2.1 Small linear categories

An $S$-linear category $\mathcal {C}$ is a small idempotent-complete stable $\infty$-category equipped with a module structure over $\operatorname {D_{{perf}}}(S)$. The collection of all $S$-linear categories is organized into an $\infty$-category $\mathrm {Cat}_S$, which admits a symmetric monoidal structure. For $\mathcal {C}, \mathcal {D} \in \mathrm {Cat}_S$ we denote by

\[ \mathcal{C} \otimes_{\operatorname{D_{{perf}}}(S)} \mathcal{D} \in \mathrm{Cat}_S \]

their tensor product. A morphism $\mathcal {C} \to \mathcal {D}$ in $\mathrm {Cat}_S$, also called an $S$-linear functor, is an exact functor that suitably commutes with the action of $\operatorname {D_{{perf}}}(S)$; these morphisms form the objects of an $S$-linear category $\mathrm {Fun}_S(\mathcal {C}, \mathcal {D})$, which is the internal mapping object in the category $\mathrm {Cat}_S$.

If $\mathcal {C} \in \mathrm {Cat}_S$ and $T \to S$ is a morphism of schemes, then the tensor product

\[ \mathcal{C}_{T} = \mathcal{C} \otimes_{\operatorname{D_{{perf}}}(S)} \operatorname{D_{{perf}}}(T) \in \mathrm{Cat}_{T} \]

is naturally a $T$-linear category, called the base change of $\mathcal {C}$ along $T \to S$. If $s \in S$ is a point with residue field $\kappa (s)$, then we write $\mathcal {C}_s$ for the $\kappa (s)$-linear category obtained by base change along $\operatorname {Spec}(\kappa (s)) \to S$, and call it the fiber of $\mathcal {C}$ over $s \in S$. In this way, an $S$-linear category $\mathcal {C}$ can be thought of as a family of categories parameterized by $S$.

Example 2.1 Let $f \colon X \to S$ be a morphism of schemes. Then $\mathcal {C} = \operatorname {D_{{perf}}}(X)$ has the structure of an $S$-linear category, with action functor $\mathcal {C} \times \operatorname {D_{{perf}}}(S) \to \operatorname {D_{{perf}}}(X)$ given by $(E, F) \mapsto E \otimes f^{*}F$. If $T \to S$ is a morphism of schemes, then by [Reference Ben-Zvi, Francis and NadlerBZFN10, Theorem 1.2] there is a $T$-linear equivalence

\[ \mathcal{C}_{T} \simeq \operatorname{D_{{perf}}}(X_{T}) \]

where $X_{T} = X \times _{S} T \to T$ denotes the derived fiber product, which agrees with the usual fiber product of schemes if $X \to S$ and $T \to S$ are $\mathrm {Tor}$-independent over $S$.

2.2 Semiorthogonal decompositions

The above example can be amplified using the following observation. If $\mathcal {C} \in \mathrm {Cat}_S$, a semiorthogonal decomposition

(2.1)\begin{equation} \mathcal{C} = \left \langle \mathcal{C}_1, \dots, \mathcal{C}_m \right \rangle \end{equation}

is called $S$-linear if the $\operatorname {D_{{perf}}}(S)$-action preserves each of the components $\mathcal {C}_i$. In this case, the $\mathcal {C}_i$ inherit the structure of $S$-linear categories. In particular, if $X$ is an $S$-scheme, then $S$-linear semiorthogonal components of $\operatorname {D_{{perf}}}(X)$ are $S$-linear categories. This will be our main source of examples in the paper.

By [Reference PerryPer19, Lemma 3.15], given an $S$-linear semiorthogonal decomposition (2.1) and a morphism $T \to S$, there is an induced $T$-linear semiorthogonal decomposition

\[ \mathcal{C}_T = \left \langle (\mathcal{C}_1)_T, \dots, (\mathcal{C}_m)_T \right \rangle. \]

If $\mathcal {C} = \operatorname {D_{{perf}}}(X)$ and $X$ and $T$ are $\mathrm {Tor}$-independent over $S$, the base changes $(\mathcal {C}_i)_T$ can be expressed without the use of higher categories and derived algebraic geometry, by working inside the ambient category $\operatorname {D_{{perf}}}(X_T)$, see [Reference KuznetsovKuz11].

The property that an $S$-linear subcategory $\mathcal {A} \subset \mathcal {C}$ forms part of a semiorthogonal decomposition can be characterized in terms of the embedding functor $\alpha \colon \mathcal {A} \to \mathcal {C}$. Namely, we say $\mathcal {A} \subset \mathcal {C}$ is left admissible if $\alpha$ admits a left adjoint, right admissible if $\alpha$ admits a right adjoint, and admissible if $\alpha$ admits both adjoints. Then if $\mathcal {A}, \mathcal {B} \subset \mathcal {C}$ are $S$-linear subcategories, we have a semiorthogonal decomposition $\mathcal {C} = \left \langle \mathcal {A}, \mathcal {B} \right \rangle$ if and only if $\mathcal {A}$ is left admissible and $\mathcal {B} = {{}^{\perp }}\mathcal {A}$, if and only if $\mathcal {B}$ is right admissible and $\mathcal {A} = \mathcal {B}^{\perp }$.

2.3 Presentable linear categories

For technical reasons it is sometimes useful to work with ‘large’ versions of linear categories, which we review here; for clarity we sometimes say ‘small $S$-linear category’ to mean an $S$-linear category in the sense of § 2.1. Large categories will only be needed for our discussion of Hochschild (co)homology in §§ 3 and 4.

A presentable $S$-linear category $\mathcal {C}$ is a presentable stable $\infty$-category $\mathcal {C}$ equipped with a module structure over $\operatorname {D_{qc}}(S)$. As in the case of small linear categories, the collection of all such categories is organized into a symmetric monoidal $\infty$-category $\mathrm {PrCat}_S$, whose tensor product is denoted by

\[ \mathcal{C} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{D} \in \mathrm{PrCat}_S. \]

A morphism $\mathcal {C} \to \mathcal {D}$ in $\mathrm {PrCat}_S$ is a cocontinuous $S$-linear functor; these morphisms form the objects of a presentable $S$-linear category $\mathrm {Fun}_{S}(\mathcal {C}, \mathcal {D})$, which is the internal mapping object in the category $\mathrm {PrCat}_S$.

Many presentable $S$-linear categories which arise in practice are compactly generated, for example, $\operatorname {D_{qc}}(S)$ is so by our assumption that $S$ is quasi-compact and quasi-separated [Reference Bondal and van den BerghBvdB03, Theorem 3.1.1]. We denote by $\mathrm {PrCat}_{S}^{\omega }$ the $\infty$-category of compactly generated presentable $S$-linear categories, with morphisms the cocontinuous $S$-linear functors which preserve compact objects. Again, $\mathrm {PrCat}_{S}^{\omega }$ admits a symmetric monoidal structure and an internal mapping object $\mathrm {Fun}_{S}^{\omega }(\mathcal {C}, \mathcal {D})$ for $\mathcal {C}, \mathcal {D} \in \mathrm {PrCat}_S^{\omega }$.

The various versions of linear categories $\mathrm {Cat}_S$, $\mathrm {PrCat}_S$, and $\mathrm {PrCat}_S^{\omega }$ are related as follows. By definition, $\mathrm {PrCat}_S^{\omega }$ is a nonfull subcategory of $\mathrm {PrCat}_S$. Moreover, for any $\mathcal {C} \in \mathrm {Cat}_S$ there is a category $\operatorname {Ind}(\mathcal {C}) \in \mathrm {PrCat}_S^{\omega }$ called its Ind-completion, which roughly is obtained from $\mathcal {C}$ by freely adjoining all filtered colimits. This gives a functor

\[ \operatorname{Ind} \colon \mathrm{Cat}_S \to \mathrm{PrCat}_{S}^{\omega} \]

which is in fact a symmetric monoidal equivalence with inverse the functor

\[ (-){}^{\rm c} \colon \mathrm{PrCat}_S^{\omega} \to \mathrm{Cat}_S \]

taking $\mathcal {C} \in \mathrm {PrCat}_S$ to its subcategory $\mathcal {C}^{\rm c}$ of compact objects.

2.4 Mapping objects

For objects $E,F \in \mathcal {C}$ of an $\infty$-category, we write $\operatorname {Map}_{\mathcal {C}}(E,F)$ for the space of maps from $E$ to $F$. If $\mathcal {C}$ is a presentable $S$-linear category, then there is a mapping object

\begin{align*} \mathcal{H}\!{\it om}_S(E,F) \in \operatorname{D_{qc}}(S) \end{align*}

characterized by equivalences

\[ \operatorname{Map}_{\operatorname{D_{qc}}(S)}(G, \mathcal{H}\!{\it om}_S(E,F)) \simeq \operatorname{Map}_{\mathcal{C}}( E \otimes G, F ) \]

for $G \in \operatorname {D_{qc}}(S)$.

If instead $\mathcal {C}$ is an $S$-linear category, we write $\mathcal {H}\!{\it om}_S(E, F) \in \operatorname {D_{qc}}(S)$ for the mapping object between $E$ and $F$ regarded as objects of the presentable $S$-linear category $\operatorname {Ind}(\mathcal {C})$; equivalently, $\mathcal {H}\!{\it om}_S(E, F)$ can be characterized by equivalences

\[ \operatorname{Map}_{\operatorname{D_{qc}}(S)}(G, \mathcal{H}\!{\it om}_S(E,F)) \simeq \operatorname{Map}_{\mathcal{C}}( E \otimes G, F) \]

for $G \in \operatorname {D_{{perf}}}(S)$. For $i \in \mathbf {Z}$ we write $\operatorname {\mathcal {E}\!{\it xt}}^{i}_S(E,F)$ for the degree $i$ cohomology sheaf of $\mathcal {H}\!{\it om}_S(E,F)$, and $\operatorname {Ext}^{i}_S(E,F)$ for the degree $i$ hypercohomology of $\mathcal {H}\!{\it om}_S(E,F)$.

Example 2.3 Let $f \colon X \to S$ be a morphism of schemes. Then, for $E,F \in \operatorname {D_{qc}}(X)$, we have

\[ \mathcal{H}\!{\it om}_S(E,F) \simeq f_* \mathcal{H}\!{\it om}_X(E,F) \]

where $\mathcal {H}\!{\it om}_X(E,F) \in \operatorname {D_{qc}}(X)$ denotes the derived sheaf Hom on $X$.

2.5 Dualizable categories

Let $(\mathcal {A}, \otimes, \mathbf {1})$ be a symmetric monoidal $\infty$-category. An object $A \in \mathcal {A}$ is called dualizable if there exist an object $A^{\scriptscriptstyle \vee } \in \mathcal {A}$ and morphisms

\[ \mathrm{coev}_A \colon \mathbf{1} \to A \otimes A^{\scriptscriptstyle\vee} \quad \text{and} \quad \mathrm{ev}_A \colon A^{\scriptscriptstyle\vee} \otimes A \to \mathbf{1} \]

such that the compositions

\begin{gather*} A \xrightarrow{ \, \mathrm{coev}_A \otimes \mathrm{id}_{A} \,} A \otimes A^{\scriptscriptstyle\vee} \otimes A \xrightarrow{\, \mathrm{id}_{A} \otimes \mathrm{ev}_A \,} A , \\ A^{\scriptscriptstyle\vee} \xrightarrow{ \, \mathrm{id}_{A^{\scriptscriptstyle\vee}} \otimes \mathrm{coev}_A \,} A^{\scriptscriptstyle\vee} \otimes A \otimes A^{\scriptscriptstyle\vee} \xrightarrow{\, \mathrm{ev}_A \otimes \mathrm{id}_{A^{\scriptscriptstyle\vee}} \,} A^{\scriptscriptstyle\vee} \end{gather*}

are equivalent to the identity morphisms of $A$ and $A^{\scriptscriptstyle \vee }$.

The following gives a large source of dualizable presentable linear categories.

Lemma 2.5 [Reference PerryPer19, Lemma 4.3]

Let $\mathcal {C}$ be a compactly generated presentable $S$-linear category. Then $\mathcal {C}$ is dualizable as an object of $\mathrm {PrCat}_S$, with dual given by

\[ \mathcal{C}^{\scriptscriptstyle\vee} = \operatorname{Ind}((\mathcal{C}{}^{\rm c})^{\mathrm{op}}) \]

where $(\mathcal {C}^{\rm c})^{\mathrm {op}}$ denotes the opposite of the category $\mathcal {C}^{\rm c}$ of compact objects in $\mathcal {C}$. There is a canonical equivalence $\mathcal {C} \otimes _{\operatorname {D_{qc}}(S)} \mathcal {C}^{\scriptscriptstyle \vee } \simeq \mathrm {Fun}_S(\mathcal {C}, \mathcal {C})$ under which the coevaluation morphism

\[ \mathrm{coev}_{\mathcal{C}} \colon \operatorname{D_{qc}}(S) \to \mathcal{C} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{C}^{\scriptscriptstyle\vee} \]

is the canonical functor sending $\mathcal {O}_S \in \operatorname {D_{qc}}(S)$ to $\mathrm {id}_{\mathcal {C}} \in \mathrm {Fun}_S(\mathcal {C}, \mathcal {C})$. The evaluation morphism

\[ \mathrm{ev}_{\mathcal{C}} \colon \mathcal{C}^{\scriptscriptstyle\vee} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{C} \to \operatorname{D_{qc}}(S) \]

is induced by the functor $\mathcal {H}\!{\it om}_{S}(-,-) \colon (\mathcal {C}^{\rm c})^{\mathrm {op}} \times \mathcal {C}^{\rm c} \to \operatorname {D_{qc}}(S)$.

In particular, the lemma implies the following corollary, recalling that by convention all schemes are quasi-compact and quasi-separated.

Dualizability of a small $S$-linear category is more restrictive. Recall that if $\mathcal {C}$ is a small $S$-linear category, then $\mathcal {C}$ is called:

• • proper (over $S$) if $\mathcal {H}\!{\it om}_S(E,F) \in \operatorname {D_{{perf}}}(S) \subset \operatorname {D_{qc}}(S)$ for all $E,F \in \mathcal {C}$; and

• • smooth (over $S$) if $\mathrm {id}_{\operatorname {Ind}(\mathcal {C})} \in \mathrm {Fun}_{\operatorname {D_{qc}}(S)}(\operatorname {Ind}(\mathcal {C}), \operatorname {Ind}(\mathcal {C}))$ is a compact object.

Moreover, $\mathcal {C}$ is dualizable as an object of $\mathrm {Cat}_S$ if and only if $\mathcal {C}$ is smooth and proper over $S$, in which case the dual is given by $\mathcal {C}^{\scriptscriptstyle \vee } = \mathcal {C}^{\mathrm {op}}$ [Reference PerryPer19, Lemma 4.8].

This is closely related to the usual notions of smoothness and properness in geometry. For instance, if $f \colon X \to S$ is a smooth and proper morphism, then $\operatorname {D_{{perf}}}(X)$ is smooth and proper over $S$ [Reference PerryPer19, Lemma 4.9]. Further, semiorthogonal components of a smooth proper $S$-linear category are automatically smooth, proper, and admissible [Reference PerryPer19, Lemma 4.15]. Putting these observations together gives the following key examples of smooth and proper linear categories for this paper.

Smooth and proper categories enjoy many nice properties. For instance, a smooth proper $S$-linear category $\mathcal {C}$ always admits a Serre functor $\mathrm {S}_{\mathcal {C}/S}$ over $S$ [Reference PerryPer19, Lemma 4.19]. By definition, this means $\mathrm {S}_{\mathcal {C}/S}$ is an autoequivalence of $\mathcal {C}$ such that there are natural equivalences

\[ \mathcal{H}\!{\it om}_S(E, \mathrm{S}_{\mathcal{C}/S}(F)) \simeq \mathcal{H}\!{\it om}_S(F,E)^{\scriptscriptstyle\vee} \]

for $E, F \in \mathcal {C}$. For example, if $f \colon X \to S$ is a smooth proper morphism of relative dimension $n$, then $\mathrm {S}_{\operatorname {D_{{perf}}}(X)/S} = - \otimes \omega _{X/S}[n]$ is a Serre functor over $S$.

3.1 Functoriality

Hochschild homology (with coefficients) is suitably functorial. This functoriality exists in the general context of traces of dualizable objects in a symmetric monoidal $(\infty,2)$-category (see, for example, [Reference Ben-Zvi and NadlerBZN19, Reference Toën and VezzosiTV15, Reference Hoyois, Scherotzke and SibillaHSS17, Reference Kondyrev and PrikhodkoKoPr20]), but here we only recall the relevant details in the case of Hochschild homology of categories.

Let $(\mathcal {C}, F)$ be a pair where $\mathcal {C}$ is a dualizable presentable $S$-linear category and $F \colon \mathcal {C} \to \mathcal {C}$ is an endomorphism. Let $(\mathcal {D}, G)$ be another such pair. We define a morphism $(\mathcal {C}, F) \to (\mathcal {D}, G)$ to be a pair $(\Phi, \gamma )$ where $\Phi \colon \mathcal {C} \to \mathcal {D}$ is morphism that admits a cocontinuous right adjoint $\Phi ^{!}$ (which is thus also a morphism in $\mathrm {PrCat}_S$), and $\gamma \colon \Phi \circ F \to G \circ \Phi$ is a natural transformation of functors; in other words, a morphism is a (not necessarily commutative) diagram

Given such a morphism $(\Phi, \gamma )$, we consider the diagram

(3.1)

where:

• • $(\Phi ^{!})^{\scriptscriptstyle \vee } \colon \mathcal {C}^{\scriptscriptstyle \vee } \to \mathcal {D}^{\scriptscriptstyle \vee }$ is the dual of the functor $\Phi ^{!} \colon \mathcal {D} \to \mathcal {C}$, defined as the composition

  \[ \mathcal{C}^{\scriptscriptstyle\vee} \xrightarrow{ \mathrm{id}_{\mathcal{C}^{\scriptscriptstyle\vee}} \otimes \mathrm{coev}_{\mathcal{D}}} \mathcal{C}^{\scriptscriptstyle\vee} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{D} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{D}^{\scriptscriptstyle\vee} \xrightarrow{ \mathrm{id}_{\mathcal{C}^{\scriptscriptstyle\vee}} \otimes \Phi^{!} \otimes \mathrm{id}_{\mathcal{D}^{\scriptscriptstyle\vee}}} \mathcal{C}^{\scriptscriptstyle\vee} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{C} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{D}^{\scriptscriptstyle\vee} \xrightarrow{\mathrm{ev}_{\mathcal{C}} \!\otimes \mathrm{id}_{\mathcal{D}^{\scriptscriptstyle\vee}}} \mathcal{D}^{\scriptscriptstyle\vee} ; \]

• • the $2$-morphism in the first square is the natural transformation

  \[ (\Phi \otimes (\Phi^{!})^{\scriptscriptstyle\vee}) \circ \mathrm{coev}_{\mathcal{C}} \simeq ((\Phi \circ \Phi^{!}) \otimes \mathrm{id}_{\mathcal{D}^{\scriptscriptstyle\vee}}) \circ \mathrm{coev}_{\mathcal{D}} \to \mathrm{coev}_{\mathcal{D}} \]
  induced by the counit of the adjunction between $\Phi$ and $\Phi ^{!}$;

• • the $2$-morphism in the last square is the natural transformation

  \[ \mathrm{ev}_{\mathcal{C}} \to \mathrm{ev}_{\mathcal{C}} \circ ((\Phi^{!} \circ \Phi) \otimes \mathrm{id}_{\mathcal{C}^{\scriptscriptstyle\vee}}) \simeq \mathrm{ev}_{\mathcal{D}} \circ (\Phi \otimes (\Phi^{!})^{\scriptscriptstyle\vee}) \]
  induced by the unit of the adjunction between $\Phi$ and $\Phi ^{!}$.

The compositions along the top and bottom of (3.1) are by definition the traces $\operatorname {Tr}(F)$ and $\operatorname {Tr}(G)$, so the composition of the $2$-morphisms in the diagram gives a natural transformation

\[ \operatorname{Tr}(\Phi, \gamma) \colon \operatorname{Tr}(F) \to \operatorname{Tr}(G). \]

In particular, applying this to $\mathcal {O}_S$, we obtain a morphism on Hochschild homology

\[ \operatorname{HH}_*(\Phi, \gamma) \colon \operatorname{HH}_*(\mathcal{C}/S, F) \to \operatorname{HH}_*(\mathcal{D}/S, G). \]

The functoriality of Hochschild homology implies the following result; cf. [Reference KuznetsovKuz09] which treats the case of semiorthogonal decompositions of varieties.

Lemma 3.3 Let $\mathcal {C} = \left \langle \mathcal {C}_1, \dots, \mathcal {C}_m \right \rangle$ be an $S$-linear semiorthogonal decomposition with admissible components. Then there is an equivalence

\[ \operatorname{HH}_*(\mathcal{C}/S) \simeq \operatorname{HH}_*(\mathcal{C}_1/S) \oplus \cdots \oplus \operatorname{HH}_*(\mathcal{C}_m/S), \]

where the map $\operatorname {HH}_*(\mathcal {C}/S) \to \operatorname {HH}_*(\mathcal {C}_i/S)$ is induced by the projection functor onto the component $\mathcal {C}_i$.

3.2 Chern characters

The functoriality of Hochschild homology can be used to define a theory of Chern characters as follows.

Let $\mathcal {C}$ be a presentable $S$-linear category. Then any object $E \in \mathcal {C}$ determines an $S$-linear functor $\Phi _{E} \colon \operatorname {D_{qc}}(S) \to \mathcal {C}$ determined by $\Phi _{E}(\mathcal {O}_S) = E$, whose right adjoint

\[ \Phi_{E}^{!} = \mathcal{H}\!{\it om}_S(E, -) \colon \mathcal{C} \to \operatorname{D_{qc}}(S) \]

is cocontinuous if and only if $E$ is a compact object of $\mathcal {C}$.

Now we assume that $E$ is compact, $\mathcal {C}$ is dualizable, and $F \colon \mathcal {C} \to \mathcal {C}$ is an endomorphism; for instance, $\mathcal {C}$ could be of the form $\mathcal {C} = \operatorname {Ind}(\mathcal {C}_0)$ for a small $S$-linear category $\mathcal {C}_0$ and $E \in \mathcal {C}_0$. In this setup, we will construct a morphism

\[ \operatorname{ch}_{E,F} \colon \mathcal{H}\!{\it om}_S(E, F(E)) \to \operatorname{HH}_*(\mathcal{C}/S, F) \]

in $\operatorname {D_{qc}}(S)$, called the Chern character of $E$ with coefficients in $F$; in practice, we often drop the subscripts $E$ and $F$ in $\operatorname {ch}_{E,F}$ when they are clear from context. By Yoneda's lemma, it suffices to construct functorially in $G \in \operatorname {D_{qc}}(S)$ a map

(3.2)\begin{equation} \operatorname{Map}_{\operatorname{D_{qc}}(S)}(G, \mathcal{H}\!{\it om}_S(E, F(E))) \to \operatorname{Map}_{\operatorname{D_{qc}}(S)}(G, \operatorname{HH}_*(\mathcal{C}/S, F)). \end{equation}

The left-hand side is identified with $\operatorname {Map}_{\mathcal {C}}(E \otimes G, F(E))$. This mapping space is in turn identified with the space of natural transformations $\gamma \colon \Phi _{E} \circ (- \otimes G) \to F \circ \Phi _E$. The pair $(\Phi _E, \gamma )$ is then a morphism of pairs $(\operatorname {D_{qc}}(S), - \otimes G) \to (\mathcal {C}, F)$ as considered in § 3.1, and hence determines a morphism on Hochschild homology

\[ G \simeq \operatorname{HH}_*(\operatorname{D_{qc}}(S)/S, G) \to \operatorname{HH}_*(\mathcal{C}/S, F). \]

All together, this gives the required map (3.2).

3.3 Base change

Hochschild homology satisfies base change in the following sense.

Lemma 3.4 Let $\mathcal {C}$ be a dualizable presentable $S$-linear category and let $F \colon \mathcal {C} \to \mathcal {C}$ be an endomorphism. Let $g \colon T \to S$ be a morphism of schemes. Let $F_T \colon \mathcal {C}_T \to \mathcal {C}_T$ be the base change of $F$ along $g$. Then there is a canonical equivalence

\[ g^{*}\operatorname{HH}_*(\mathcal{C}/S, F) \simeq \operatorname{HH}_*(\mathcal{C}_T/T, F_T) . \]

For smooth proper categories in characteristic $0$, the individual Hochschild homology groups are vector bundles and satisfy base change.

Theorem 3.5 [Reference KaledinKal08, Reference KaledinKal17, Reference MathewMat20]

Let $\mathcal {C}$ be a smooth proper $S$-linear category, where $S$ is a $\mathbf {Q}$-scheme. Then $\operatorname {HH}_i(\mathcal {C}/S)$ is a finite locally free sheaf on $S$ for any $i \in \mathbf {Z}$. Further, if $g \colon T \to S$ is a morphism of schemes, then for any $i \in \mathbf {Z}$ there is a canonical isomorphism

\[ g^{*}\operatorname{HH}_i(\mathcal{C}/S) {\cong} \operatorname{HH}_i(\mathcal{C}_T/T). \]

3.4 Mukai pairing

In the smooth and proper case, Hochschild homology carries a canonical nondegenerate pairing, known as the Mukai pairing. This pairing has been studied from many points of view in the literature [Reference Căldăraru and WillertonCW10, Reference CăldăraruCăl05, Reference ShklyarovShk13, Reference MarkarianMar09].

Proof. We sketch a short proof following [Reference Antieau and VezzosiAV20]. (We already observed $\operatorname {HH}_*(\mathcal {C}/S) \in \operatorname {D_{{perf}}}(S)$ in Remark 3.2, but give another proof here.) The functor $\operatorname {HH}_*(-/S) \colon \mathrm {Cat}_S \to \operatorname {D_{qc}}(S)$ is symmetric monoidal; see, for instance, [Reference Antieau and VezzosiAV20, Proposition 2.1] (where the result is stated for $S$ affine, but from which the general case follows by Lemma 3.4). If $\mathcal {C}$ is smooth and proper over $S$, then it is dualizable as an object of $\mathrm {Cat}_S$. Since $\operatorname {HH}_*(-/S)$ is symmetric monoidal, its value on $\mathcal {C}$ is also dualizable as an object of $\operatorname {D_{qc}}(S)$, and hence belongs to $\operatorname {D_{{perf}}}(S)$. The evaluation morphism for $\operatorname {HH}_*(\mathcal {C}/S)$ is obtained by applying the functor $\operatorname {HH}_*(-/S)$ to the evaluation morphism for $\mathcal {C}$, and hence takes the form

\[ \operatorname{HH}_*(\mathcal{C}^{\scriptscriptstyle\vee}/S) \otimes \operatorname{HH}_*(\mathcal{C}/S) \to \mathcal{O}_S . \]

But it follows from the definition of Hochschild homology that there is a canonical identification $\operatorname {HH}_*(\mathcal {C}^{\scriptscriptstyle \vee }/S) \simeq \operatorname {HH}_*(\mathcal {C}/S)$. This completes the proof.

We will need a compatibility between the Mukai pairing, Serre duality, and Chern characters, which we formulate in the case where the base is a field.

Lemma 3.7 Let $\mathcal {C}$ be a smooth proper $k$-linear category, where $k$ is a field. Then for $i \in \mathbf {Z}$ there is an isomorphism

\[ \operatorname{HH}_i(\mathcal{C}/k) \cong \operatorname{Ext}^{-i}_{k}(\mathrm{id}_{\mathcal{C}}, \mathrm{S}_{\mathcal{C}}) \]

where $\mathrm {S}_{\mathcal {C}}$ is the Serre functor for $\mathcal {C}$ over $k$ and the $\operatorname {Ext}$ group is considered in the category of $k$-linear endofunctors of $\mathcal {C}$. Moreover, for $E \in \mathcal {C}$ if we denote by

\[ \eta_E \colon \operatorname{HH}_i(\mathcal{C}/k) \to \operatorname{Ext}^{-i}_k(E, \mathrm{S}_{\mathcal{C}}(E)) \]

the natural map arising from the above isomorphism, then there is a commutative diagram

where the left vertical arrow is given by Serre duality and the right vertical arrow is give by the Mukai pairing.

3.5 HKR isomorphism

The Hochschild–Kostant–Rosenberg (HKR) isomorphism identifies the Hochschild homology of the derived category of a scheme in terms of Hodge cohomology. This subject has been studied by many authors (see, for example, [Reference Hochschild, Kostant and RosenbergHKR62, Reference SwanSwa96, Reference YekutieliYek02]); the form in which we state the result is a consequence of Yekutieli's work [Reference YekutieliYek02]. For a morphism $X \to S$ and an endomorphism $F \colon \operatorname {D_{{perf}}}(X) \to \operatorname {D_{{perf}}}(X)$, we use the notation $\operatorname {HH}_*(X/S, F) = \operatorname {HH}_*(\operatorname {D_{{perf}}}(X)/S, F)$.

Theorem 3.8 [Reference YekutieliYek02]

Let $f \colon X \to S$ be smooth morphism of relative dimension $n$, where $n!$ is invertible on $S$. Let $F \in \operatorname {D_{{perf}}}(S)$. Then there is an equivalence

\[ \operatorname{HH}_*(X/S, F) \simeq \bigoplus_{p = 0}^{n} F \otimes f_* \Omega^{p}_{X/S}[p] . \]

4.1 Functoriality

As recalled in § 3.1, Hochschild homology is functorial with respect to functors that admit a right adjoint. Hochschild cohomology, however, is only functorial with respect to functors which are also fully faithful.

Let $(\mathcal {C}, F)$ be a small or presentable $S$-linear category, and let $F \colon \mathcal {C} \to \mathcal {C}$ be an $S$-linear endofunctor. Let $(\mathcal {D}, G)$ be another such pair. We consider pairs $(\Phi, \delta )$ where $\Phi \colon \mathcal {C} \to \mathcal {D}$ is a morphism that admits a right adjoint morphism $\Phi ^{!} \colon \mathcal {D} \to \mathcal {D}$ (so $\Phi ^{!}$ is required to be cocontinuous in case $\mathcal {C}$ and $\mathcal {D}$ are presentable), and $\delta \colon G \circ \Phi \to \Phi \circ F$ is a natural transformation of functors; in other words, we consider a (not necessarily commutative) diagram

To distinguish from the notion of a morphism $(\mathcal {C}, F) \to (\mathcal {D}, G)$ introduced in § 3.1, we will call such a pair $(\Phi, \delta )$ a comorphism from $(\mathcal {C}, F)$ to $(\mathcal {D}, G)$. We say that $(\Phi, \delta )$ is fully faithful if $\Phi$ is fully faithful.

Now assume that $(\Phi, \delta ) \colon (\mathcal {C}, F) \to (\mathcal {D}, G)$ is a fully faithful comorphism. In this setup, we will construct a morphism

\[ \operatorname{HH}^{*}(\Phi, \delta) \colon \operatorname{HH}^{*}(\mathcal{D}, G) \to \operatorname{HH}^{*}(\mathcal{C}, F) \]

in $\operatorname {D_{qc}}(S)$. By the definition of Hochschild cohomology and Yoneda, it suffices to construct functorially in $A \in \operatorname {D_{qc}}(S)$ a map

\[ \operatorname{Map}_{\mathrm{Fun}_{S}(\mathcal{D}, \mathcal{D})}((- \otimes A), G) \to \operatorname{Map}_{\mathrm{Fun}_{S}(\mathcal{C}, \mathcal{C})}((- \otimes A), F) . \]

For this, we send $\alpha \colon (- \otimes A) \to G$ on the left-hand side to the morphism $(- \otimes A) \to F$ given by the composition

\[ (- \otimes A) \xrightarrow{ \sim } \Phi^{!} \circ \Phi \circ (- \otimes A) \xrightarrow{\sim} \Phi^{!} \circ (- \otimes A) \circ \Phi \xrightarrow{\Phi^{!} \alpha \Phi} \Phi^{!} \circ G \circ \Phi \xrightarrow{\Phi^{!} \delta} \Phi^{!} \circ \Phi \circ F \xrightarrow{\sim} F , \]

where the first and last equivalences come from full faithfulness of $\Phi$ and the second equivalence from the $S$-linearity of $\Phi$.

4.2 Base change

Like Hochschild homology, Hochschild cohomology satisfies base change. This will not be needed in the paper, but we include it for completeness.

Lemma 4.3 Let $\mathcal {C}$ be a dualizable presentable $S$-linear category and let $F \colon \mathcal {C} \to \mathcal {C}$ be an endomorphism. Let $g \colon T \to S$ be a morphism of schemes. Let $F_T \colon \mathcal {C}_T \to \mathcal {C}_T$ be the base change of $F$ along $g$. Then there is a canonical equivalence

\[ g^{*}\operatorname{HH}^{*}(\mathcal{C}/S, F) \simeq \operatorname{HH}^{*}(\mathcal{C}_T/T, F_T) . \]

Proof. We claim that the natural functor

\[ \mathrm{Fun}_{S}(\mathcal{C}, \mathcal{C}) \otimes_{\operatorname{D_{qc}}(S)} \operatorname{D_{qc}}(T) \to \mathrm{Fun}_T(\mathcal{C}_T, \mathcal{C}_T) \]

is an equivalence of $T$-linear categories. From this, the lemma follows from the definition of Hochschild cohomology and base change for mapping objects in linear categories (see [Reference PerryPer19, Lemma 2.10]).

To prove the claim, note that by dualizability of $\mathcal {C}$ we have an equivalence

\[ \mathrm{Fun}_{S}(\mathcal{C}, \mathcal{C}) \simeq \mathcal{C}^{\scriptscriptstyle\vee} \otimes_{\operatorname{D_{qc}}(S)} \mathcal{C} . \]

Base change along $T \to S$ preserves dualizability of $\mathcal {C}$ and $(\mathcal {C}_T)^{\scriptscriptstyle \vee } \simeq (\mathcal {C}^{\scriptscriptstyle \vee })_T$, so we similarly have an equivalence

\begin{align*} \mathrm{Fun}_{T}(\mathcal{C}_T, \mathcal{C}_T) \simeq (\mathcal{C}^{\scriptscriptstyle\vee})_T \otimes_{\operatorname{D_{qc}}(T)} \mathcal{C}_T . \end{align*}

Since these descriptions of the functor categories are compatible with base change along $T$, the claim follows.

4.3 Action on homology

Hochschild cohomology acts on Hochschild homology. More generally, suppose $\mathcal {C}$ is a dualizable presentable $S$-linear category, and $F, G \colon \mathcal {C} \to \mathcal {C}$ are endomorphisms. Then there is an action functor

\begin{align*} \mathcal{H}\!{\it om}_S(F, G) \otimes \operatorname{HH}_*(\mathcal{C}/S, F) \to \operatorname{HH}_*(\mathcal{C}/S, G) \end{align*}

where the first term is the mapping object from $F$ to $G$ in $\mathrm {Fun}_S(\mathcal {C},\mathcal {C})$. This boils down to assigning to any natural transformation $\gamma \colon F \to G$ a morphism $\operatorname {HH}_*(\mathcal {C}/S, F) \to \operatorname {HH}_*(\mathcal {C}/S, G)$; since $(\mathrm {id}_{\mathcal {C}}, \gamma ) \colon (\mathcal {C}, F) \to (\mathcal {C}, G)$ is a morphism of pairs in the sense of § 3.1, we can simply take $\operatorname {HH}_*(\mathrm {id}_{\mathcal {C}}, \gamma )$.

Note that as a particular case, we have an action

\[ \operatorname{HH}^{*}(\mathcal{C}/S, F) \otimes \operatorname{HH}_*(\mathcal{C}/S) \to \operatorname{HH}_*(\mathcal{C}/S, F), \]

and hence for any $i, j \in \mathbf {Z}$ an action

\[ \operatorname{HH}^{i}(\mathcal{C}/S, F) \otimes \operatorname{HH}_{j}(\mathcal{C}/S) \to \operatorname{HH}_{j-i}(\mathcal{C}/S, F). \]

For any $E \in \mathcal {C}$, there is also an evident action

\[ \operatorname{HH}^{*}(\mathcal{C}/S, F) \otimes \mathcal{H}\!{\it om}_S(E,E) \to \mathcal{H}\!{\it om}_S(E,F(E)). \]

Lemma 4.4 For $E \in \mathcal {C}$ the diagram

commutes.

4.4 HKR isomorphism

The HKR isomorphism for Hochschild cohomology identifies this group in the case of the derived category of a scheme with polyvector field cohomology. Like the HKR isomorphism for Hochschild homology, the following form of this result can be deduced from [Reference YekutieliYek02]. For a morphism $X \to S$ and an endomorphism $F \colon \operatorname {D_{{perf}}}(X) \to \operatorname {D_{{perf}}}(X)$, we use the notation $\operatorname {HH}^{*}(X/S, F) = \operatorname {HH}^{*}(\operatorname {D_{{perf}}}(X)/S, F)$.

Theorem 4.5 Let $f \colon X \to S$ be smooth morphism of relative dimension $n$, where $n!$ is invertible on $S$. Let $F \in \operatorname {D_{{perf}}}(S)$. Then there is an equivalence

\begin{align*} \operatorname{HH}^{*}(X/S, F) \simeq \bigoplus_{t = 0}^{n} F \otimes f_*(\wedge^{t} \mathrm{T}_{X/S})[-t] . \end{align*}

4.5 Deformation theory

Let $0 \to I \to A' \to A \to 0$ be a square-zero extension of rings, and let $X \to \operatorname {Spec}(A)$ be a smooth morphism of schemes. A deformation of $X$ over $A'$ is a smooth scheme $X'$ over $A'$ equipped with an isomorphism $X'_{A} \cong X$. Recall that, provided one exists, the set of isomorphism classes of such deformations form a torsor under $\mathrm {H}^{1}(X, \mathrm {T}_{X/\operatorname {Spec}(A)} \otimes I)$ (where we abusively write $I$ for the pullback of $I$ to $X$). If $A' \to A$ is a trivial square-zero extension, that is, admits a section $A \to A'$, then there is a trivial deformation $X_{A'}$ obtained by base change along the section, so there is a canonical identification of the set of deformations of $X$ over $A'$ with $\mathrm {H}^{1}(X, \mathrm {T}_{X/\operatorname {Spec}(A)} \otimes I)$ taking the trivial deformation to $0$; in this case, for a deformation $X' \to \operatorname {Spec}(A')$ we write

\[ \kappa(X') \in \mathrm{H}^{1}(X, \mathrm{T}_{X/\operatorname{Spec}(A)} \otimes I) \]

for the corresponding element, called the Kodaira–Spencer class.

We will need a generalization of the Kodaira–Spencer class to the setting of categories. Note that by Theorem 4.5, if $\dim (X/A)!$ is invertible on $A$ (where $\dim (X/A)$ denotes the relative dimension of $X \to \operatorname {Spec}(A)$), then we have an isomorphism

\[ \operatorname{HH}^{2}(X/A, I) \cong \mathrm{H}^{0}(X, \wedge^{2}\mathrm{T}_{X/\operatorname{Spec}(A)} \otimes I) \oplus \mathrm{H}^{1}(X, \mathrm{T}_{X/\operatorname{Spec}(A)} \otimes I) \oplus \mathrm{H}^{2}(X, I) , \]

and, in particular, a natural inclusion

(4.1)\begin{equation} \mathrm{H}^{1}(X, \mathrm{T}_{X/\operatorname{Spec}(A)} \otimes I) \hookrightarrow \operatorname{HH}^{2}(X/A, I). \end{equation}

This suggests that when we replace $X$ by an $A$-linear category, the role of the cohomology of $\mathrm {T}_{X/\operatorname {Spec}(A)}$ in deformation theory should be replaced by Hochschild cohomology.

If $A' \to A$ is a square-zero extension and $\mathcal {C}$ is an $A$-linear category, then a deformation of $\mathcal {C}$ over $A'$ is an $A'$-linear category $\mathcal {C}'$ equipped with an equivalence $\mathcal {C}'_{A} \simeq \mathcal {C}$. If $\Phi \colon \mathcal {C} \to \mathcal {D}$ is a morphism of $A$-linear categories, then a deformation of $\Phi$ over $A'$ is a morphism $\Phi ' \colon \mathcal {C}' \to \mathcal {D}'$ where $\mathcal {C}'$ and $\mathcal {D}'$ are deformations of $\mathcal {C}$ and $\mathcal {D}$ over $A'$ and the base change $\Phi '_{A}$ is equipped with an equivalence $\Phi '_{A} \simeq \Phi$.

Lemma 4.6 Let $0 \to I \to A' \to A \to 0$ be a trivial square-zero extension of rings, and let $\mathcal {C}$ be an $A$-linear category. Then, for any deformation $\mathcal {C}'$ of $\mathcal {C}$ over $A'$, there is an associated Kodaira–Spencer class

\begin{align*} \kappa(\mathcal{C}') \in \operatorname{HH}^{2}(\mathcal{C}/A, I) \end{align*}

with the following properties.

1. (1) Let $\Phi \colon \mathcal {C} \to \mathcal {D}$ be a fully faithful morphism of $A$-linear categories which admits a right adjoint. Let $\Phi ' \colon \mathcal {C}' \to \mathcal {D}'$ be a deformation of $\Phi$ over $A'$. Then the map

   \[ \operatorname{HH}^{2}(\mathcal{D}/A, I) \to \operatorname{HH}^{2}(\mathcal{C}/A, I) \]
   induced by $\Phi$ (see § 4.1) takes $\kappa (\mathcal {D}')$ to $\kappa (\mathcal {C}')$.

2. (2) Let $X \to \operatorname {Spec}(A)$ be a smooth morphism of schemes with $\dim (X/A)!$ invertible on $A$. Let $X' \to \operatorname {Spec}(A')$ be a deformation of $X$ over $A'$. Then the inclusion (4.1) takes $\kappa (X')$ to $\kappa (\operatorname {D_{{perf}}}(X'))$.

We can also describe the deformation theory of objects along a deformation of a category. If $A' \to A$ is a square-zero extension of rings, $\mathcal {C}$ is an $A$-linear category, $\mathcal {C}'$ is a deformation of $\mathcal {C}$ over $A'$, and $E \in \mathcal {C}$ is an object, then a deformation of $E$ to $\mathcal {C}'$ is an object $E' \in \mathcal {C}'$ equipped with an equivalence $E'_A \simeq E \in \mathcal {C}$ (where we have used the given identification $\mathcal {C}'_A \simeq \mathcal {C}$).

Lemma 4.8 Let $0 \to I \to A' \to A \to 0$ be a square-zero extension of rings. Let $\mathcal {C}$ be an $A$-linear category, $\mathcal {C}'$ a deformation of $\mathcal {C}$ over $A'$, and $E \in \mathcal {C}$ an object. Then there is an obstruction class

\[ \omega(E) \in \operatorname{Ext}^{2}_A(E, E \otimes I) \]

with the following properties.

1. (1) $\omega (E)$ vanishes if and only if a deformation of $E$ to $\mathcal {C}'$ exists, in which case the set of isomorphism classes of deformations of $E$ to $\mathcal {C}'$ forms a torsor under $\operatorname {Ext}^{1}_A(E, E \otimes I)$.

2. (2) Assume the extension $A' \to A$ is trivial, so that by Lemma 4.6 we have a Kodaira–Spencer class $\kappa (\mathcal {C}') \in \operatorname {HH}^{2}(\mathcal {C}/A, I)$, which by the definition of Hochschild cohomology corresponds to a natural transformation $\mathrm {id}_{\mathcal {C}} \to (- \otimes I)[2]$. Then, writing $\kappa (\mathcal {C}')(E) \in \operatorname {Ext}_A^{2}(E, E \otimes I)$ for the class obtained by applying this natural transformation to $E$, we have an equality

   \[ \omega(E) = \kappa(\mathcal{C}')(E). \]

Proof. Similarly to Lemma 4.6, the result can be proved using the arguments and results of [Reference LurieLur18, Chapter 16]; cf. [Reference LurieLur18, Remark 16.0.0.3].

More concretely, in this paper we shall only need the result in the case where $\mathcal {C} \hookrightarrow \operatorname {D_{{perf}}}(X)$ is a semiorthogonal component for $X$ a noetherian scheme smooth over $A$, $\mathcal {C}' \hookrightarrow \operatorname {D_{{perf}}}(X')$ is a semiorthogonal component of a deformation $X'$ of $X$ over $A'$, and everything is defined over a base field. In this setting, the result can be proved as follows. First consider the purely geometric case where $\mathcal {C} = \operatorname {D_{{perf}}}(X)$ and $\mathcal {C}' = \operatorname {D_{{perf}}}(X')$. Then [Reference LieblichLie06, Theorem 3.1.1] gives the existence of a class $\omega (E)$ satisfying property (1). If the extension $A' \to A$ is trivial, then by the main theorem of [Reference Huybrechts and ThomasHT10] we haveFootnote ¹

\[ \omega(E) = ( \mathrm{id}_{E} \otimes \kappa(X') ) \circ A(E), \]

where $A(E) \in \operatorname {Ext}^{1}(E, E \otimes \Omega _{X/\operatorname {Spec}(A)})$ is the Atiyah class of $E$ and $\kappa (X')$ is regarded as an element of $\operatorname {Ext}^{1}(\Omega _{X/\operatorname {Spec}(A)}, I)$. One checks

\[ ( \mathrm{id}_{E} \otimes \kappa(X') ) \circ A(E) = \kappa(\operatorname{D_{{perf}}}(X))(E) , \]

so that (2) holds. Now the case where $\mathcal {C} \hookrightarrow \operatorname {D_{{perf}}}(X)$ and $\mathcal {C}' \hookrightarrow \operatorname {D_{{perf}}}(X')$ are not necessarily equalities follows from two observations: an object $E'$ is a deformation of $E$ to $\operatorname {D_{{perf}}}(X')$ if and only if $E'$ is a deformation of $E$ to $\mathcal {C}'$; and we have $\kappa (\operatorname {D_{{perf}}}(X))(E) = \kappa (\mathcal {C}')(E)$ by Lemma 4.6(1).

5.1 Topological K-theory

Blanc [Reference BlancBla16] constructed a lax symmetric monoidal topological K-theory functor

\[ \mathrm{K}_{}^{\mathrm{top}} \colon \mathrm{Cat}_{\mathbf{C}} \to \operatorname{Sp} \]

from $\mathbf {C}$-linear categories to the $\infty$-category of spectra. The following theorem summarizes the results about this construction that are relevant to this paper.

For $\mathcal {C} \in \mathrm {Cat}_{\mathbf {C}}$ and an integer $n$, we write

\[ \mathrm{K}_{n}^{\mathrm{top}}(\mathcal{C}) = \pi_{n} \mathrm{K}_{}^{\mathrm{top}}(\mathcal{C}) \]

for the $n$th homotopy group of $\mathrm {K}_{}^{\mathrm {top}}(\mathcal {C})$. These groups carry canonical pairings in the proper case.

Proof. As the functor $\mathrm {K}_{}^{\mathrm {top}} \colon \mathrm {Cat}_{\mathbf {C}} \to \operatorname {Sp}$ is lax monoidal, we have a natural map

\[ \mathrm{K}_{}^{\mathrm{top}}(\mathcal{C}^{\mathrm{op}}) \otimes \mathrm{K}_{}^{\mathrm{top}}(\mathcal{C}) \to \mathrm{K}_{}^{\mathrm{top}}(\mathcal{C}^{\mathrm{op}} \otimes_{\operatorname{D_{{perf}}}(\operatorname{Spec}(\mathbf{C}))} \mathcal{C}). \]

There is a canonical identification $\mathrm {K}_{}^{\mathrm {top}}(\mathcal {C}^{\mathrm {op}}) = \mathrm {K}_{}^{\mathrm {top}}(\mathcal {C})$; indeed, this follows from the definition of $\mathrm {K}_{}^{\mathrm {top}}(-)$ in [Reference BlancBla16] and the corresponding identification for algebraic K-theory. Therefore, passing to homotopy groups, we obtain a map

\[ \mathrm{K}_{n}^{\mathrm{top}}(\mathcal{C}) \otimes \mathrm{K}_{n}^{\mathrm{top}}(\mathcal{C}) \to \mathrm{K}_{2n}^{\mathrm{top}}(\mathcal{C}^{\mathrm{op}} \otimes_{\operatorname{D_{{perf}}}(\operatorname{Spec}(\mathbf{C}))} \mathcal{C}). \]

As $\mathcal {C}$ is proper over $\mathbf {C}$, we have an evaluation functor

\[ \mathcal{C}^{\mathrm{op}} \otimes_{\operatorname{D_{{perf}}}(\operatorname{Spec}(\mathbf{C}))} \mathcal{C} \to \operatorname{D_{{perf}}}(\operatorname{Spec}(\mathbf{C})) \]

induced by the functor $\mathcal {H}\!{\it om}_{\mathbf {C}}(-,-) \colon \mathcal {C}^{\mathrm {op}} \times \mathcal {C} \to \operatorname {D_{{perf}}}(\operatorname {Spec}(\mathbf {C}))$. Taking the topological K-theory of this functor and composing with the above map, we obtain the desired map

\[ \chi^{\mathrm{top}}(-,-) \colon \mathrm{K}_{n}^{\mathrm{top}}(\mathcal{C}) \otimes \mathrm{K}_{n}^{\mathrm{top}}(\mathcal{C}) \to \mathrm{K}_{2n}^{\mathrm{top}}(\operatorname{D_{{perf}}}(\operatorname{Spec}(\mathbf{C}))) \cong \mathbf{Z}. \]

All of the claimed properties follow directly from the definition. For instance, to show the semiorthogonality claimed in (1), note that restriction of the pairing to $\mathrm {K}_{n}^{\mathrm {top}}(\mathcal {C}_i) \otimes \mathrm {K}_{n}^{\mathrm {top}}(\mathcal {C}_j)$ is induced by the functor

\[ \mathcal{C}_i^{\mathrm{op}} \otimes_{\operatorname{D_{{perf}}}(\operatorname{Spec}(\mathbf{C}))} \mathcal{C}_j \to \operatorname{D_{{perf}}}(\operatorname{Spec}(\mathbf{C})) , \]

which is in turn induced by the functor $\mathcal {H}\!{\it om}_{\mathbf {C}}(-,-) \colon \mathcal {C}_i^{\mathrm {op}} \times \mathcal {C}_j \to \operatorname {D_{{perf}}}(\operatorname {Spec}(\mathbf {C}))$; but if $i > j$, then this functor vanishes by semiorthogonality.

Proof. First note that $\mathrm {K}_{n}^{\mathrm {top}}(\mathcal {C})$ is a summand of the finitely generated abelian group $\mathrm {K}_{n}^{\mathrm {top}}(X^{\mathrm {an}})$, and hence finitely generated. The noncommutative Hodge-to-de Rham spectral sequence and its degeneration [Reference KaledinKal08, Reference KaledinKal17, Reference MathewMat20] give a canonical filtration of $\operatorname {HP}_n(\mathcal {C})$ whose $p$th graded piece is $\operatorname {HH}_{n+2p}(\mathcal {C})$. Consider the Chern character map $\mathrm {K}_{n}^{\mathrm {top}}(\mathcal {C}) \otimes \mathbf {C} \to \operatorname {HP}_n(\mathcal {C})$ from Theorem 5.1. We claim that this map is an isomorphism for $\mathcal {C}$ as in the proposition, and the above filtration provides the desired Hodge structure on $\mathcal {C}$. This claim is preserved under passing to semiorthogonal components, so we may assume that $\mathcal {C} = \operatorname {D_{{perf}}}(X)$.

In this case, it is well known that the Chern character indeed provides an isomorphism

\[ \mathrm{K}_{n}^{\mathrm{top}}(\operatorname{D_{{perf}}}(X)) \otimes \mathbf{Q} \cong \bigoplus_{k \in \mathbf{Z}} \mathrm{H}^{2k - n}(X, \mathbf{Q}) \]

of abelian groups. Recall [Reference WeibelWei97] that we have an identification

\[ \operatorname {HP}_n(\operatorname {D_{{perf}}}(X)) \cong \bigoplus _{k \in \mathbf {Z}} \mathrm {H}_{\mathrm {dR}}^{2k-n}(X)\]

with $2$-periodic de Rham cohomology, under which the noncommutative Hodge-to-de Rham filtration agrees with the $2$-periodic Hodge-to-de Rham filtration, that is, the filtration corresponding to the Hodge structure on $\bigoplus _{k \in \mathbf {Z}} \mathrm {H}^{2k - n}(X, \mathbf {Q})(k)$ under the comparison isomorphism $\mathrm {H}^{2k - n}(X, \mathbf {C}) \cong \mathrm {H}_{\mathrm {dR}}^{2k-n}(X)$. We conclude that $\mathrm {K}_{n}^{\mathrm {top}}(\operatorname {D_{{perf}}}(X)) \otimes \mathbf {C} \xrightarrow {\sim } \operatorname {HP}_n(\operatorname {D_{{perf}}}(X))$ is an isomorphism, the noncommutative Hodge-to-de Rham filtration defines a Hodge structure of weight $-n$, and the above isomorphism of abelian groups provided by the Chern character is in fact an isomorphism of rational Hodge structures.

We will need a generalization of Proposition 5.4 to families of categories. This relies on a relative version of Blanc's topological K-theory, due to Moulinos [Reference MoulinosMou19]. Namely, for a scheme $S$ over $\mathbf {C}$, Moulinos constructs a functor

\begin{align*} \mathrm{K}_{}^{\mathrm{top}}(-/S) \colon \mathrm{Cat}_S \to \mathrm{Shv}_{\operatorname{Sp}}(S^{\mathrm{an}}) \end{align*}

from $S$-linear categories to the $\infty$-category of sheaves of spectra on the analytification $S^{\mathrm {an}}$.

For $\mathcal {C} \in \mathrm {Cat}_{S}$ and an integer $n$, we write

\[ \mathrm{K}_{n}^{\mathrm{top}}(\mathcal{C}/S) = \pi_{n} \mathrm{K}_{}^{\mathrm{top}}(\mathcal{C}/S) \]

for the $n$th homotopy sheaf of $\mathrm {K}_{}^{\mathrm {top}}(\mathcal {C}/S)$, which is a sheaf of abelian groups on $S^{\mathrm {an}}$.

5.2 The noncommutative Hodge conjecture and its variants

Using the above, we can formulate a natural notion of Hodge classes on a category.

Definition 5.9 Let $\mathcal {C} \subset \operatorname {D_{{perf}}}(X)$ be a $\mathbf {C}$-linear admissible subcategory, where $X$ is a smooth proper complex variety. The group of integral Hodge classes $\operatorname {Hdg}(\mathcal {C}, \mathbf {Z})$ on $\mathcal {C}$ is the subgroup of Hodge classes in $\mathrm {K}_{0}^{\mathrm {top}}(\mathcal {C})$ for the Hodge structure given by Proposition 5.4. More explicitly, $\operatorname {Hdg}(\mathcal {C}, \mathbf {Z})$ consists of all classes in $\mathrm {K}_{0}^{\mathrm {top}}(\mathcal {C})$ which map to $\operatorname {HH}_0(\mathcal {C})$ under the Hodge decomposition

\[ \mathrm{K}_{0}^{\mathrm{top}}(\mathcal{C}) \otimes \mathbf{C} \cong \bigoplus_{p+q = 0} \operatorname{HH}_{p-q}(\mathcal{C}) . \]

The group of rational Hodge classes is defined by $\operatorname {Hdg}(\mathcal {C}, \mathbf {Q}) = \operatorname {Hdg}(\mathcal {C}, \mathbf {Z}) \otimes \mathbf {Q}$. We say an element $v \in \mathrm {K}_{0}^{\mathrm {top}}(\mathcal {C})$ is algebraic if it is in the image of $\mathrm {K}_0(\mathcal {C}) \to \mathrm {K}_{0}^{\mathrm {top}}(\mathcal {C})$; similarly, an element $v \in \mathrm {K}_{0}^{\mathrm {top}}(\mathcal {C}) \otimes \mathbf {Q}$ is algebraic if it is in the image of $\mathrm {K}_0(\mathcal {C}) \otimes \mathbf {Q} \to \mathrm {K}_{0}^{\mathrm {top}}(\mathcal {C}) \otimes \mathbf {Q}$.

By Proposition 5.4(2), if $X$ is a smooth proper complex variety, then the Chern character identifies $\operatorname {Hdg}(\operatorname {D_{{perf}}}(X), \mathbf {Q})$ with the usual group of rational Hodge classes $\operatorname {Hdg}^{*}(X, \mathbf {Q})$, that is, the group of Hodge classes for $\bigoplus _{k \in \mathbf {Z}} \mathrm {H}^{2k}(X, \mathbf {Q})(k)$. Recall that the cycle class map from the Chow ring

\[ \operatorname{CH}^{*}(X) \otimes \mathbf{Q} \to \mathrm{H}^{*}(X, \mathbf{Q}) \]

factors through $\operatorname {Hdg}^{*}(X, \mathbf {Q})$, and the usual Hodge conjecture predicts that this map surjects onto $\operatorname {Hdg}^{*}(X, \mathbf {Q})$. Since the Chern character gives an isomorphism

\[ \mathrm{K}_0(\operatorname{D_{{perf}}}(X)) \otimes \mathbf{Q} \cong \operatorname{CH}^{*}(X) \otimes \mathbf{Q}, \]

we conclude that the map

\[ \mathrm{K}_0(\operatorname{D_{{perf}}}(X)) \to \mathrm{K}_{0}^{\mathrm{top}}(\operatorname{D_{{perf}}}(X)) \]

factors through $\operatorname {Hdg}(\operatorname {D_{{perf}}}(X), \mathbf {Z})$, and the usual Hodge conjecture is equivalent to the surjectivity of the map $\mathrm {K}_0(\operatorname {D_{{perf}}}(X)) \otimes \mathbf {Q} \to \operatorname {Hdg}(\operatorname {D_{{perf}}}(X), \mathbf {Q})$. Now using additivity under semiorthogonal decompositions of all the invariants in sight leads to the following lemma and conjecture.

Conjecture 5.11 (Noncommutative Hodge conjecture)

Let $\mathcal {C} \subset \operatorname {D_{{perf}}}(X)$ be a $\mathbf {C}$-linear admissible subcategory, where $X$ is a smooth proper complex variety. Then the map

\[ \mathrm{K}_0(\mathcal{C}) \otimes \mathbf{Q} \to \operatorname{Hdg}(\mathcal{C}, \mathbf{Q}) \]

is surjective.

We record the following observation from above.

There is also an obvious integral variant of the Hodge conjecture for categories.

Conjecture 5.13 (Noncommutative integral Hodge conjecture)

Let $\mathcal {C} \subset \operatorname {D_{{perf}}}(X)$ be a $\mathbf {C}$-linear admissible subcategory, where $X$ is a smooth proper complex variety. Then the map

\begin{align*} \mathrm{K}_0(\mathcal{C}) \to \operatorname{Hdg}(\mathcal{C}, \mathbf{Z}) \end{align*}

is surjective.

The integral Hodge conjectures for varieties and categories are closely related, but not so simply as in the rational case. The result can be conveniently formulated in terms of Voisin groups. Recall from § 1 that for a smooth proper complex variety $X$, the degree $n$ Voisin group $\mathrm {V}^{n}(X)$ is defined as the cokernel of the cycle class map $\operatorname {CH}^{n}(X) \to \operatorname {Hdg}^{n}(X, \mathbf {Z})$.

Definition 5.15 Let $\mathcal {C} \subset \operatorname {D_{{perf}}}(X)$ be a $\mathbf {C}$-linear admissible subcategory, where $X$ is a smooth proper complex variety. The Voisin group of $\mathcal {C}$ is the cokernel

\[ \mathrm{V}(\mathcal{C}) = \mathrm{coker}(\mathrm{K}_0(\mathcal{C}) \to \operatorname{Hdg}(\mathcal{C}, \mathbf{Z})). \]

Note that the integral Hodge conjecture holds for $\mathcal {C}$ if and only if $\mathrm {V}(\mathcal {C}) = 0$.

The (integral) Hodge conjecture for categories behaves well under semiorthogonal decompositions. This will be important in our applications to the integral Hodge conjecture for varieties with CY2 semiorthogonal components.

Lemma 5.20 Let $\mathcal {C} \subset \operatorname {D_{{perf}}}(X)$ be a $\mathbf {C}$-linear admissible subcategory, where $X$ is a smooth proper complex variety. Let $\mathcal {C} = \left \langle \mathcal {C}_1, \dots, \mathcal {C}_m \right \rangle$ be a $\mathbf {C}$-linear semiorthogonal decomposition. Then there is an isomorphism of Voisin groups

\[ \mathrm{V}(\mathcal{C}) \cong \mathrm{V}(\mathcal{C}_1) \oplus \cdots \oplus \mathrm{V}(\mathcal{C}_{m}) . \]

In particular, the (integral) Hodge conjecture holds for $\mathcal {C}$ if and only if the (integral) Hodge conjecture holds for all of the semiorthogonal components $\mathcal {C}_1, \dots, \mathcal {C}_m$.

We can also formulate a version of the variational Hodge conjecture for categories.

Note that, as in Lemma 5.12, for $\mathcal {C} = \operatorname {D_{{perf}}}(X)$ the noncommutative variational Hodge conjecture is equivalent to the usual variational Hodge conjecture. In general, this conjecture is extremely difficult. One of the main results of this paper, Theorem 1.3, is an integral version of the noncommutative variational Hodge conjecture for families of CY2 categories.

We note that, using deep known results for varieties, it is easy to prove the statement obtained by replacing ‘algebraic’ with ‘Hodge’ in the noncommutative variational Hodge conjecture.

5.3 Odd-degree cohomology and intermediate Jacobians

The following gives conditions under which the odd topological K-theory of categories recovers the odd integral cohomology of varieties. For an abelian group $A$, we write $A_{\mathrm {tf}}$ for the quotient by its torsion subgroup.

Proposition 5.23 Let $X$ be a smooth proper complex variety of odd dimension $n$, such that $\mathrm {H}^{k}(X, \mathbf {Z}) = 0$ for all odd $k < n$. Let $\operatorname {D_{{perf}}}(X) = \left \langle \mathcal {C}_1, \dots, \mathcal {C}_m \right \rangle$ be a semiorthogonal decomposition. Then the Chern character induces an isometry of weight $n$ Hodge structures

\[ \operatorname{ch} \colon \mathrm{K}_{-n}^{\mathrm{top}}(\mathcal{C}_1)_{\mathrm{tf}} \oplus \cdots \oplus \mathrm{K}_{-n}^{\mathrm{top}}(\mathcal{C}_m)_{\mathrm{tf}} \xrightarrow{\sim} \mathrm{H}^{n}(X, \mathbf{Z})_{\mathrm{tf}} \]

where the left-hand side is the orthogonal sum of the Hodge structures $\mathrm {K}_{-n}^{\mathrm {top}}(\mathcal {C}_i)_{\mathrm {tf}}$ of Proposition 5.4 equipped with their Euler pairings of Lemma 5.2, and the right-hand side is equipped with its standard Hodge structure and pairing. If, moreover, $\mathrm {H}^{k}(X, \mathbf {Z}) = 0$ for all odd $k \neq n$, then the above isomorphism holds before quotienting by torsion, that is,

\[ \mathrm{K}_{-n}^{\mathrm{top}}(\mathcal{C}_1) \oplus \cdots \oplus \mathrm{K}_{-n}^{\mathrm{top}}(\mathcal{C}_m) \xrightarrow{\sim} \mathrm{H}^{n}(X, \mathbf{Z}) . \]

Recall that if $H$ is a Hodge structure of odd weight $n$, then its intermediate Jacobian is

\[ J(H) = \frac{H_{\mathbf{C}}}{F^{{(n+1)}/{2}}(H_{\mathbf{C}}) \oplus H_{\mathbf{Z}} }, \]

where $H_{\mathbf {Z}}$ is the underlying abelian group of $H$, $H_{\mathbf {C}}$ is its complexification, and $F^{\bullet }(H_{\mathbf {C}})$ is the Hodge filtration. In general, $J(H)$ only has the structure of a complex torus, but if $H$ is polarized and its Hodge decomposition only has two terms (i.e. $H_{\mathbf {C}} = H^{p,q} \oplus H^{q,p}$ for some $p,q$), then $J(H)$ is a principally polarized abelian variety. If $X$ is a smooth proper complex variety, for an odd integer $k$ we write $J^{k}(X)$ for the intermediate Jacobian associated to the Hodge structure $\mathrm {H}^{k}(X, \mathbf {Z})$.

Definition 5.24 Let $\mathcal {C} \subset \operatorname {D_{{perf}}}(X)$ be a $\mathbf {C}$-linear admissible subcategory, where $X$ is a smooth proper complex variety. The intermediate Jacobian of $\mathcal {C}$ is the complex torus

\[ J(\mathcal{C}) = J(\mathrm{K}_{1}^{\mathrm{top}}(\mathcal{C})) \]

given by the intermediate Jacobian of the weight $-1$ Hodge structure $\mathrm {K}_{1}^{\mathrm {top}}(\mathcal {C})$.

Using Proposition 5.23, we can prove Theorem 1.6 and Corollary 1.7.

Proof of Theorem 1.6 By Proposition 5.23 and our assumptions on the cohomology of $X$ and $Y_i$, we have isomorphisms $J(\operatorname {D_{{perf}}}(X)) \cong J^{n}(X)$ and $J(\operatorname {D_{{perf}}}(Y_i)) \cong J^{n_i}(Y_i)$ if $n_i$ is odd. Moreover, Proposition 5.4(2) and our cohomological assumption imply that if $n_i$ is even then $\mathrm {K}_{1}^{\mathrm {top}}(\operatorname {D_{{perf}}}(Y_i))$ is torsion, so $J(\operatorname {D_{{perf}}}(Y_i)) = 0$. Thus, by Proposition 5.23 applied to the semiorthogonal decomposition of $\operatorname {D_{{perf}}}(X)$, we obtain an isomorphism of complex tori

(5.2)\begin{equation} J^{n}(X) \cong \bigoplus_{n_i \, \mathrm{odd}} J^{n_i}(Y_i) . \end{equation}

Under the further assumption that $\mathrm {H}^{n_i}(Y_i, \mathbf {C}) = \mathrm {H}^{p_i, q_i}(Y_i) \oplus \mathrm {H}^{q_i, p_i}(Y_i)$, the HKR isomorphism shows that $\mathrm {H}^{n}(X, \mathbf {C}) = \mathrm {H}^{p,q}(X) \oplus \mathrm {H}^{q,p}(X)$, where $p_i, q_i, p, q$ are as in the statement of Theorem 1.6. The fact that the isomorphisms of Proposition 5.23 respect the pairings on each side then implies that the above isomorphism (5.2) respects the principal polarizations on each side.

Proof of Corollary 1.7 Note that the category $\left \langle E_j \right \rangle \subset \operatorname {D_{{perf}}}(X)$ generated by $E_j$ is equivalent to $\operatorname {D_{{perf}}}(\operatorname {Spec}(\mathbf {C}))$. Hence the first part of Corollary 1.7 is just a special case of Theorem 1.6. Under the further assumption that $\mathrm {H}^{5}(X, \mathbf {Z}) = 0$, we can apply the second part of Proposition 5.23 (and the $2$-periodicity of $\mathrm {K}_{n}^{\mathrm {top}}(-)$) to conclude there are isomorphisms

\begin{align*} \mathrm{H}^{3}(X, \mathbf{Z}) & \cong \mathrm{K}_{-3}^{\mathrm{top}}(\operatorname{D_{{perf}}}(X)) \\ & \cong \mathrm{K}_{-3}^{\mathrm{top}}(\operatorname{D_{{perf}}}(C_1)) \oplus \cdots \oplus \mathrm{K}_{-3}^{\mathrm{top}}(\operatorname{D_{{perf}}}(C_r)) \\ & \cong \mathrm{H}^{1}(C_1, \mathbf{Z}) \oplus \cdots \oplus \mathrm{H}^{1}(C_r, \mathbf{Z}). \end{align*}

In particular, $\mathrm {H}^{3}(X, \mathbf {Z})_{\mathrm {tors}} = 0$.

Example 5.26 Let $V_6$ be a six-dimensional vector space, and consider the Plücker embedded Grassmannian $\mathrm {Gr}(2,V_6) \subset \mathbf {P}(\wedge ^{2}V_6)$ and the Pfaffian cubic hypersurface $\mathrm {Pf}(4, V_6^{\scriptscriptstyle \vee }) \subset \mathbf {P}(\wedge ^{2} V_6^{\scriptscriptstyle \vee })$ parameterizing forms of rank at most $4$. Let $L \subset \wedge ^{2}V_6$ be a codimension $3$ linear subspace, and let $L^{\perp } = \ker (\wedge ^{2}V_6^{\scriptscriptstyle \vee } \to L^{\scriptscriptstyle \vee })$ be its orthogonal. We assume $L$ is generic so that the intersections

\[ X_L = \mathrm{Gr}(2, V_6) \cap \mathbf{P}(L) \quad \text{and} \quad Y_L = \mathrm{Pf}(4, V_6^{\scriptscriptstyle\vee}) \cap \mathbf{P}(L^{\perp}) \]

are smooth of expected dimension, in which case $X_L$ is a Fano fivefold and $Y_L$ is an elliptic curve. Then there is an isomorphism

(5.3)\begin{equation} J^{5}(X_L) \cong J^{1}(Y_L) \end{equation}

of principally polarized abelian varieties. Indeed, by [Reference KuznetsovKuz06a, § 10] there is a semiorthogonal decomposition of $\operatorname {D_{{perf}}}(X_L)$ consisting of $\operatorname {D_{{perf}}}(Y_L)$ and exceptional objects, and by the Lefschetz hyperplane theorem $\mathrm {H}^{k}(X_L, \mathbf {Z}) = 0$ for all odd $k < 5$, so Theorem 1.6 gives the result.

Homological projective geometry [Reference KuznetsovKuz07, Reference Jiang, Leung and XieJLX21, Reference Kuznetsov and PerryKuPe21, Reference Kuznetsov and PerryKuPe19] is a general theory for producing varieties whose derived categories have a semiorthogonal component in common, and gives many examples to which our methods apply. Example 5.26 is a simple instance of this. To explain a more interesting example, we need some terminology.