simpler description:

(4.9) $$ \begin{align} HH_*(D_{C(g)}(X), g) \xrightarrow[\mathrm{res}]{\sim} HH_*(D_{C(g)}(X^g), g). \end{align} $$

Notice that the action of g on $X^g$ is trivial. If $(E, \{\phi _h^E\}_{h\in G})$ is a G-equivariant sheaves on X, then $\varphi ^E_g = \left (\phi _g^E\right )^{-1}$ restricts to the central automorphism $\mathfrak {can} _{E\vert _{X_g}}$ of $E\vert _{X^g}$ . In fact, any $C(g)$ -equivariant object $(F, \{\phi ^F_h\}_{h\in C(g)})$ on $X^g$ carries a distinguished automophism $\mathfrak {can}_F = \left (\phi ^F_g\right )^{-1}$ . This assignment is viewed as a natural transformation between identity functors or an element of zeroth Hochschild cohomology;

$$\begin{align*}[\mathfrak{can}] \in HH^0(D_{C(g)}(X^g)).\end{align*}$$

The map $\mathfrak {tw}$ on Hochschild chains is a cap product with $[\mathfrak {can}]$ .

Lastly, observe that $D_{C(g)}(X^g)$ is generated by $\mathcal O_{X^g}$ . Notice that g-action on $\mathcal O_{X^g}$ is trivial, so $\mathfrak {can}$ could be ignored. This implies

(4.10) $$ \begin{align} HH_*(A_g)^{C(g)} \xrightarrow[\mathrm{inc}]{\sim}HH_*(D_{C(g)}(X^g), g). \end{align} $$

4.7 Mixed complex case

In general, the map $\tau $ is not a morphism of mixed complexes. In this subsection, we modify $\tau $ to get a morphism of mixed complexes.

For $\chi \in \hat {\mu _r}$ , let $q\mathrm {MF}_{dg} ^{\chi } (I_{\mu _r}{\mathcal {X}}, w)$ be the full subcategory of $q\mathrm {MF}_{dg} (I_{\mu _r}{\mathcal {X}}, w)$ consisting $\chi $ -eigenobjects of $q\mathrm {MF}_{dg} (I_{\mu _r}{\mathcal {X}}, w)$ . The map $\mathfrak {tw}$ restricted to the subcomplex $C ( q\mathrm {MF}_{dg} ^{\chi } (I_{\mu _r}{\mathcal {X}}, w) )$ , denoted by $\mathfrak {tw} _{\chi }$ , is nothing but the multiplication by $\chi (e^{2\pi i/r})$ . Hence,

$$\begin{align*}\mathfrak{tw}_{\chi} : \mathrm{MC} (q\mathrm{MF}_{dg} ^{\chi} (I_{\mu_r}{\mathcal{X}}) ) \to \mathrm{MC} ( q\mathrm{MF}_{dg} ^{\chi} (I_{\mu_r}{\mathcal{X}}) ) \end{align*}$$

is an automorphism of the mixed Hochschild complex.

Consider the composition $\tau _m$ of a sequence of morphisms of mixed complexes

$$ \begin{align*} \tau _m: &\overline{\mathrm{MC}}^{II} (q\mathrm{MF}_{dg} ({\mathcal{X}}, w)) \xrightarrow{pullback} \oplus _{r, \chi} \overline{\mathrm{MC}}^{II} (q\mathrm{MF}_{dg} ^{\chi} (I_{\mu_r}{\mathcal{X}}, w)) \\ &\qquad \hspace{1cm} \xrightarrow{\oplus \mathfrak{tw}_{\chi} } \oplus _{r, \chi} \overline{\mathrm{MC}}^{II} (q\mathrm{MF}_{dg} ^{\chi} (I_{\mu_r}{\mathcal{X}}, w))\xrightarrow{nat} \oplus _{r} \Gamma (I_{\mu_r}{\mathcal{X}}, \overline{\mathrm{MC}}^{II} (q\mathrm{MF}_{I_{\mu_r}{\mathcal{X}}, w} (-) ) ) , \end{align*} $$

where the natural map $nat$ is defined by setting

$$\begin{align*}nat \left(\sum _{\chi} a^{\chi}_0 [\overline{a}^{\chi}_1 | \ldots | \overline{a}^{\chi}_n ] ) = (U\mapsto \sum_{\chi} a^{\chi}_{0|_{IU}} [\overline{a}^{\chi}_{1|_{IU}} | \ldots | \overline{a}^{\chi}_{n|_{IU}}] \right). \end{align*}$$

Remark 4.16. While the cochain map

$$\begin{align*}C ^{II} ( q\mathrm{MF}_{dg} (I_{\mu_r}{\mathcal{X}}, w) ) \xrightarrow{\mathrm{Tr}} \oplus _{\chi} C ^{II} ( q\mathrm{MF}_{dg} ^{\chi} (I_{\mu_r}{\mathcal{X}}, w) ) \end{align*}$$

is an isomorphism, $ \overline {C} ^{II} ( q\mathrm {MF}_{dg} (I_{\mu _r}{\mathcal {X}}, w) ) \xrightarrow {\overline {\mathrm {Tr}}} \oplus _{\chi } \overline {C} ^{II} ( q\mathrm {MF}_{dg} ^{\chi } (I_{\mu _r}{\mathcal {X}}, w) ) $ is not an isomorphism in general but a quasi-isomorphism from the facts that $C ^{II}\to \overline {C}^{II}$ is a quasi-isomorphism and the above $\mathrm {Tr}$ is an isomorphism.

4.8 Global case

Let $\underline {{\mathcal {X}}}$ denote the coarse moduli space of ${\mathcal {X}}$ . For an étale map $V\to \underline {{\mathcal {X}}}$ , let ${\mathcal {X}}_V := V \times _{ \underline {{\mathcal {X}}} } {\mathcal {X}}$ . We take the sheafification $\underline {\overline {\mathrm {MC}}} ^{II} (q\mathrm {MF}_{dg} ({\mathcal {X}}, w))$ (resp. $\underline {\overline {\mathrm {MC}}} ^{II} (q\mathrm {MF} _{I{\mathcal {X}}, w})$ ) of the presheaf

$$\begin{align*}V\mapsto \overline{\mathrm{MC}} ^{II} (q\mathrm{MF}_{dg} ({\mathcal{X}} _V, w)) \text{ resp. } (V\mapsto \Gamma (I{\mathcal{X}} _V, \overline{\mathrm{MC}} ^{II} (q\mathrm{MF}_{I{\mathcal{X}}, w} (-))) \end{align*}$$

both on the étale site of $\underline {{\mathcal {X}}}$ . We take the sheaf homomorphism $\underline {\tau }_m$ induced from $\tau _m$

$$\begin{align*}\underline{\tau}_m : \underline{\overline{\mathrm{MC}}}^{II} (q\mathrm{MF}_{dg} ({\mathcal{X}} ,w ) ) \to \underline{\overline{\mathrm{MC}}}^{II} (q\mathrm{MF} _{I{\mathcal{X}}, w} (-)). \end{align*}$$

Lemma 4.18. Suppose that ${\mathcal {X}}$ is smooth over k. Then the induced morphism ${\mathbb R}\Gamma (\underline {\tau }_m)$ fits into a diagram of isomorphisms in the derived category of mixed complexes:

(4.11)

Theorem 4.19. Suppose that ${\mathcal {X}}$ is smooth over k. Then the isomorphism

(4.12) $$ \begin{align} \overline{\mathrm{MC}} (\mathrm{MF}_{dg} ({\mathcal{X}} , w)) \cong {\mathbb R}\Gamma ( \underline{\overline{\mathrm{MC}}}^{II} ( {\mathcal{O}} _{I{\mathcal{X}}} , -w|_{I{\mathcal{X}}})) \end{align} $$

in the derived category of mixed complexes induces an isomorphism

$$\begin{align*}\overline{\mathrm{MC}} (\mathrm{MF}_{dg} ({\mathcal{X}} , w)) \cong {\mathbb R}\Gamma (\Omega ^{\bullet}_{I{\mathcal{X}}}, -dw|_{I{\mathcal{X}}}, ud) .\end{align*}$$

5.1 A formula via Čech model and Chern–Weil theory

We fix an affine étale surjective morphism $\mathfrak {p} : \mathfrak {U} \to {\mathcal {X}}$ from a k-scheme $\mathfrak {U}$ as in § 4.2. Since $\mathfrak {U}$ is an affine scheme over k, every $E|_{\mathfrak {U}}$ has a connection

$$\begin{align*}\nabla _{E|_{\mathfrak{U}}} : E|_{\mathfrak{U}} \to E|_{\mathfrak{U}} \otimes \Omega _{\mathfrak{U}}^1. \end{align*}$$

Define a connection

$$\begin{align*}\nabla _{E |_{\mathfrak{U}^r}} : E|_{\mathfrak{U}^r} \to E|_{\mathfrak{U}^r} \otimes \Omega _{\mathfrak{U}^r}^1 \end{align*}$$

by letting $\nabla _{E |_{\mathfrak {U}^r}} = p_1^* \nabla _{E |_{\mathfrak {U}}}$ , where $p_1$ is the first projection $\mathfrak {U}^r \to \mathfrak {U}$ . This gives rise to a connection

$$\begin{align*}\nabla _E : \check{\text{C}} (E) \to \Omega ^1_{{\mathcal{X}}} \otimes \check{\text{C}} (E), \end{align*}$$

where $\check{\text{C}} (E) := ( \bigoplus _{r \ge 0} \check{\text{C}} ^r (E), d_{\scriptscriptstyle \check{\text{C}}\text{ech}} )$ and $\check{\text{C}} ^r (E)= \mathfrak {p}_{r*} \mathfrak {p}_{r}^{*} E $ . For every E, fix such a connection once and for all.

Let $I\mathfrak {U}$ denote the affine scheme $\mathfrak {U} \times _{{\mathcal {X}}} I{\mathcal {X}}$ . Using this affine covering of $I{\mathcal {X}}$ , we have the Čech resolution $\check{\text{C}} (E|_{I{\mathcal {X}}})$ and the connection

$$\begin{align*}\nabla _{ E|_{I{\mathcal{X}}}} : \check{\text{C}} (E|_{I{\mathcal{X}}} ) \to \Omega ^1_{I{\mathcal{X}}} \otimes \check{\text{C}} (E|_{I{\mathcal{X}}}) .\end{align*}$$

In general, for every vector bundle F on $I{\mathcal {X}}$ , we can choose a connection $\nabla _F : \check{\text{C}} (F ) \to \Omega ^1_{I{\mathcal {X}}} \otimes \check{\text{C}} (F) $ .

For each $P\in q\mathrm {MF}_{dg} (I{\mathcal {X}}, w) )$ , choose a connection $\nabla _{P}$ as above once and for all. Let $R = u \nabla _{\nabla }^2 + [\nabla _{P} , \delta _{\check{\text{C}} (P)} ]$ a kind of the total curvature of $\nabla _P$ . By a straightforward generalization of the definition of a chain map ${\mathrm {tr}} _{\nabla }$ in [Reference Chung, Kim and Kim10] to the stacky case, we obtain a $k[[u]]$ -linear map

$$\begin{align*}{\mathrm{tr}} _{\nabla, I{\mathcal{X}}} : C (q\mathrm{MF}_{dg} (I{\mathcal{X}}, w) ) [[u]] \to \Gamma ( \check{\text{C}} (\Omega ^{\bullet}_{I{\mathcal{X}}} ) ) [[u]] \end{align*}$$

mapping $a_0[a_1| \cdots | a_n]$ for $a_i \in \mathrm {End} _{\check{\text{C}} (P)} (P)$ , $P\in q\mathrm {MF}_{dg} (I{\mathcal {X}}, w) )$ to

$$\begin{align*}\sum _{(j_0, \ldots , j_n) : j_i \in {\mathbb Z} _{\ge 0}} \frac{(-1)^{|j_0 + \cdots + j_n| }{\mathrm{tr}} (a_0 R^{j_0} [\nabla _{P}, a_1] R^{j_1} [\nabla _{P}, a_2] \cdots [\nabla _{P}, a_n] R^{j_n} ) } {(n+|j_0 + \cdots + j_n|)!}. \end{align*}$$

It is clear how to map an arbitrary element of $C (q\mathrm {MF}_{dg} (I{\mathcal {X}}, w) ) [[u]]$ . By the same proof in [Reference Chung, Kim and Kim10, Appendix B], the map ${\mathrm {tr}} _{\nabla , I{\mathcal {X}}} $ is a chain map. Likewise, we have a chain map

$$\begin{align*}{\mathrm{tr}}_{\nabla} : \underline{\overline{C}}^{II} (q\mathrm{MF}_{I{\mathcal{X}}, w} ) [[u]] \to p_*\check{\text{C}} (\Omega ^{\bullet}_{I{\mathcal{X}}} ) [[u]] , \end{align*}$$

where $p : I{\mathcal {X}} \to \underline {{\mathcal {X}}}$ is the natural morphism.

Consider a diagram of chain maps in negative cyclic type complexes

(5.1)

Note that the diagram is commutative, and all arrows may possibly be quasi-isomorphisms but the two top horizontal arrows are quasi-isomorphisms. Hence, we have the following corollaries.

Corollary 5.1. The chain map

$$\begin{align*}{\mathrm{tr}}_{\nabla, I{\mathcal{X}}} \circ \mathfrak{tw} \circ \rho _{{\mathcal{X}}}^*: C (q\mathrm{MF}_{dg} ({\mathcal{X}}, w) ) [[u]] \to \Gamma ( \check{\text{C}} (\Omega ^{\bullet}_{I{\mathcal{X}}} ) ) [[u]] \end{align*}$$

is a quasi-isomorphism.

Corollary 5.2. Under the isomorphism in equation (4.11), the Chern character $\mathrm {ch} _{HN} (P)$ of $P \in \mathrm {MF}_{dg} ({\mathcal {X}}, w)$ is the class represented by Čech cocycle

$$ \begin{align*} {\mathrm{tr}} \left( \mathfrak{can} _{P |_{I{\mathcal{X}}}} \circ \exp ( - u \nabla _{P |_{I{\mathcal{X}}}}^2 - [\nabla_{P |_{I{\mathcal{X}}}} , \delta _{P|_{I{\mathcal{X}}}} + d_{\scriptscriptstyle \check{\text{C}}\text{ech}} ] ) \right) \end{align*} $$

in $\check {H} (I\mathfrak {U}, (\Omega ^{\bullet }_{I{\mathcal {X}}}, -dw|_{I{\mathcal {X}}})) $ .

Example 5.3. Consider a DM stack ${\mathcal {X}}$ of the form $[X/G]$ with X quasi-projective and G a linearly reductive group. Then there is a finite collection $\mathfrak {U}=\{ U_i \}_{i \in I}$ of G-invariant affine open subset $U_i$ of X such that $\bigcup _i U_i = X$ . On the other hand, there is a finite subset S of G such that

$$\begin{align*}I{\mathcal{X}} = \sqcup _{g \in S} [ X^g / \mathrm{C} _G (g ) ]. \end{align*}$$

Note that $I\mathfrak {U} = \{ U_i ^g : i \in I, g \in S \} $ . Instead of affine étale covering, we may use the affine smooth covering $\mathfrak {U}$ for a Čech model of $q\mathrm {MF}_{dg} ({\mathcal {X}}, w)$ and the chain map ${\mathrm {tr}} _{\nabla , I{\mathcal {X}}}$ . Since G is linearly reductive, each $P|_{U_i}$ has a G-equivariant connection, which in turn gives a $\mathrm {C} _{G}(g)$ -equivariant connection $\nabla _{i, g}$ on $P|_{U^g_i}$ . This is because of the surjection of the canonical map ${\mathrm {Hom}} ^G_{{\mathcal {O}} _{X}} (E, J(E)) \to {\mathrm {Hom}} ^G_{{\mathcal {O}} _{X}} (E, E)$ induced from the jet sequence

$$\begin{align*}0 \to \Omega ^1_X \otimes_{{\mathcal{O}}_X} E \to J(E) \to E \to 0. \end{align*}$$

We have

$$\begin{align*}\mathrm{ch}_{HN} (P) = \oplus _{g\in S} {\mathrm{tr}} \left( g \circ \exp ( - u \Pi _{i \in I} \nabla _{i, g}^2 - [\Pi _{i\in I} \nabla_{i, g}, \delta _{P|_{U^g}} + d_{\scriptscriptstyle \check{\text{C}}\text{ech}} ] ) \right). \end{align*}$$

When X itself is affine, then the formula simplifies to

$$\begin{align*}\mathrm{ch}_{HN} (P) = \oplus _{g\in S} {\mathrm{tr}} \left( g \circ \exp ( - u \nabla _{g}^2 - [ \nabla_{ g}, \delta _{P|_{U^g}} ] ) \right) , \end{align*}$$

taking into account the fact that $[\nabla _{g}, d_{\scriptscriptstyle \check{\text{C}}\text{ech}} ] = 0$ .

Let $a \in \oplus _i {\mathbb R} ^i\mathrm {End} (P)$ , then it determines a class in $H^* (I{\mathcal {X}} , (\Omega ^{\bullet }_{I{\mathcal {X}}}, -dw|_{I{\mathcal {X}}} ))$ under the HKR isomorphism. Denote the class by $\tau (a)$ . The assignment $a \mapsto \tau (a)$ is called the boundary-bulk map.

Corollary 5.4 (The boundary-bulk map formula)

$$\begin{align*}\tau (a) = {\mathrm{tr}} \left( a \circ \mathfrak{can} _{P |_{I{\mathcal{X}}}} \circ \exp ( - [\nabla_{P |_{I{\mathcal{X}}}} , \delta _{P|_{I{\mathcal{X}}}} + d_{\scriptscriptstyle \check{\text{C}}\text{ech}} ] ) \right). \end{align*}$$

5.2 A formula via Atiyah class

Let $P \in \mathrm {MF}_{dg} ({\mathcal {X}}, w)$ , and let

be the matrix factorization for $({\mathcal {X}}, 0)$ located at amplitude $[-1, 0 ]$ . The Atiyah class $\hat {\mathrm {at}} (P)$ defined in [Reference Favero and Kim16, Appendix B] is a suitable element of

$$\begin{align*}{\mathrm{Ext}} ^1 (P, P \otimes \Omega ^{-dw} _{{\mathcal{X}}} ). \end{align*}$$

When $w=0$ , we have the decomposition

$$\begin{align*}{\mathrm{Ext}} ^1 (P, P \otimes \Omega ^{-dw} _{{\mathcal{X}}} ) = {\mathrm{Ext}} ^0 (P, P) \oplus {\mathrm{Ext}} ^1 (P, P\otimes \Omega ^1_{{\mathcal{X}}}) \end{align*}$$

and let

$$\begin{align*}\mathrm{at} (P) = proj \circ \hat{\mathrm{at}} (P) \in {\mathrm{Ext}} ^1 (P, P\otimes \Omega ^1_{{\mathcal{X}}}) ,\end{align*}$$

where $proj$ is the projection. For example, when P is a vector bundle F and ${\mathcal {X}}$ is nonstacky, then $\mathrm {at} (F)$ is the usual Atiyah class [Reference Atiyah2]. In this case $\hat {\mathrm {at}} (F) = 1_F + \mathrm {at} (F)$ .

Definition 5.5. Taking into account the convention of the exponential $\underline {\exp }$ of $\hat {P}$ as explained in [Reference Favero and Kim16, Reference Kim and Polishchuk24], we define a naive Chern character of P by

$$\begin{align*}\mathrm{ch} (P) := {\mathrm{tr}} \big( \underline{\exp} (\hat{\mathrm{at}} (P)) \big) \in H^* ({\mathcal{X}}, ( \Omega ^{\bullet}_{{\mathcal{X}}}, -dw)). \end{align*}$$

For simplicity, we abuse notation writing $\exp (\hat {\mathrm {at}} (P))$ for $\underline {\exp } (\hat {\mathrm {at}} (P))$ .

The correct formula for $\mathrm {ch} _{HH} (P)$ in [Reference Kim and Polishchuk25] is

(5.2) $$ \begin{align} \mathrm{ch} _{HH} (P) & = {\mathrm{tr}} \left( \mathfrak{can}_{P|_{I{\mathcal{X}}}} \circ \exp (\hat{\mathrm{at}} (P|_{I{\mathcal{X}}}) ) \right) \\ \nonumber & = \mathrm{ch} (P) + \text{ twisted part }. \end{align} $$

We note that this formula agrees with the formula in Corollary 5.2 since Atiyah class $\hat {\mathrm {at}} (P|_{I{\mathcal {X}}})$ is representable as

$$\begin{align*}(\mathrm{id}_P, - [\nabla_{P |_{I{\mathcal{X}}}} , \delta _{P |_{I{\mathcal{X}}}} + d_{\scriptscriptstyle \check{\text{C}}\text{ech}} ] ) \in \Gamma (I{\mathcal{X}}, \mathrm{End} (P|_{I{\mathcal{X}}}) \otimes \Omega ^{-dw} _{{\mathcal{X}}}\otimes \check{\text{C}} ({\mathcal{O}} _{IX})) \end{align*}$$

in Čech cohomology $\check {H} (I\mathfrak {U}, \Omega ^{\bullet }_{I{\mathcal {X}}}, -dw|_{I{\mathcal {X}}}) $ (see [Reference Kim and Polishchuk24, Proposition 1.3]) and

$$\begin{align*}\underline{\exp} (\mathrm{id}_P, - [\nabla_{P |_{I{\mathcal{X}}}} , \delta _{P |_{I{\mathcal{X}}}} + d_{\scriptscriptstyle \check{\text{C}}\text{ech}} ] ) = \exp (- [\nabla_{P |_{I{\mathcal{X}}}} , \delta _{P |_{I{\mathcal{X}}}} + d_{\scriptscriptstyle \check{\text{C}}\text{ech}} ]). \end{align*}$$

The boundary bulk map formula can also be written in terms of the Atiyah class:

$$\begin{align*}\tau (a) = {\mathrm{tr}} \left( a \circ \mathfrak{can}_{P|_{I{\mathcal{X}}}} \circ \exp (\hat{\mathrm{at}} (P|_{I{\mathcal{X}}}) ) \right). \end{align*}$$

Definition 5.6. For a vector bundle E on $I{\mathcal {X}}$ , we define

$$\begin{align*}\mathrm{ch} _{tw} (E) := {\mathrm{tr}} (\mathfrak{can} _E \exp (\mathrm{at} (E)) \end{align*}$$

and Todd class $\mathrm {td} _{tw} (E)$ of E by the expression of Todd class in terms of the Chern character $\mathrm {ch} _{tw} (E)$ . For example, $\mathrm {td} _{tw} (T_{I{\mathcal {X}}})$ is defined. Since $T_{I{\mathcal {X}}}$ is fixed under the canonical automorphism, we simply write $\mathrm {td} (T_{I{\mathcal {X}}})$ for $\mathrm {td} _{tw} (T_{I{\mathcal {X}}})$ .

5.3 Proof of Theorem 1.1

The first statement of Theorem 1.1 is Theorem 4.19. The second statement follows from equation (5.2).

5.4 Compactly supported case

Let Z be a closed substack of ${\mathcal {X}}$ proper over k. Let P be a matrix factorization for $({\mathcal {X}}, w)$ which is coacyclic over ${\mathcal {X}} - Z$ . Note that

$$\begin{align*}\hat{\mathrm{at}} (P|_{I{\mathcal{X}}}) \in {\mathrm{Ext}} ^1 (P|_{I{\mathcal{X}}}, P|_{I{\mathcal{X}}} \otimes \Omega ^{-dw|_{I{\mathcal{X}}}} _{I{\mathcal{X}}} ) = {\mathrm{Ext}} ^1 _{IZ} (P|_{I{\mathcal{X}}}, P|_{I{\mathcal{X}}} \otimes \Omega ^{-dw|_{I{\mathcal{X}}}} _{I{\mathcal{X}}} ). \end{align*}$$

To emphasize that $\hat {\mathrm {at}} (P|_{I{\mathcal {X}}}) $ can be considered as an $IZ$ -supported extension class, write $\hat {\mathrm {at}}_Z (P|_{I{\mathcal {X}}}) $ for $\hat {\mathrm {at}} (P|_{I{\mathcal {X}}}) $ . Let $\mathrm {MF}_{dg} ({\mathcal {X}}, w))_Z$ be the full subcategory of $\mathrm {MF}_{dg} ({\mathcal {X}}, w)$ consisting of all matrix factorization for $({\mathcal {X}}, w)$ that are coacyclic over ${\mathcal {X}} - Z$ .

Corollary 5.7. There is an isomorphism

$$\begin{align*}\mathrm{MC} (\mathrm{MF}_{dg} ({\mathcal{X}}, w))_Z \cong {\mathbb R}\Gamma _Z (\Omega ^{\bullet}_{I{\mathcal{X}}}, -dw|_{I{\mathcal{X}}} , d) \end{align*}$$

in the derived category of mixed complexes. Under the isomorphism $\mathrm {ch} ^Z_{HH}, (P)$ is equal to

$$\begin{align*}{\mathrm{tr}} \big( \mathfrak{can} _{P|_{I{\mathcal{X}}}} \exp ( \hat{\mathrm{at}}_Z (P |_{I{\mathcal{X}}} )) \big) \end{align*}$$

in ${\mathbb H} ^*_{IZ} ( I{\mathcal {X}}, (\Omega ^{\bullet }_{I{\mathcal {X}}}, -dw|_{I{\mathcal {X}}} ) )$ .

6.1 The categorical HRR

The results in this subsection are taken from [Reference Polishchuk and Vaintrob35, Reference Shklyarov41] with a weaker condition on a dg category ${\mathcal {A}}$ . Instead of assuming that ${\mathcal {A}}$ is saturated, we assume that ${\mathcal {A}}$ is locally proper and smooth.

Definition 6.1. Let ${\mathcal {A}}$ be a locally proper dg category: That is, for every $x, y\in {\mathcal {A}}$ , the dimension $\sum _{i\in {\mathbb G}} \dim H^i {\mathrm {Hom}} _{{\mathcal {A}}} (x, y)$ of total cohomology of ${\mathrm {Hom}} _{{\mathcal {A}}} (x, y)$ is finite. Let

$$\begin{align*}\langle , \rangle _{can} : HH _* ({\mathcal{A}}) \times HH_* ({\mathcal{A}} ^{op}) \to k \end{align*}$$

be the canonical pairing of ${\mathcal {A}}$ defined by Shklyarov. It is a k-linear pairing.

6.1.1 Transformations by bimodules

Definition 6.2. For a dg category ${\mathcal {C}}$ , we take the projective model structure on the category $\mathrm {Mod} ({\mathcal {C}})$ of right ${\mathcal {C}}$ -modules. The cofibrant objects are exactly the summands of semifree dg-modules. A right ${\mathcal {C}}$ -module N is called perfect if N is a cofibrant object which is compact in the derived category $D({\mathcal {C}} )$ of right ${\mathcal {C}}$ -modules. Let $\mathrm {Mod} _{dg}({\mathcal {C}})$ be the dg category of right ${\mathcal {C}}$ -modules. Let $\mathrm {Perf} ({\mathcal {C}})$ be the full subcategory of $\mathrm {Mod} _{dg}({\mathcal {C}})$ consisting of all perfect ${\mathcal {C}}$ -modules. We call a dg category ${\mathcal {C}}$ is smooth if the diagonal bimodule $\Delta _{{\mathcal {A}}}$ is a perfect bimodule.

From now on, let ${\mathcal {A}}$ and $\mathcal {B}$ be locally perfect and smooth dg categories unless otherwise stated.

Lemma 6.3. The total dimension of Hochschild homology $HH_*({\mathcal {A}})$ of ${\mathcal {A}}$ is finite. The dg category $\mathrm {Perf} ({\mathcal {A}} \otimes \mathcal {B})$ is locally perfect and smooth.

Proof. The first claim amounts $ \Delta _{{\mathcal {A}}} \otimes _{{\mathcal {A}} ^{op} \otimes {\mathcal {A}}}^{\mathbb {L}} \Delta _{{\mathcal {A}}} $ is a perfect dg k-module, which follows from tensor-hom adjunction and the conditions on ${\mathcal {A}}$ . The second claim follows from [Reference Lunts and Schnürer30, Lemma 2.13, 2.14, 2.15] since k is a field.

For a right ${\mathcal {A}} ^{op} \otimes \mathcal {B}$ -module M, there is a dg functor $T_M : \mathrm {Perf} ({\mathcal {A}}) \to \mathrm {Mod}_{dg} (\mathcal {B}) $ sending N to $N\otimes _{{\mathcal {A}}} M$ . If M is representable, then $T_M$ factors though $\mathrm {Perf} (\mathcal {B})$ since ${\mathcal {A}}$ is locally proper. Hence, this is the case for every perfect ${\mathcal {A}} ^{op} \otimes \mathcal {B}$ -module M.

Let $M \in \mathrm {Perf} ({\mathcal {A}} ^{op} \otimes \mathcal {B})$ , and let $\mathrm {Ch} (M) = \sum _i \gamma _i \otimes \gamma ^i $ under the Künneth isomorphism $ HH_* (\mathrm {Perf} ({\mathcal {A}} ^{op} \otimes \mathcal {B}) ) \cong HH_* (\mathrm {Perf} ({\mathcal {A}} ^{op})) \otimes HH_* (\mathrm {Perf} (\mathcal {B}) ) .$ Let $HH (T_M) : HH_* (\mathrm {Perf} ({\mathcal {A}})) \to HH_* (\mathrm {Perf} (\mathcal {B}))$ be the homomorphism induced by $T_M : \mathrm {Perf} ({\mathcal {A}}) \to \mathrm {Perf} (\mathcal {B})$ .

Proposition 6.4. If $\langle , \rangle _{can}$ denotes the canonical pairing of $\mathrm {Perf} ({\mathcal {A}} ^{op} \otimes \mathcal {B})$ , then for every $\sigma \in HH_* (\mathrm {Perf} ({\mathcal {A}}))$ we have

$$\begin{align*}HH (T_M) (\sigma ) = \sum _i \langle \sigma, \gamma _i \rangle _{can} \gamma ^i. \end{align*}$$

6.1.2 The characteristic property

There are natural isomorphisms

$$\begin{align*}HH_* (\mathrm{Perf} ({\mathcal{A}} ^{op} \otimes {\mathcal{A}} )) \cong HH_* ({\mathcal{A}} ^{op} \otimes {\mathcal{A}}) \cong HH_* ({\mathcal{A}} ^{op}) \otimes HH_*( {\mathcal{A}} ) \end{align*}$$

by the Morita invariance and the Künneth isomorphism. Write

$$\begin{align*}\mathrm{Ch} _{HH}(\Delta _{{\mathcal{A}}}) = \sum _i T^i \otimes T_i \in HH_*({\mathcal{A}} ^{op}) \otimes HH_* ({\mathcal{A}} ). \end{align*}$$

Then by Proposition 6.4 we obtain this.

Corollary 6.5. The canonical pairing $\langle , \rangle _{can}$ is characterized by two conditions: (1) it is nondegenerate and (2) it satisfies the ‘diagonal decomposition’ property:

$$\begin{align*}\sum _i \langle \gamma , T^i \rangle \langle T_i , \gamma ' \rangle = \langle \gamma, \gamma ' \rangle \end{align*}$$

for every $\gamma \in HH_*({\mathcal {A}}), \gamma ' \in HH_*({\mathcal {A}} ^{op})$ .

6.1.3 The Cardy condition

Consider objects $x, y \in {\mathcal {A}}$ . Let a and b be closed endomorphisms of x and y, respectively. Let

$$\begin{align*}L_b \circ R_a : {\mathrm{Hom}} _{{\mathcal{A}}} (x, y) \to {\mathrm{Hom}} _{{\mathcal{A}}} (x, y) , \ (-1)^{|a||c|} \mapsto b \circ c \circ a. \end{align*}$$

Theorem 6.6. We have

$$\begin{align*}{\mathrm{tr}} (L_b \circ R_a) = \langle [b], [a] \rangle _{can}. \end{align*}$$

For the identities $a=1_x$ , $b=1_y$ , it is specialized to

$$\begin{align*}\chi (x, y) = \langle \mathrm{Ch}_{HH} (y), \mathrm{Ch}_{HH} (x) \rangle _{can}. \end{align*}$$

6.2 On $\mathrm {MF}_{dg} ({\mathcal {X}}, w)$

Consider two proper LG models $({\mathcal {X}}, w)$ and $({\mathcal {Y}}, v)$ . We want to show that $\mathrm {MF}_{dg} ({\mathcal {X}}, w)$ is locally proper, smooth; and the following $(\dagger )$ and $(\star )$ hold:

$(\dagger )$ There is a natural dg functor

$$ \begin{align*} \mathrm{MF}_{dg} ({\mathcal{X}} \times {\mathcal{Y}}, w \boxplus v) & \to \mathrm{Perf} (\mathrm{MF}_{dg}({\mathcal{X}}, w) \otimes \mathrm{MF}_{dg} ({\mathcal{Y}}, v)) \quad \text{ defined by } \\ E & \mapsto \Psi (E) : x \otimes y \mapsto {\mathrm{Hom}} _{\mathrm{MF}_{dg} ({\mathcal{X}} \times {\mathcal{Y}}, w \boxplus v) } (x\boxtimes y, E). \end{align*} $$

Here, $w \boxplus v$ denotes $w \otimes 1 + 1 \otimes v$ .

$(\star )$ The triangulated category $[\mathrm {MF}_{dg} ({\mathcal {X}} \times {\mathcal {Y}}, w \boxplus v)]$ is the smallest full triangulated subcategory containing all exterior products closed under finite coproducts and summands. Here, an object of $[\mathrm {MF}_{dg} ({\mathcal {X}} \times {\mathcal {Y}}, w \boxplus v)]$ is called an exterior product if it is isomorphic to $x\boxtimes y$ for some $x \in \mathrm {MF}_{dg}({\mathcal {X}}, w) , y \in \mathrm {MF}_{dg}({\mathcal {Y}}, v) $ .

Since $\mathrm {MF}_{dg} ({\mathcal {X}}, w)$ is clearly locally proper, it is enough to show $(\star )$ . We check this when ${\mathcal {X}}$ is a stack quotient $[X/G]$ of a smooth variety by an action of an affine algebraic group G. When ${\mathbb G} = {\mathbb Z}$ , then $w=0$ . Note that $(\star )$ holds by Theorem 2.29 and Corollary 4.21 of [Reference Ballard, Favero and Katzarkov3]. When ${\mathbb G}={\mathbb Z}/2$ , then w is a G-invariant function on X, not identically zero on any component of X. Note that $(\star )$ holds by Theorem 2.29 and Lemma 4.23 of [Reference Ballard, Favero and Katzarkov3].

6.3 A geometric realization of the diagonal module

Consider two proper LG models $({\mathcal {X}}, w), ({\mathcal {Y}}, v)$ . Suppose that ${\mathcal {X}}$ , ${\mathcal {Y}}$ are stack quotients of smooth varieties by actions of affine algebraic groups. Let $f: {\mathcal {X}} \to {\mathcal {Y}}$ be a proper morphism with $f^* v = w$ . We call $f : ({\mathcal {X}}, w) \to ({\mathcal {Y}}, v)$ an proper LG morphism. Choose an affine étale cover $\mathfrak {U} \to {\mathcal {X}}$ and $\mathfrak {U} ' \to {\mathcal {Y}}$ . Denote

$$\begin{align*}{\mathcal{A}} := \mathrm{MF}_{dg} ({\mathcal{X}}, w) , \ \mathcal{B} := \mathrm{MF}_{dg} ({\mathcal{Y}} , v ). \end{align*}$$

They are locally proper and smooth as seen in § 6.2.

Let $-w\boxplus v := - w\otimes 1 + 1\otimes v$ , and let $ \mathrm {MF}_{dg} ({\mathcal {X}} \times {\mathcal {Y}} , -w\boxplus v )$ be the Čech dg model of the matrix factorizations for $({\mathcal {X}} \times {\mathcal {Y}} , -w\boxplus v )$ with respect to the affine cover $\mathfrak {U} \times \mathfrak {U} ' \to {\mathcal {X}} \times {\mathcal {Y}}$ . Then by $(\dagger )$ we have a natural dg functor

$$\begin{align*}\Psi : \mathrm{MF}_{dg} ({\mathcal{X}} \times {\mathcal{Y}} , -w\boxplus v ) \to \mathrm{Perf} ({\mathcal{A}} ^{op} \otimes \mathcal{B}). \end{align*}$$

Let $D: {\mathcal {A}} ^{op} \to \mathrm {MF}_{dg} ({\mathcal {X}}, -w)$ be the duality functor. Then we have a commuting diagram of isomorphisms

(6.1)

where HKR and Künneth are the HKR-type isomorphisms in § 4.8 and the Künneth isomorphisms, respectively.

Consider a matrix factorization K for $({\mathcal {X}} \times {\mathcal {Y}}, -w\boxplus v)$ . For example, we have a coherent factorization

$$\begin{align*}{\mathcal{O}} _{\Gamma _f} := (\Gamma _f )_* {\mathcal{O}} _{{\mathcal{X}}} \text{ for } ({\mathcal{X}} \times {\mathcal{Y}}, -w\boxplus v). \end{align*}$$

Since ${\mathcal {X}} \times {\mathcal {Y}}$ satisfies the resolution property by [Reference Ballard, Favero and Katzarkov3, Theorem 2.29], ${\mathcal {O}} _{\Gamma _f}$ is quasi-isomorphic to a matrix factorization.

For all $x \in {\mathcal {A}}, y\in \mathcal {B}$ , there is a natural quasi-isomorphism

$$\begin{align*}{\mathbb R}{\mathrm{Hom}} ( y, q_* ( K \otimes p^*x )) \sim_{qiso} {\mathrm{Hom}} _{\mathrm{MF}_{dg} ({\mathcal{X}} \times {\mathcal{Y}} , -w\boxplus v)} ( x^{\vee} \boxtimes y, K ) \end{align*}$$

functorial under the morphisms in the categories $\mathcal {B}$ and ${\mathcal {A}}$ . This shows the following, which will be used later.

The second statement in the above lemma is also in Lemma 5.24 of [Reference Ballard, Favero and Katzarkov3].

6.4 An explicit realization of the canonical pairing

Theorem 6.9. Let $({\mathcal {X}}, w)$ be a proper LG model. Assume that ${\mathcal {X}}$ is a smooth quotient DM stack which satisfies the resolution property. Then the canonical pairing coincides with the pairing defined by

(6.2) $$ \begin{align} \int _{I{\mathcal{X}}} (-1)^{\binom{\dim _{I{\mathcal{X}}}+1}{2}} \cdot \wedge \cdot \wedge \frac{\mathrm{td} (T_{I{\mathcal{X}}})}{\mathrm{ch}_{tw} (\lambda _{-1} ( N^{\vee}_{I{\mathcal{X}} / {\mathcal{X}}}))} , \end{align} $$

where $\dim _{I{\mathcal {X}}}$ is the locally constant dimension function of $I{\mathcal {X}}$ .

Proof. We prove the characteristic property in Corollary 6.5 for the pair (6.2). The nondegeneracy follows from Serre duality [Reference Nironi32] as argued in [Reference Favero and Kim16, § 4.1]. By Lemma 6.8 the ‘diagonal decomposition’ is

(6.3) $$ \begin{align} \sum _i \int _{I{\mathcal{X}}} \gamma \cdot t^i \cdot \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) \int _{I{\mathcal{X}}} t^i \cdot \gamma ' \cdot \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) = \int _{I{\mathcal{X}}} (-1)^{\binom{\dim _{I{\mathcal{X}}}+1}{2}} \gamma " \cdot \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}), \end{align} $$

where

(6.4) $$ \begin{align} \sum_i t^i \otimes t_i & = \mathrm{ch} _{HH} (\Delta _* {\mathcal{O}} _{{\mathcal{X}}}) \in {\mathbb H} ^* (I{\mathcal{X}}, (\Omega ^{\bullet}_{I{\mathcal{X}}}, -dw|_{I{\mathcal{X}}})) \otimes {\mathbb H}^* (I{\mathcal{X}}, (\Omega ^{\bullet}_{I{\mathcal{X}}}, dw|_{I{\mathcal{X}}})) \nonumber \\ \gamma & \in {\mathbb H} ^* (I{\mathcal{X}}, (\Omega ^{\bullet}_{I{\mathcal{X}}}, dw|_{I{\mathcal{X}}})), \gamma ' \in {\mathbb H}^* (I{\mathcal{X}}, (\Omega ^{\bullet}_{I{\mathcal{X}}}, -dw|_{I{\mathcal{X}}})) \nonumber \\ & \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) := \frac{\mathrm{td} (T_{I{\mathcal{X}}})}{\mathrm{ch}_{tw} (\lambda _{-1} ( N_{I{\mathcal{X}}/{\mathcal{X}}}^{\vee}))}. \end{align} $$

To show equation (6.3), we use the deformation to the normal cone.

Over ${\mathbb P}^1$ , there is a deformation stack ${\mathcal {M}}^{\circ }$ to the normal cone $N_{{\mathcal {X}} / {\mathcal {X}} ^2}$ : The general fiber is ${\mathcal {X}} ^2$ and the special fiber, say over $\infty $ , is the vector bundle stack $N_{{\mathcal {X}} / {\mathcal {X}} ^2} \cong T_{{\mathcal {X}}}$ . It comes with a natural morphism $h: {\mathcal {M}} ^{\circ } \to {\mathcal {X}} ^2$ , a flat morphism $pr: {\mathcal {M}} ^{\circ } \to {\mathbb P} ^1$ , and a morphism $\widetilde {\Delta } : {\mathcal {X}} \times {\mathbb P} ^1 \to {\mathcal {M}}^{\circ }$ such that $(h, pr) \circ \widetilde {\Delta } = \Delta \times \mathrm {id}_{{\mathbb P} ^1}$ . Consider the fiber square diagram

Here, we use facts that $I{\mathcal {X}} ^2 \cong I {\mathcal {X}} \times I {\mathcal {X}}$ and $T_{I{\mathcal {X}}} \overset {}{\cong } IT_{{\mathcal {X}}}$ by Lemma 3.1.

Let

$$\begin{align*}\pi _{{\mathcal{X}}} : T_{{\mathcal{X}}} \to {\mathcal{X}} \text{ and } \pi _{I{\mathcal{X}}} : T_{I{\mathcal{X}}} \to I{\mathcal{X}} \end{align*}$$

be the projections from vector bundles. Then left-hand side of equation (6.3) becomes

(6.5) $$ \begin{align} \int _{N_{I{\mathcal{X}}/ I{\mathcal{X}} ^2 }} \pi_{I{\mathcal{X}}}^* (\gamma ") (\mathrm{ch}_{tw} (\mathbb{L} \rho ^*_{T_{\mathcal{X}}} \delta_* {\mathcal{O}} _{{\mathcal{X}}})) \cdot \pi_{I{\mathcal{X}}}^* (\widetilde{\mathrm{td}}_{I{\mathcal{X}}} )^2 \end{align} $$

by the Tor independence of the pair $({\mathcal {X}} \times {\mathbb P} ^1, {\mathcal {M}} ^{\circ } \times _{{\mathbb P} ^1} p)$ over ${\mathcal {M}} ^{\circ }$ for $p=0, \infty $ and the base change II in § 7.0.1; for details, see the proof of [Reference Kim23, § 3.3]. Let $\sigma $ be the diagonal section of the vector bundle $\pi _{{\mathcal {X}}}^* T_{\mathcal {X}}$ on $T_{{\mathcal {X}}}$ , and let $\mathrm {Kos} (\sigma )$ denote the Koszul complex associated to the section $\sigma $ . Then equation (6.5) becomes

$$ \begin{align*} \int _{N_{I{\mathcal{X}}/ I{\mathcal{X}} ^2 }} \pi_{I{\mathcal{X}}}^* (\gamma ") (\mathrm{ch}_{tw} ( \rho ^* _{T_{{\mathcal{X}}}}\mathrm{Kos} (\sigma))) \cdot \pi_{I{\mathcal{X}}}^* (\widetilde{\mathrm{td}}_{I{\mathcal{X}}})^2, \end{align*} $$

which equals, by the functoriality and the projection formula § 7.0.1,

(6.6) $$ \begin{align} \int _{I{\mathcal{X}} } \left( ( \gamma " \cdot \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) \int _{\pi_{I{\mathcal{X}}}} (\mathrm{ch}_{tw} ( \rho ^*_{T_{{\mathcal{X}}}} \mathrm{Kos} (\sigma))) \cdot \pi_{I{\mathcal{X}}}^* (\widetilde{\mathrm{td}}_{I{\mathcal{X}}} ) \right). \end{align} $$

Let $I\sigma $ be the diagonal section of the vector bundle $\pi _{I{\mathcal {X}}} ^* T_{I{\mathcal {X}}} $ on $T_{I{\mathcal {X}}}$ . From the short exact sequence in § 3.4.2, we have a short exact sequence

(6.7) $$ \begin{align} 0 \to \pi_{I{\mathcal{X}}} ^* T_{I{\mathcal{X}}} \xrightarrow{\iota} \pi_{I{\mathcal{X}}} ^*( T_{{\mathcal{X}}} |_{I{\mathcal{X}}}) \to \pi_{I{\mathcal{X}}} ^* N_{I{\mathcal{X}} / {\mathcal{X}}} \to 0; \\ \nonumber \text{ with } \iota (I\sigma) = \pi_{I{\mathcal{X}}}^* \sigma \end{align} $$

and an equality

(6.8) $$ \begin{align} T_{{\mathcal{X}}} |_{I{\mathcal{X}}} ^{\mathrm{{fixed}}} = T_{I{\mathcal{X}}}. \end{align} $$

Then equation (6.6) becomes, by equations (6.7) and (6.8),

$$ \begin{align*} \int _{I{\mathcal{X}}} \left( ( \gamma " \cdot \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) ) \int _{\pi_{I{\mathcal{X}}}} \mathrm{ch} (\mathrm{Kos} (I\sigma) ) \pi_{I{\mathcal{X}}}^* \mathrm{td} (T_{I{\mathcal{X}}}) \right) \end{align*} $$

which becomes, by § 7.0.2,

$$ \begin{align*} \int _{I{\mathcal{X}}} (-1)^{\binom{\dim _{I{\mathcal{X}}}+1}{2}} \gamma " \cdot \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}). \end{align*} $$

This completes the proof.

6.5 Proof of Theorem 1.2

This follows from Theorems 6.6 and 6.9.

6.6 GRR

Consider a proper morphism $f: {\mathcal {X}} \to {\mathcal {Y}}$ with $f^* v = w$ as in § 6.3. Let $K_0 ({\mathcal {A}})$ be the Grothendieck group of the homotopy category of a pretriangulated dg category ${\mathcal {A}}$ . Denote $f_! : K_0 (\mathrm {MF}_{dg} ({\mathcal {X}}, w) ) \to K_0 (\mathrm {MF}_{dg} ({\mathcal {Y}} , v) ) $ be the homomorphism in the Grothendieck groups induced from ${\mathbb R} f_*$ .

Theorem 6.10 (=Theorem 1.3)

The diagram

is commutative. Here, $\widetilde {\mathrm {td}} (T_{If}) : = \widetilde {\mathrm {td}} (T_{I{\mathcal {X}}}) / If^* \widetilde {\mathrm {td}} (T_{I{\mathcal {Y}}})$ and $\dim _{If} = \dim _{I{\mathcal {X}}} - \dim _{I{\mathcal {Y}}} $ , where $\widetilde {\mathrm {td}} (T_{?})$ is $\widetilde {\mathrm {td}}$ for $T_?$ in equation (6.4).

Proof. The proof is parallel to that of Theorem 3.6 of [Reference Kim23]. The upper rectangle is clearly commutative. We show the commutativity of the lower rectangle. For $\gamma \in HH_* (\mathrm {MF}_{dg} ({\mathcal {X}}, w)) $ , let ${\alpha } := I_{HKR} (\gamma )$ , ${\alpha } ' := I_{HKR} (HH ({\mathbb R} f_* ) (\gamma )) $ , and let

$$\begin{align*}\mathrm{ch} (\Psi ( {\mathcal{O}} _{\Gamma _f})) = \sum_i T^i \otimes T_i \in {\mathbb H} ^{*} (I{\mathcal{X}}, (\Omega ^{\bullet}_{I{\mathcal{X}}}, dw|_{I{\mathcal{X}}})) \otimes {\mathbb H} ^{*} (I{\mathcal{Y}}, (\Omega ^{\bullet}_{I{\mathcal{Y}}}, -dv|_{I{\mathcal{Y}}})) , \end{align*}$$

then by Proposition 6.4 and Theorem 6.9 we have for $\beta \in {\mathbb H} ^{-*} (I{\mathcal {Y}}, (\Omega ^{\bullet }_{I{\mathcal {Y}}}, dv|_{I{\mathcal {Y}}}))$

(6.9) $$ \begin{align} \int _{I{\mathcal{Y}}} {\alpha} ' \wedge \beta \wedge \widetilde{\mathrm{td}} (T_{I{\mathcal{Y}}}) = \sum _i \int _{I{\mathcal{X}}} (-1)^{{\dim _{I{\mathcal{X}}}+1}\choose{2}} {\alpha} \wedge T^i \wedge \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) \int _{I{\mathcal{Y}}} T_i \wedge \beta \wedge \widetilde{\mathrm{td}} (T_{I{\mathcal{Y}}}). \end{align} $$

Let $\pi $ denote the projection $If^*T_{I{\mathcal {Y}}} \to I{\mathcal {X}}$ , and let s be the diagonal section of $\pi ^*If^*T_{I{\mathcal {Y}}}$ on $If^*T_{I{\mathcal {Y}}} $ . Then the deformation argument for $\Gamma _f : {\mathcal {X}} \to {\mathcal {X}} \times {\mathcal {Y}}$ as in the proof of Theorem 6.9 shows that

$$ \begin{align*} & \text{right-hand side of equation (6.9)} \\ & = \int _{I{\mathcal{X}}\times I{\mathcal{Y}} } (-1)^{{\dim _{I{\mathcal{X}}}+1}\choose{2}} ({\alpha} \otimes \beta) \wedge \mathrm{ch} ({\mathcal{O}} ^{w\boxminus v}_{\Gamma _f }) \wedge ( \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) \otimes \widetilde{\mathrm{td}} (T_{I{\mathcal{Y}}})) \\ & = \int _{f^*T_{I{\mathcal{Y}}}}(-1)^{{\dim _{I{\mathcal{X}}}+1}\choose{2}} \pi ^* ({\alpha} \wedge If^* \beta \wedge \widetilde{\mathrm{td}} (If^* T_{I{\mathcal{Y}}}) \wedge \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) ) \wedge \mathrm{ch} (\mathrm{Kos} (s)) \\ & = \int _{I{\mathcal{X}}} (-1)^{{\dim _{If}+1}\choose{2}} {\alpha} \wedge f^* \beta \wedge \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}) = \int _{I{\mathcal{Y}}} (\int _{If} (-1)^{{\dim _{If}+1}\choose{2}} {\alpha} \wedge \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}})) \wedge \beta .\end{align*} $$

This completes the proof.

Remark 6.11. We briefly discuss how the GRR for $\Delta $ would compute the canonical pairing, which shows some relationship between GRR and the canonical pairing.

Consider the Riemann–Roch map

$$ \begin{align*} \mathrm{ch} ^{\tau}: K_0 ({\mathcal{X}} , w) & \to {\mathbb H} ^* (I{\mathcal{X}}, (\Omega ^{\bullet}_{I{\mathcal{X}}} , -dw)) \\ E &\mapsto \mathrm{ch} _{HH} (E) \widetilde{\mathrm{td}} (T_{I{\mathcal{X}}}). \end{align*} $$

Suppose that we have a GRR type theorem for the diagonal map $\Delta : {\mathcal {X}} \to {\mathcal {X}} ^2$ :

(6.10) $$ \begin{align} \Delta _* \mathrm{ch} ^{\tau} ({\mathcal{O}} _X) = \mathrm{ch} ^{\tau} (\Delta _* {\mathcal{O}} _{{\mathcal{X}}}) = \frac{\mathrm{td} (I{\mathcal{X}} ^2) \mathrm{ch}_{HH} (\Delta _* {\mathcal{O}} _{{\mathcal{X}}})}{\mathrm{ch}_{tw} (N_{I{\mathcal{X}} ^2/{\mathcal{X}} ^2})}. \end{align} $$

This yields a formula

$$ \begin{align*} \mathrm{ch}_{HH}(\Delta _* ( {\mathcal{O}} _{{\mathcal{X}}})) = \Delta _* \left( \frac{ \mathrm{ch}_w(N_{I{\mathcal{X}}/ {\mathcal{X}}}) }{ \mathrm{td} (I{\mathcal{X}}) } \mathrm{ch} _{HH} ({\mathcal{O}} _{{\mathcal{X}}} )\right) = \Delta _* \frac{ \mathrm{ch}_w(N_{I{\mathcal{X}}/ {\mathcal{X}}}) }{ \mathrm{td} (I{\mathcal{X}}) } \end{align*} $$

since $\mathrm {ch} _{HH} ({\mathcal {O}} _{{\mathcal {X}}} ) = 1.$ Denote $ \widetilde {\mathrm {td}} = \widetilde {\mathrm {td}} {T_{I{\mathcal {X}}}} $ . Then

$$ \begin{align*} \int _{I{\mathcal{X}}} (-1)^{\binom{\dim _{I{\mathcal{X}}}+1}{2}} \gamma \cdot t^i \cdot \widetilde{\mathrm{td}} \ &\int _{I{\mathcal{X}}} (-1)^{\binom{\dim _{I{\mathcal{X}}}+1}{2}} t_i \cdot \gamma' \cdot \widetilde{\mathrm{td}} \\ &= \int _{I{\mathcal{X}} ^2} \gamma \otimes \gamma ' \cdot \Delta _* \frac{ \mathrm{ch}_w(N_{I{\mathcal{X}}/ {\mathcal{X}}}) }{ \mathrm{td} (I{\mathcal{X}}) } \cdot \widetilde{\mathrm{td}} \otimes \widetilde{\mathrm{td}} = \int _{I{\mathcal{X}}} (-1)^{\binom{\dim _{I{\mathcal{X}}}+1}{2}} \gamma \cdot \gamma ' \cdot \widetilde{\mathrm{td}} , \end{align*} $$

which is the characteristic property of the canonical pairing. Thus, equation (6.10) implies that the canonical pairing is $\int _{I{\mathcal {X}}} (-1)^{\binom {\dim _{I{\mathcal {X}}}+1}{2}} \cdot \wedge \cdot \wedge \widetilde {\mathrm {td}} (T_{I{\mathcal {X}}})$ .

7.0.1 Basic properties

In this section, all stacks are assumed to be smooth separated DM stacks of finite type over k. Let $f: {\mathcal {X}} \to {\mathcal {Y}}$ be a morphism. Assume that they are pure dimensional, and let d be $\dim {\mathcal {X}} - \dim {\mathcal {Y}}$ .

Definition 7.1. Once we have the right adjoint functor $f^!$ of ${\mathbb R} f_*$ , as in [Reference Kim23] we can define

$$\begin{align*}\int _f: H ^{q}_{\sharp _1} ({\mathcal{X}}, \Omega ^p_{{\mathcal{X}}}) \to H^{q-d}_{\sharp _2} ({\mathcal{Y}}, \Omega _{{\mathcal{Y}}}^{p-d}), \end{align*}$$

where $(\sharp _1, \sharp _2)$ is either $(c, c)$ or $(cf, \emptyset )$ . When ${\mathcal {Y}}$ is ${\mathrm {Spec}}\,k$ , write $\int _{{\mathcal {X}}}$ for $\int _f$ .

The following can be straightforwardly proven as in [Reference Kim23, § 3.6].

1. 1. (Base change I) Consider a Cartesian diagram (7.1) below. Assume that f is a flat, proper and locally complete intersection morphism. Then

   $$ \begin{align*} \int _{{\mathcal{X}} '} v^* (\gamma ) = u^* \left(\int _f \gamma \right). \end{align*} $$

2. 2. (Base change II) Consider a Cartesian diagram (7.1) below. Assume that f is a flat morphism, ${\mathcal {Y}}$ is a connected one-dimensional smooth scheme, and u is the embedding of a closed point ${\mathcal {Y}}'$ of ${\mathcal {Y}}$ . Then

   $$ \begin{align*} \int _{{\mathcal{X}} '} v^* (\gamma ) = u^* \left(\int _f \gamma \right) \in k. \end{align*} $$

3. 3. (Functoriality) Let ${\mathcal {X}} \xrightarrow {f} {\mathcal {Y}} \xrightarrow {g} {\mathcal {Z}}$ be morphisms. Then

   $$\begin{align*}\int _{g \circ f} = \int _g \circ \int _f. \end{align*}$$

4. 4. (Projection formula) Let ${\mathcal {X}} \xrightarrow {f} {\mathcal {Y}}$ be a morphisms. Then

   $$\begin{align*}\int _f (f^* \sigma \wedge \gamma) = \sigma \wedge \int _f \gamma \end{align*}$$

   for $\gamma \in H^d_{cf} ({\mathcal {X}}, \Omega ^d_{{\mathcal {X}}})$ and $\sigma \in H^q ({\mathcal {Y}}, \Omega _{{\mathcal {Y}}}^p)$ .

Remark 7.3. The base change formula I and II relies on the following form of a base change formula; suppose that we have a tor-independent Cartesian diagram of DM stack

(7.1)

such that f is proper. One can ask whether the canonical base change map

$$\begin{align*}\beta: v^*f^! \to g^!u^*. \end{align*}$$

is an isomorphism. It is known to be true in scheme cases and generalized to stacks when u is representable affine. (See [Reference Huybrechts19, § 2, § 3] [Reference Lipman and Hashimoto28]).

7.0.2 Computation

Let E be a vector bundle on ${\mathcal {X}}$ of rank n, let $\pi : E \to {\mathcal {X}}$ be the projection and let s be the diagonal section of $\pi ^*E$ . Since $\pi $ is representable, we have

$$\begin{align*}\int _{\pi} \mathrm{ch} (\mathrm{Kos} (s)) \mathrm{td} (\pi ^* E) = (-1)^{{n+1}\choose{2}} \end{align*}$$

by the base change I in § 7.0.1 and the computation of [Reference Kim23, § 3.6.6].