2.1 $\infty$-Categories of dg-categories

We denote by $A$ a commutative ring. We recall here some basic facts about the $\infty$-category of dg-categories, its monoidal structure and its theory of monoids and modules.

We consider the category $dgCat_{A}$ of small $A$-linear dg-categories and $A$-linear dg-functors. We recall that an $A$-linear dg-functor $T \longrightarrow T'$ is a Morita equivalence if the induced functor of the corresponding derived categories of dg-modules $f^{*} : D(T') \longrightarrow D(T)$ is an equivalence of categories (see [Reference ToënToë07] for details). The $\infty$-category of dg-categories over $S$ is defined to be the localization of $dgCat_{A}$ along these Morita equivalences, and will be denoted by $\mathbf {dgCat}_S$ or $\mathbf {dgCat}_A$. Being the $\infty$-category associated to a combinatorial model category, $\mathbf {dgCat}_A$ is a presentable $\infty$-category [Reference LurieLur09b, A.3.7.6]. As in [Reference ToënToë07, § 4], the tensor product of $A$-linear dg-categories can be derived to a symmetric monoidal structure on the $\infty$-category $\mathbf {dgCat}_A$. This symmetric monoidal structure moreover distributes over colimits making $\mathbf {dgCat}_A$ into a presentable symmetric monoidal $\infty$-category. We have a notion of rigid or, equivalently, dualizable object in $\mathbf {dgCat}_A$. It is a well-known fact that dualizable objects in $\mathbf {dgCat}_A$ are precisely smooth and proper dg-categories over $A$ (see [Reference ToënToë12, Proposition 2.5]).

The compact objects in $\mathbf {dgCat}_A$ are the dg-categories of finite type over $A$ in the sense of [Reference Toën and VaquiéTV07]. We denote their full sub-$\infty$-category by $\mathbf {dgCat}_A^{\rm ft} \subset \mathbf {dgCat}_A$. The full sub-category $\mathbf {dgCat}_S^{\rm ft}$ is preserved by the monoidal structure and, moreover, any dg-category is a filtered colimit of dg-categories of finite type. We, thus, have a natural equivalence of symmetric monoidal $\infty$-categories

\[ \mathbf{dgCat}_A \simeq \mathbf{Ind}(\mathbf{dgCat}_S^{\rm ft}). \]

We from time to time have to work in a bigger $\infty$-category, denoted by $\mathbf {dgCAT}_A$, that contains $\mathbf {dgCat}_A$ as a non-full sub-$\infty$-category. By [Reference ToënToë12], we have a symmetric monoidal $\infty$-category $\mathbf {dgCat}_A^{lp}$ of presentable dg-categories over $A$ with morphisms given by colomit-preserving functors. We define $\mathbf {dgCAT}_A$ the full sub-$\infty$-category of $\mathbf {dgCat}_A^{lp}$ consisting of all compactly generated dg-categories. The $\infty$-category $\mathbf {dgCat}_A$ can be identified with the non-full sub-$\infty$-category of $\mathbf {dgCAT}_A$ which consists of compact objects preserving dg-functors. This provides a faithful embeddingFootnote ⁵ of symmetric monoidal $\infty$-categories

\[ \mathbf{dgCat}_A \hookrightarrow \mathbf{dgCAT}_A. \]

At the level of objects, this embedding sends a small dg-category $T$ to the compactly generated dg-category $\hat {T}$ of dg-modules over $T^{\rm op}$. An equivalent description of $\mathbf {dgCAT}_A$ is as the $\infty$-category of small dg-categories together with the mapping spaces given by the classifying space of all bi-dg-modules between small dg-categories. Objects and morphisms in $\mathbf {dgCAT}_A$ (respectively, in $\mathbf {dgCat}_A$) will be called big (respectively, small).

For a $B$-module $T$, we have a morphism $\mu : B\otimes _A T \to T$ in $\mathbf {dgCat}_A$, that will be simply denoted by $(b,x) \mapsto b\otimes x$. For a monoidal $A$-dg-category $B$, we denote by $B^{\otimes \textrm {-op}}$ the monoidal $A$-dg-category where the monoid structure is the opposite to the one of $B$, that is, $b\otimes ^{\rm op} b' := b'\otimes b$. Note that $B^{\otimes \textrm {-op}}$ should not be confused with $B^{\rm op}$ (which is still a monoidal $A$-dg-category), where the ‘arrows’ and not the monoid structure have been reversed, that is, $B^{\rm op}(b,b'):= B(b', b)$. By definition, a right $B$-module is a (left) $B^{\otimes \textrm {-op}}$-module. The $\infty$-category of right $B$-modules will be denoted by $\mathbf {dgCat}_{B^{\otimes \textrm {-op}}}$, or simply by $\mathbf {dgCat}^{B}$. If $B$ is a monoidal $A$-dg-category, then $B^{\otimes \textrm {-op}}\otimes _A B$ is again a monoidal $A$-dg-category, and $B$ can be considered either as a left $B^{\otimes \textrm {-op}}\otimes _A B$ (denoted by $B^{\rm L}$), or as a right $B^{\otimes \textrm {-op}}\otimes _A B$-module (denoted by $B^{\rm R}$). For $T$ a $B$-module, and $T'$ a right $B$-module, then $T'\otimes _A T$ is naturally a right $B^{\otimes \textrm {-op}}\otimes _A B$-module, and we define

\[ T'\otimes_B T := (T'\otimes_A T)\otimes_{B^{\otimes\textrm{-op}}\otimes_A B}B{}^{\rm L} \]

which is an object in $\mathbf {dgCat}_A$.

Let $B$ be a monoidal $A$-dg-category. We can consider the symmetric monoidal embedding $\mathbf {dgCat}_A \hookrightarrow \mathbf {dgCAT}_A$, so that the image $\hat {B}$ of $B$ is a monoid in $\mathbf {dgCAT}_A$. The $\infty$-category of $\hat {B}$-modules in $\mathbf {dgCAT}_A$ is denoted by $\mathbf {dgCAT}_B$, and its objects are called big $B$-modules. The natural $\infty$-functor $\mathbf {dgCat}_B \longrightarrow \mathbf {dgCAT}_B$ is faithful and its image consists of all big $B$-modules $\hat {T}$ such that the morphism $\hat {B}\,\widehat {\otimes }_A\, \hat {T} \longrightarrow \hat {T}$ is a small morphism.

It is known (see [Reference ToënToë12]) that the symmetric monoidal $\infty$-category $\mathbf {dgCAT}_A$ is rigid, that for any $\hat {T}$ its dual is given by $\widehat {T^{\rm op}}$, and that the evaluation and coevaluation morphisms are defined by $T$ considered as $T^{\rm op}\otimes _A T$-module. This formally implies that if $\hat {T}$ is a big $B$-module, then its dual $\widehat {T^{\rm op}}$ is naturally a right big $B$-module. We thus have two big (i.e. morphisms in the category of big dg-categories) morphisms

\[ \mu : \hat{B} \,\widehat{\otimes}_A\, \hat{T} \longrightarrow \hat{T},\quad \mu^{\rm op} : \widehat{T^{\rm op}} \,\widehat{\otimes}_A\, \hat{B} \longrightarrow \widehat{T^{\rm op}}. \]

Here $\mu$ has a right adjoint $\mu ^{*}$ (the restriction at the level of modules) which is presentable but not, in general, continuous. In fact, $\mu ^{*}$ is continuous if and only if $\mu$ is a small morphisms. When this is the case these morphisms also provide a third big morphism

\[ h : \widehat{T^{\rm op}} \,\widehat{\otimes}_A\, \hat{T} \longrightarrow \hat{B} \]

obtained by duality from

\[ \mu^{*} : \hat{T} \longrightarrow\hat{B} \,\widehat{\otimes}_A\, \hat{T}. \]

We now make the following definitions.

We can make the above definition more explicit as follows. Let $B$ and $T$ be as above; for two objects $b \in B$ and $x \in T$, we can consider the dg-functor

\[ x^{b} : T^{\rm op} \longrightarrow L(A) \]

sending $y \in T$ to $T(b\otimes y),x)$ where $(b, x)\mapsto b\otimes x: B \otimes _A T \longrightarrow T$ is the $B$-module structure on $T$. Then, $T$ is co-tensored over $B$ if and only if for all $b$ and $x$ the above dg-module $x^{b}: T^{\rm op} \longrightarrow L(A)$ is compact in the derived category $D(T^{\rm op})$ of all $T^{\rm op}$-dg-modules. When $T$ is furthermore assumed to be triangulated, then this is equivalent to ask for the dg-module to be representable by an object $x^{b} \in T$. In a similar manner, we can phrase properness of $T$ over $B$ by saying that, for any $x\in T$ and $y\in T$, the dg-module

\[ B^{\rm op} \longrightarrow L(A) \]

sending $b$ to $T(b\otimes x,y)$ is compact (or representable, if $B$ is furthermore assumed to be triangulated).

Remark 2.1.3 It is important to note that, by definition, when $T$ is co-tensored, then the big right $B$-module $\widehat {T^{\rm op}}$ is, in fact, a small right $B$-module (i.e. it is in the essential image of $\mathbf {dgCat}_B \longrightarrow \mathbf {dgCAT}_B$). Equivalently, when $T$ is co-tensored, then $T^{\rm op}$ is naturally a right $B$-module inside $\mathbf {dgCat}_A$, with the right module structure given by the morphism in $\mathbf {dgCat}_A$

\[ \mu^{\rm op} : T^{\rm op}\otimes_A B \longrightarrow T^{\rm op} \]

sending $(x,b) \in T^{\rm op}\otimes _A B$ to the co-tensor $x^{b} \in T^{\rm op}$.

We end this section by recalling some facts about existence of tensor products of modules over monoidal dg-categories. As a general fact, because $\mathbf {dgCat}_A$ is a presentable symmetric monoidal $\infty$-category, for any monoidal dg-category $B$ there exists a tensor product $\infty$-functor

\[ \otimes_B : \mathbf{dgCat}_B \times \mathbf{dgCat}^{B} \longrightarrow \mathbf{dgCat}_A, \]

sending a left $B$-module $T$ and a right $B$-module $T'$ to $T'\otimes _B T$ (see [Reference LurieLur16]). Now, if $T$ is a $B$-module which is also co-tensored (over $B$) in the sense of Definition 2.1.2, we have that $T^{\rm op}$ is a right $B$-module and we can, thus, form

\[ T^{\rm op} \otimes_B T \in \mathbf{dgCat}_A. \]

When $T$ is not co-tensored, the object $T \otimes _B T^{\rm op}$ does not exist anymore. However, we can always consider the presentable dg-categories $\hat {T}$ and $\widehat {T^{\rm op}}$ as left and right modules over $\hat {B}$, respectively, and their tensor product $\widehat {T^{\rm op}} \,\widehat {\otimes }_{\hat {B}}\,\hat {T}$ now only makes sense as a presentable dg-category which has no reason to be compactly generated, in general. Of course, when $T$ is co-tensored, this presentable dg-category is compactly generated and we have

\[ \widehat{T^{\rm op} \otimes_B T}\simeq \widehat{T^{\rm op}}\, \widehat{\otimes}_{\hat{B}}\, \hat{T}. \]

2.2 The $\ell$-adic realization of dg-categories

We denote by $\mathcal {SH}_S$ the stable $\mathbb {A}^{1}$-homotopy $\infty$-category of schemes over $S$ (see [Reference VoevodskyVoe98, Definition 5.7] and [Reference RobaloRob15, § 2]). It is a presentable symmetric monoidal $\infty$-category whose monoidal structure will be denoted by $\wedge _S$. Homotopy invariant algebraic K-theory defines an $\mathbb {E}_{\infty }$-ring object in $\mathcal {SH}_S$ that we denote by $\mathbf {B}\mathbb {U}_S$ (a more standard notation is $\textit{KGL}_S$). We denote by $\mathbf {Mod}_{\mathcal {SH}_S}(\mathbf {B}\mathbb {U}_S)$ the $\infty$-category of $\mathbf {B}\mathbb {U}_S$-modules in $\mathcal {SH}_S$. It is a presentable symmetric monoidal $\infty$-category whose monoidal structure will be denoted by $\wedge _{\mathbf {B}\mathbb {U}_S}$.

As proved in [Reference Blanc, Robalo, Toën and VezzosiBRTV18], there exists a lax symmetric monoidal $\infty$-functor

\[ \mathsf{M}^{-}: \mathbf{dgCat}_S \longrightarrow \mathbf{Mod}_{\mathcal{SH}_S}(\mathbf{B}\mathbb{U}_S), \]

which will be denoted by $T \mapsto \mathsf {M}^{T}$ (while it is denoted by $T \mapsto \mathcal {M}_S^{\vee }(T)$ in [Reference Blanc, Robalo, Toën and VezzosiBRTV18]). The precise construction of the $\infty$-functor $\mathsf {M}^{-}$ is rather involved and uses in an essential manner the theory of non-commutative motives of [Reference RobaloRob15] as well as the comparison with the stable homotopy theory of schemes. Intuitively, the $\infty$-functor $\mathsf {M}^{-}$ sends a dg-category $T$ to the homotopy invariant K-theory functor $S' \mapsto \mathsf {HK}(S' \otimes _S T)$. To be more precise, there is an obvious forgetful $\infty$-functor

\[ \mathsf{U}: \mathbf{Mod}_{\mathcal{SH}_S}(\mathbf{B}\mathbb{U}_S) \longrightarrow \mathbf{Fun}^{\infty}(Sm_S^{\rm op},\mathbf{Sp}), \]

to the $\infty$-category of presheaves of spectra on the category $Sm_S$ of smooth $S$-schemes. For a given dg-category $T$ over $S$, the presheaf $\mathsf {U}(\mathsf {M}^{T})$ is defined by sending a smooth $S$-scheme $S'=\mathbf {Spec}\, A' \rightarrow \mathbf {Spec}\, A=S$ to $\mathsf {HK}(A'\otimes _A T)$, the homotopy invariant non-connective K-theory spectrum of $A'\otimes _A T$ (see [Reference RobaloRob15, 4.2.3]).

The $\infty$-functor $\mathsf {M}^{-}$ satisfies some basic properties which we recall here.

1. 1. The $\infty$-functor $\mathsf {M}^{-}$ is a localizing invariant, that is, for any short exact sequence $T_0 \hookrightarrow T \longrightarrow T/T_0$ of dg-categories over $A$, the induced sequence

   
   exhibits $\mathsf {M}^{T_0}$ has the fiber of the morphism $\mathsf {M}^{T} \rightarrow \mathsf {M}^{T/T_0}$ in $\mathbf {Mod}(\mathbf {B}\mathbb {U}_S)$.

2. 2. The natural morphism $\mathbf {B}\mathbb {U}_S \longrightarrow \mathsf {M}^{A},$ induced by the lax-monoidal structure of $\mathsf {M}^{-}$, is an equivalence of $\mathbf {B}\mathbb {U}_S$-modules.

3. 3. The $\infty$-functor $T \mapsto \mathsf {M}^{T}$ commutes with filtered colimits.

4. 4. For any quasi-compact and quasi-separated scheme $X$, and any morphism $p : X \longrightarrow S$, we have a natural equivalence of $\mathbf {B}\mathbb {U}_S$-modules

   \[ \mathsf{M}^{\mathsf{Perf}(X)} \simeq p_*(\mathbf{B}\mathbb{U}_X), \]
   where $p_* : \mathbf {Mod}_{\mathcal {SH}_X}(\mathbf {B}\mathbb {U}_X) \longrightarrow \mathbf {Mod}_{\mathcal {SH}_S}(\mathbf {B}\mathbb {U}_S)$ is the direct image of $\mathbf {B}\mathbb {U}$-modules, and $\mathsf {Perf}(X)$ is the dg-category of perfect complexes on $X$.

We now let $\ell$ be a prime number invertible in $A$. We denote by $Sh^{\mathsf {ct}}_{\mathbb {Q}_{\ell }}(S)$ the $\infty$-category of constructible $\mathbb {Q}_{\ell }$-adic complexes on the étale site $S_{\unicode{x00E9} {\textrm {t}}}$ of $S$. It is a symmetric monoidal $\infty$-category, and we denote by

\[ Sh_{\mathbb{Q}_{\ell}}(S):=\mathbf{Ind}(Sh^{\mathsf{ct}}_{\mathbb{Q}_{\ell}}(S)) \]

its completion under filtered colimits (see [Reference Gaitsgory and LurieGL, Definition 4.3.26]). According to [Reference RobaloRob15, Corollary 2.3.9], there exists an $\ell$-adic realization $\infty$-functor

\[ R_{\ell} : \mathcal{SH}_S \longrightarrow Sh_{\mathbb{Q}_{\ell}}(S). \]

By construction, $R_{\ell }$ is a symmetric monoidal $\infty$-functor sending a smooth scheme $p : X \longrightarrow S$ to $p_!p^{!}(\mathbb {Q}_{\ell })$ or, in other words, to the relative $\ell$-adic homology of $X$ over $S$.

We let $\mathbb {T}:=\mathbb {Q}_{\ell }[2](1)$, and we consider the $\mathbb {E}_{\infty }$-ring object in $Sh_{\mathbb {Q}_{\ell }}(S)$,

\[ \mathbb{Q}_{\ell}(\beta):=\oplus_{n\in \mathbb{Z}}\mathbb{T}^{\otimes n}. \]

In this notation, $\beta$ stands for $\mathbb {T}$, and $\mathbb {Q}_{\ell }(\beta )$ for the algebra of Laurent polynomials in $\beta$, so we might as well write

\[ \mathbb{Q}_{\ell}(\beta)=\mathbb{Q}_{\ell}[\beta,\beta^{-1}]. \]

As shown in [Reference Blanc, Robalo, Toën and VezzosiBRTV18], there exists a canonical equivalence $R_{\ell }(\mathbf {B}\mathbb {U}_S) \simeq \mathbb {Q}_{\ell }(\beta )$ of $\mathbb {E}_{\infty }$-ring objects in $Sh_{\mathbb {Q}_{\ell }}(S)$, that is induced by the Chern character from algebraic K-theory to étale cohomology. We thus obtain a well-defined symmetric monoidal $\infty$-functor

\[ \mathsf{R}_{\ell} : \mathbf{Mod}_{\mathcal{SH}_S}(\mathbf{B}\mathbb{U}_S) \longrightarrow \mathbf{Mod}_{Sh_{\mathbb{Q}_{\ell}}(S)}(\mathbb{Q}_{\ell}(\beta)), \]

from $\mathbf {B}\mathbb {U}_S$-modules in $\mathcal {SH}_S$ to $\mathbb {Q}_{\ell }(\beta )$-modules in $Sh_{\mathbb {Q}_{\ell }}(S)$. By pre-composing with the functor $T \mapsto \mathsf {M}^{T}$, we obtain a lax-monoidal $\infty$-functor

\[ r_{\ell} := \mathsf{R}_{\ell} \circ \mathsf{M}^{-}: \mathbf{dgCat}_S \longrightarrow \mathbf{Mod}_{Sh_{\mathbb{Q}_{\ell}}(S)}(\mathbb{Q}_{\ell}(\beta)). \]

Definition 2.2.1 The $\infty$-functor defined above

\[ r_{\ell} : \mathbf{dgCat}_S \longrightarrow \mathbf{Mod}_{Sh_{\mathbb{Q}_{\ell}}(S)}(\mathbb{Q}_{\ell}(\beta)) \]

is called the $\ell$-adic realization functor for dg-categories over $S$.

From the standard properties of the functor $T \mapsto \mathsf {M}^{T}$, recalled above, we obtain the following properties for the $\ell$-adic realization functor $T \mapsto r_{\ell }(T)$.

1. 1. The $\infty$-functor $r_{\ell }$ is a localizing invariant, that is, for any short exact sequence $T_0 \hookrightarrow T \longrightarrow T/T_0$ of dg-categories over $A$, the induced sequence

   
   is a fibration sequence in $\mathbf {Mod}(\mathbb {Q}_{\ell }(\beta ))$.

2. 2. The natural morphism

   \[ \mathbb{Q}_{\ell}(\beta) \longrightarrow r_{\ell}(A), \]
   induced by the lax-monoidal structure, is an equivalence in $\mathbf {Mod}_{Sh_{\mathbb {Q}_{\ell }}(S)}(\mathbb {Q}_{\ell }(\beta ))$.

3. 3. The $\infty$-functor $r_{\ell }$ commutes with filtered colimits.

4. 4. For any separated morphism of finite type $p : X \longrightarrow S$, we have a natural morphism of $\mathbb {Q}_{\ell }(\beta )$-modules

   \[ r_{\ell}(\mathsf{Perf}(X)) \longrightarrow p_*(\mathbb{Q}_{\ell}(\beta)), \]
   where $p_* : \mathbf {Mod}_{Sh_{\mathbb {Q}_{\ell }}(S)}(\mathbb {Q}_{\ell, X}(\beta )) \longrightarrow \mathbf {Mod}_{Sh_{\mathbb {Q}_{\ell }}(S)}(\mathbb {Q}_{\ell }(\beta ))$ is induced by the direct image $Sh^{\mathsf {ct}}_{\mathbb {Q}_{\ell }}(X) \longrightarrow Sh^{\mathsf {ct}}_{\mathbb {Q}_{\ell }}(S)$ of constructible $\mathbb {Q}_{\ell }$-complexes. If either $p$ is proper or $A$ is a field, this morphism is an equivalence.

Remark 2.2.2 The $\ell$-adic realization functor of Definition 2.2.1 can be easily ‘sheafified’, as follows. For $X$ a separated, excellent scheme, and $T$ a sheaf of dg-categories on $X_{\textrm {zar}}$ (i.e. $T \in \mathbf {dgCat}_X := \textrm {lim}\,\mathbf {dgCat}_{\mathcal {U}^{\bullet }}$ where $\mathcal {U}^{\bullet }$ is the Cech nerve of any Zariski open cover $\mathcal {U}$ of $X$), by taking the limit along the Cech nerve of a Zariski open cover of $X$, we define $\mathsf {M}_{X}^{-}: \mathbf {dgCat}_X \to \mathbf {Mod}_{\mathcal {SH}_X}(\mathbf {B}\mathbb {U}_X)$ and $\mathsf {R}_{\ell, X} : \mathbf {Mod}_{\mathcal {SH}_X}(\mathbf {B}\mathbb {U}_X) \longrightarrow \mathbf {Mod}_{Sh_{\mathbb {Q}_{\ell }}(X)}(\mathbb {Q}_{\ell, X}(\beta ))$. The composite

\[ r_{\ell,X}:= \mathsf{R}_{\ell, X} \circ \mathsf{M}_{X}^{-} : \mathbf{dgCat}_X \longrightarrow \mathbf{Mod}_{Sh_{\mathbb{Q}_{\ell}}(X)}(\mathbb{Q}_{\ell, X}(\beta)) \]

is called the $\ell$-adic realization functor for dg-categories over $X$. Obviously we have $r_{\ell } = r_{\ell, S}$, for $r_{\ell }$ as in Definition 2.2.1. Moreover, for $X$ as above, any separated morphism of finite type $p : X \longrightarrow S$ induces a commutative diagram as follows.Footnote ⁶

2.3 The $\ell$-adic Chern character

As explained in [Reference Blanc, Robalo, Toën and VezzosiBRTV18], there is a symmetric monoidal $\infty$-category $\mathcal {SH}_A^{\rm nc}$ of non-commutative motives over $A$. As an $\infty$-category it is the full sub-$\infty$-category of $\infty$-functors of (co)presheaves of spectra

\[ \mathbf{dgCat}_A^{\rm ft} \longrightarrow \mathbf{Sp}, \]

satisfying Nisnevich descent and $\mathbb {A}^{1}$-homotopy invariance.

We consider $\Gamma : Sh_{\mathbb {Q}_{\ell }}(S) \longrightarrow \mathbf {dg}_{\mathbb {Q}_{\ell }},$ the global section $\infty$-functor, taking an $\ell$-adic complex on $S_{\unicode{x00E9} {\textrm {t}}}$ to its hyper-cohomology. Composing this with the Dold–Kan construction $\mathbb {R}\mathbf {Map}_{\mathbf {dg}_{\mathbb {Q}_{\ell }}}(\mathbb {Q}_{\ell },-) : \mathbf {dg}_{\mathbb {Q}_{\ell }} \longrightarrow \mathbf {Sp}$ we obtain an $\infty$-functor

\[ |-| : Sh_{\mathbb{Q}_{\ell}}(S) \longrightarrow \mathbf{Sp}, \]

which computes hyper-cohomology of $S_{\unicode{x00E9} {\textrm {t}}}$ with $\ell$-adic coefficients, that is, for any $E \in Sh_{\mathbb {Q}_{\ell }}(S)$, we have natural isomorphisms

\[ H^{i}(S_{\unicode{x00E9} {\textrm {t}}},E) \simeq \pi_{-i}(|E|),\quad i\in \mathbb{Z}. \]

By what we have seen in our last paragraph, the composite functor $T \mapsto |r_{\ell }(T)|$ provides a (co)presheaf of spectra

\[ \mathbf{dgCat}_A^{\rm ft} \longrightarrow \mathbf{Sp}, \]

satisfying Nisnevich descent and $\mathbb {A}^{1}$-homotopy invariance. It thus defines an object $|r_{\ell }| \in \mathcal {SH}_A^{\rm nc}$. The fact that $r_{\ell }$ is lax symmetric monoidal implies, moreover, that $|r_{\ell }|$ is endowed with a natural structure of a $\mathbb {E}_\infty$-ring object in $\mathcal {SH}_A^{\rm nc}$.

Each $T \in \mathbf {dgCat}_A^{\rm ft}$ defines a corepresentable object $h^{T} \in \mathcal {SH}_A^{\rm nc}$, characterized by the ($\infty$-)functorial equivalence

\[ \mathbb{R}\mathbf{Map}_{\mathcal{SH}_A^{\rm nc}}(h^{T},F) \simeq F(T), \]

for any $F \in \mathcal {SH}_A^{\rm nc}$. The existence of $h^{T}$ is a formal statement, however the main theorem of [Reference RobaloRob15] implies that we have a natural equivalence of spectra

\[ \mathbb{R}\mathbf{Map}_{\mathcal{SH}_A^{\rm nc}}(h^{T},h^{A}) \simeq \mathsf{HK}(T), \]

where $\mathsf {HK}(T)$ stands for non-connective homotopy invariant algebraic $K$-theory of the dg-category $T$. In other words, $T \mapsto \mathsf {HK}(T)$ defines an object in $\mathcal {SH}_A^{\rm nc}$ which is isomorphic to $h^{B}$. By the Yoneda lemma, we thus obtain an equivalence of spaces

\[ \mathbb{R}\mathbf{Map}^{{\rm lax}\text{-}\otimes}(\mathsf{HK},|r_\ell|) \simeq \mathbb{R}\mathbf{Map}_{\mathbb{E}_\infty-\mathbf{Sp}}(\mathbb{S},|r_{\ell}(A)|)\simeq *. \]

In other words, there exists a unique (up to a contractible space of choices) lax symmetric monoidal natural transformation

\[ \mathsf{HK} \longrightarrow |r_\ell|, \]

between lax-monoidal $\infty$-functors from $\mathbf {dgCat}^{\rm ft}_A$ to $\mathbf {Sp}$. We extend this to all dg-categories over $A$, as usual, by passing to Ind-completion $\mathbf {dgCat}_A \simeq \mathbf {Ind}(\mathbf {dgCat}_A^{\rm ft})$.

Definition 2.3.1 The natural transformation defined above is called the $\ell$-adic Chern character. It is denoted by

\[ \textit{Ch}_{\ell} : \mathsf{HK}(-) \longrightarrow |r_{\ell}(-)|. \]

Remark 2.3.2 Definition 2.3.1 contains a built-in, formal Grothendieck–Riemann–Roch formula. Indeed, for any $B$-linear dg-functor $f : T \longrightarrow T'$, the square of spectra

commutes up to a natural equivalence.

2.4 Trace formula for dg-categories

Let $\mathcal {C}^{\otimes }$ be a symmetric monoidal $\infty$-category (see [Reference Toën and VezzosiTV15] and [Reference LurieLur16, Definition 2.0.0.7]).

Informally, one can describe $\underline {\mathbf {Alg}}(\mathcal {C})$ as the $(\infty, 2)$-category with:

• • objects, associative unital monoids ($=:E_1$-algebras) in $\mathcal {C}^{\otimes }$;

• • $\textit {Map}_{\underline {\mathbf {Alg}}(\mathcal {C})}(B,B'):=\mathbf {Bimod}_{B', B} (\mathcal {C}^{\otimes })$, the $\infty$-category of $(B',B)$-bimodules;

• • the composition of $1$-morphisms (i.e. of bimodules) is given by tensor product;

• • the composition of $2$-morphisms (i.e. of morphisms between bimodules) is the usual composition.

Unraveling the definition, we find that a right dual of $X$ is a left $B^{\otimes \textrm {-op}}$-module $Y$, where $(-)^{\otimes \textrm {-op}}$ denotes the opposite $E_1$-algebra structure, the unit $u$ of adjunction (or coevaluation) is a map $coev: 1_{\mathcal {C}} \to Y\otimes _B X$ in $\mathcal {C}$, the counit $v$ of adjunction (or evaluation) is a map ${ev: X\otimes Y \to B}$ of $(B,B)$-bimodules; $u$ and $v$ satisfy usual compatibilities.

Note that, if a right $B$-dual of $X$ exists, then it is unique up to an equivalence.

If the right $B$-module $Y$ is the right $B$-dual of the left $B$-module $X$, then we can define the trace of any map $f: X\to X$ of left $B$-modules, as follows.

Recall that we have a coevaluation map in $\mathcal {C}$ $coev: 1_{\mathcal {C}} \to Y \otimes _B X$ in $\mathcal {C}$ and an evaluation map of $(B,B)$-bimodules $ev: X\otimes Y \to B$.

Consider the graph $\Gamma _f$ defined as the composite

We now elaborate on the evaluation map. Observe that:

• • $B \in \mathcal {C}$ has a left $B\otimes B^{\otimes \textrm {-op}}$-module structure that we denote by $B^{\rm L}$;

• • $B \in \mathcal {C}$ has a right $B\otimes B^{\otimes \textrm {-op}}$-module structure (i.e. a left $B^{\otimes \textrm {-op}} \otimes B$-module), that we denote by $B^{\rm R}$;

• • $ev: X \otimes Y \to B^{\rm L}$ is a map of left $B\otimes B^{\otimes \textrm {-op}}$- modules;

• • the composite is a map of left $(B^{\otimes \textrm {-op}}\otimes B)$-modules.

Apply $(-) \otimes _{B \otimes B^{\otimes \textrm {-op}}}B^{\rm L}$ to the composite

to obtain

Now observe that $(Y\otimes X) \otimes _{B \otimes B^{\otimes \textrm {-op}}}B^{\rm L} \simeq Y \otimes _B X$ in $\mathcal {C}$.

Note that, by definition, $\mathsf {HH}_{\mathcal {C}} (B)$ is (only) an object in $\mathcal {C}$, called the Hochschild homology object of $B$.

Definition 2.4.4 The non-commutative trace of $f: X \to X$ over $B$ is defined as the composite

Here $\textit{Tr}_B(f)$ is a morphism in $\mathcal {C}$.

Remark 2.4.5 Let $B \in \mathbf {CAlg}(\mathcal {C^{\otimes }})$, and let us still denote by $B$ its image via the canonical map $\mathbf {CAlg} (\mathcal {C}^{\otimes }) \to \mathbf {Alg}_{E_1}(\mathcal {C}^{\otimes })$. In this case, $\mathbf {Mod}_B(\mathcal {C}^{\otimes })$ is a symmetric monoidal $\infty$-category, and if $X\in \mathbf {Mod}_B(\mathcal {C}^{\otimes })$ is a dualizable object (in the usual sense), then its (left and right) dual in $\mathbf {Mod}_B(\mathcal {C}^{\otimes })$ is also a right dual of $X$ according to Definition 2.4.3. In this case, any $f: X \to X$ in $\mathbf {Mod}_B(\mathcal {C}^{\otimes })$, has therefore two possible traces, a non-commutative trace (as in Definition 2.4.4)

\[ \textit{Tr}_B(f): 1_{\mathcal{C}} \longrightarrow \mathsf{HH}_{\mathcal{C}}(B), \]

which is a morphism in $\mathcal {C}$, and a more standard, commutative trace

\[ \textit{Tr}^{\textrm{c}}_B(f): B \longrightarrow B, \]

which is a morphism on $\mathbf {Mod}_B(\mathcal {C}^{\otimes })$. The two traces are related by the following commutative diagram

where $a: \mathsf {HH}_{\mathcal {C}}(B) \to B$ is the canonical augmentation (which exists because $B$ is commutative), and $u_B: 1_{\mathcal {C}} \to B$ is the unit map of the algebra $B$ in $\mathcal {C}$.

The case of dg-categories. Let us specialize the previous discussion to the case $\mathcal {C}^{\otimes }= \mathbf {dgCat}_A$.

Let $B$ be a monoidal dg-category, that is, an associative and unital monoid in the symmetric monoidal $\infty$-category $\mathbf {dgCat}_A$.

Proof. This is very similar to the argument used in [Reference ToënToë12, Proposition 2.5(1)]. We consider $\widehat {T^{\rm op}}$, and we define evaluation and coevaluation maps as follows.

The big morphism $h$ introduced right before Definition 2.1.2,

\[ h : \widehat{T^{\rm op}} \,\widehat{\otimes}_A\, \hat{T} \longrightarrow \hat{B}, \]

whose domain is naturally a $\hat {B}$-bimodule, can be canonically lifted to a morphism of bimodules. We choose this as our evaluation morphism.

The coevaluation is then obtained by duality. We start by the diagonal bimodule

\[ T : (T^{\rm op} \otimes_A T)^{\rm op} \longrightarrow L(A)=\hat{A} \]

sending $(x,y)$ to $T(y,x)$. This morphism naturally descends to $(T^{o} \otimes _B T)^{\rm op}$, providing a dg-functor

\[ (T^{\rm o} \otimes_B T)^{\rm op}\longrightarrow \hat{A}. \]

Note that $T^{\rm op}$ is naturally a right $B$-module because $T$ is assumed to be co-tensored (and note that, otherwise, $T^{\rm op} \otimes _B T$ would not make sense). This dg-functor is an object in $\widehat {T^{\rm op}\otimes _B T} \simeq \widehat {T^{\rm op}} \,\widehat {\otimes }_A\, \hat {T}$ and, thus, defines a coevaluation morphism

\[ \hat{A} \longrightarrow \widehat{T^{\rm op}} \,\widehat{\otimes}_A\, \hat{T}. \]

These two evaluation and coevaluation morphisms satisfy the required triangular identities and, thus, make $\widehat {T^{\rm op}}$ a right dual to $\hat {T}$.

According to the previous proposition, any co-tensored $B$-module $T$ has a big right dual, so it comes equipped with big evaluation and coevaluation maps.

In particular, if $T$ saturated over $B$, and $f:T \to T$ is a morphism in $\mathbf {dgCat}_\mathcal {B}$, then the trace

\[ \textit{Tr}_{B}(f:T \to T): A \to \mathsf{HH}(B/A)=B{}^{\rm R} \otimes_{B^{\otimes \textrm{-op}}\otimes_A B}B{}^{\rm L} \]

is also small, that is, it is a morphism inside $\mathbf {dgCat}_A$.

As our $\ell$-adic realization functor $r_{\ell }$ is only lax-monoidal, to establish our trace formula for an endomorphism $f: T\to T$ of a saturated $B$-module, we need to restrict to those saturated $B$-modules on which $r_{\ell }$ is, in fact, symmetric monoidal.

Definition 2.4.8 A saturated $T \in \mathbf {dgCat}_{B}$ is called $\ell ^{\otimes }$-admissible if the canonical map

\[ r_{\ell}(T^{\rm op})\otimes_{r_{\ell}(B)} r_{\ell}(T) \longrightarrow r_{\ell}(T^{\rm op}\otimes_{B} T) \]

is an equivalence in $Sh_{\mathbb {Q}_{\ell }}(S)$.

The trace $\textit {Tr}_B(f)$ is a map $A \to \mathsf {HH}(B/A)$, hence it induces a map in $\mathbf {Sp}$,

\[ K(\textit{Tr}_B (f)): K(A) \to K(\mathsf{HH}(B/A)), \]

which is actually a map of $K(A)$-modules (in spectra), because $K$ is lax-monoidal. Hence, it corresponds to an element denoted as

\[ [\mathsf{HH}(T/B,f)] \equiv \textit{Tr}_B (f) \in K_0(\mathsf{HH}(B/A)). \]

Therefore, its image by the $\ell$-adic Chern character $\textit {Ch}_{\ell, 0}:= \pi _0(\textit {Ch}_{\ell })$,

\[ \textit{Ch}_{\ell, 0}: K_0(\mathsf{HH}(B/A)) \to Hom_{D(r_{\ell}(A))}(r_{\ell}(A), r_{\ell}(\mathsf{HH}(B/A))) \simeq H^{0}(S_{\unicode{x00E9} {\textrm {t}}}, r_{\ell}(\mathsf{HH}(B/A))), \]

is an element $\textit {Ch}_{\ell, 0}([\mathsf {HH}(T/B;f)]) \in H^{0}(S_{\unicode{x00E9} {\textrm {t}}}, r_{\ell }(\mathsf {HH}(B/A)))$.

On the other hand, the trace $\textit {Tr}_{r_{\ell }(B)}(r_{\ell }(f))$ of $r_{\ell }(f)$ over $r_{\ell }(B)$ is, by definition, a morphism

\[ r_{\ell}(A) \simeq \mathbb{Q}_{\ell}(\beta)\to \mathsf{HH}(r_{\ell}(B)/ r_{\ell}(A)) \]

in $\mathbf {Mod}_{r_{\ell }(A)}(Sh_{\mathbb {Q}_{\ell }}(S))$.

We may further compose this with the canonical map

\[ \mathsf{HH}(r_{\ell}(B)/ r_{\ell}(A)) \to r_{\ell}(\mathsf{HH}(B/ A)) \]

(given by lax-monoidality of $r_{\ell }(-)$), to obtain a map

\[ r_{\ell}(A) \to r_{\ell}(\mathsf{HH}(B/ A)) \]

in $\mathbf {Mod}_{r_{\ell }(A)}(Sh_{\mathbb {Q}_{\ell }}(S))$. This is the same thing as an element denoted as

\[ \textit{Tr}_{r_{\ell}(B)}(r_{\ell}(f)) \in \pi_0 (|r_{\ell}(\mathsf{HH}(B/ A))|) \simeq H^{0}(S_{\unicode{x00E9} {\textrm {t}}}, r_{\ell}(\mathsf{HH}(B/ A)) ). \]

Theorem 2.4.9 Let $B$ a monoidal dg-category over $A$, $T\in \mathbf {dgCat}_B$ a saturated and $\ell ^{\otimes }$-admissible $B$-module, and $f:T\to T$ map in $\mathbf {dgCat}_B$. Then, we have an equality

\[ \textit{Ch}_{\ell, 0}([\mathsf{HH}(T/B,f)]) =\textit{Tr}_{r_{\ell}(B)}(r_{\ell}(f)) \]

in $H^{0}(S_{\unicode{x00E9} {\textrm {t}}}, r_{\ell }(\mathsf {HH} (B/ A))$.

Lemma 2.4.10 Let

\[ F : \mathcal{C} \longrightarrow \mathcal{D} \]

be a lax symmetric monoidal $\infty$-functor between presentable symmetric monoidal $\infty$-categories. Let $B$ be a monoid in $\mathcal {C}$, $M$ a left $B$-module, and $f : M \longrightarrow M$ a morphism of $B$-modules. We assume that $M$ has a right dual $M^{o}$, and that the natural morphism

\[ F(M^{o}) \otimes_{F(B)}F(M) \longrightarrow F(M^{o}\otimes_B M) \]

is an equivalence. Then, $F(M)$ has a right dual, and we have

\[ F(\textit{Tr}(f)) = i(\textit{Tr}(F(f))) \]

as elements in $\pi _0(Hom_{\mathcal {D}}(\mathbf {1},F(\mathsf {HH}(B)))$, where $i$ is induced by the natural morphism $\mathsf {HH}(F(B)) \longrightarrow F(\mathsf {HH}(B))$.

3.1 Trivializing the Tate twist

We let $\ell$ be a prime invertible in $k$, and we denote by $p$ the characteristic exponent of $k$. As $k$ is algebraically closed, we may, and will, choose once for all a group isomorphism

\[ \mu_{\infty}(k)\simeq \mu_{\infty}(K) \simeq (\mathbb{Q}/\mathbb{Z})[p^{-1}] \]

between the group of roots of unity in $k$ and the prime-to-$p$ part of $\mathbb {Q}/\mathbb {Z}$. Equivalently, we have chosen a given group isomorphism

\[ \lim_{(n,p)=1}\mu_n(k) \simeq \hat{\mathbb{Z}}', \]

where $\hat {\mathbb {Z}}':=\lim _{(n,p)=1} \mathbb {Z}/n$. In particular, we have selected a topological generator of $\lim _{(n,p)=1}\mu _n(k)$, corresponding to the image of $1\in \mathbb {Z}$ inside $\hat {\mathbb {Z}}'$. The choice of the isomorphism above also provides a specific induced isomorphism $\mathbb {Q}_{\ell }(1) \simeq \mathbb {Q}_{\ell }$ of $\mathbb {Q}_{\ell }$-sheaves on $S$, where $(1)$ denotes, as usual, the Tate twist. By taking tensor powers of this isomorphism, we obtain various induced isomorphisms $\mathbb {Q}_{\ell }(i) \simeq \mathbb {Q}_{\ell }$ for all $i\in \mathbb {Z}$.

We recall that the absolute Galois group $\mathrm {I}$ of $K$ (which coincides with the inertia group in our case) sits in an extension of pro-finite groupsFootnote ⁷

where $P$ is a pro-$p$-group (the wild inertia subgroup). For any continuous finite-dimensional $\mathbb {Q}_{\ell }$-representation $V$ of $\mathrm {I}$, the group $P$ acts by a finite quotient $G_V$ on $V$. Moreover, the Galois cohomology of $V$ can then be explicitly identified with the two-terms complex

where $T$ is the action of the chosen topological generator of $\mathrm {I}_{\rm t}$. This easily implies that for any $\mathbb {Q}_{\ell }$-representation $V$ of $\mathrm {I}$, the natural pairing on Galois cohomology

\[ H^{i}(\mathrm{I},V) \otimes H^{1-i}(\mathrm{I},V^{\vee}) \longrightarrow H^{1}(\mathrm{I},\mathbb{Q}_{\ell})\simeq \mathbb{Q}_{\ell} \]

is non-degenerate. In other words, if we denote by $V^{\mathrm{I}}$ the complex of cohomology of $\mathrm {I}$ with coefficients in $V$, we have a natural quasi-isomorphism $(V^{\mathrm{I}})^{\vee } \simeq (V^{\vee })^{\mathrm{I}}[1]$.

3.2 Reminders on actions of the inertia group

Let $X \longrightarrow S$ be an $S$-scheme (separated and of finite type, according to our conventions). We recall from [Reference Deligne and KatzSGA7-II, Exp. XIII, 1.2] that we can associate to $X$ a vanishing topos $(X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu }$ which is defined as a (2-)fiber product of toposes

\[ (X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu}:=(X_s)^{\sim}_{\unicode{x00E9} {\textrm {t}}} \times_{s_{\unicode{x00E9} {\textrm {t}}}^{\sim}}\eta_{\unicode{x00E9} {\textrm {t}}}^{\sim}. \]

As $S$ is strictly henselian, $s_{\unicode{x00E9} {\textrm {t}}}^{\sim }$ is, in fact, the punctual topos, and the fiber product above is, in fact, a product of topos. The topos $\eta _{\unicode{x00E9} {\textrm {t}}}^{\sim }$ is equivalent to the topos of sets with continuous action of $\mathrm {I}=\mathrm {Gal}(K^{\rm sp}/k)$, where $K^{\rm sp}$ denotes a separable closure of $K$. Morally, $(X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu }$ is the topos of étale sheaves on $X_s$, endowed with a continuous action of $\mathrm {I}$ (see [Reference Deligne and KatzSGA7-II, Exp. XIII, 1.2.4]).

As explained in [Reference Blanc, Robalo, Toën and VezzosiBRTV18] we have an $\ell$-adic $\infty$-category $\mathcal {D}((X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu },\mathbb {Z}_\ell )$. Morally speaking, objects of this $\infty$-category consist of the data of an object $E \in \mathcal {D}(\bar {X}_s,\mathbb {Z}_\ell )= Sh_{\mathbb {Z}_{\ell }}(\bar {X}_s)$ together with a continuous action of $\mathrm {I}$. We say that such an object is constructible if $E$ is a constructible object in $\mathcal {D}(\bar {X}_s,\mathbb {Z}_\ell )=Sh_{\mathbb {Z}_{\ell }}(\bar {X}_s)$, and we denote by $\mathcal {D}_c((X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu },\mathbb {Z}_\ell )$ the full sub-$\infty$-category of constructible objects.

Definition 3.2.1 The $\infty$-category of ind-constructible $\mathrm {I}$-equivariant $\ell$-adic complexes on $X_s$ is defined by

\[ \mathcal{D}^{\mathrm{I}}_{\rm ic}(X_s,\mathbb{Q}_{\ell}):=\mathsf{Ind}(\mathcal{D}_{\rm c}((X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu},\mathbb{Z}_\ell) \otimes_{\mathbb{Z}_\ell}\mathbb{Q}_{\ell}). \]

The full sub-$\infty$-category of constructible objects is $\mathcal {D}^{\mathrm {I}}_{\rm c}(X_s,\mathbb {Q}_{\ell }):=\mathcal {D}_{\rm c}((X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu },\mathbb {Z}_\ell ) \otimes _{\mathbb {Z}_\ell } \mathbb {Q}_{\ell }$.

Note that because we have chosen trivializations of the Tate twists, $\mathbb {Q}_{\ell }(\beta )$ is identified with $\mathbb {Q}_{\ell }[\beta,\beta ^{-1}]$ where $\beta$ is a free variable in degree $2$. This is a graded algebra object in $\mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$, and we define the $\infty$-category $\mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell }(\beta ))$ as the $\infty$-category of $\mathbb {Q}_{\ell }(\beta )$-modules in $\mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ or, equivalently, the $2$-periodic $\infty$-category of ind-constructible $\mathbb {Q}_{\ell }$-adic complexes on $(X/S)^{\nu }_{\unicode{x00E9} {\textrm {t}}}$.

Note that strictly speaking an object of $\mathcal {D}^{\mathrm {I}}_{\rm c}(X_s,\mathbb {Q}_{\ell }(\beta ))$ is not constructible in the usual sense, as its underlying object in $\mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ is $2$-periodic.

The topos $(X/S)_{\unicode{x00E9} {\textrm {t}}}^{\nu }$ comes with a natural projection $(X/S)^{\nu }_{\unicode{x00E9} {\textrm {t}}} \longrightarrow (X_s)^{\sim }_{\unicode{x00E9} {\textrm {t}}}$ whose direct image is an $\infty$-functor denoted by

\[ (-)^{\mathrm{I}} : \mathcal{D}^{\mathrm{I}}_{\rm ic}(X_s,\mathbb{Q}_{\ell}) \longrightarrow \mathcal{D}_{\rm ic}(X_s,\mathbb{Q}_{\ell}) \]

called the $\mathrm {I}$-invariants functor. This $\infty$-functor preserves constructibility. The $\infty$-categories $\mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ and $\mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ carries natural symmetric monoidal structures and the $\infty$-functor $(-)^{\mathrm{I}}$ comes equipped with a natural lax symmetric monoidal structure (being induced by the direct image of a morphism of toposes). Moreover, $(-)^{\mathrm{I}}$ is the right adjoint of the symmetric monoidal $\infty$-functor $U: \mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell }) \longrightarrow \mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ endowing objects in $\mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ with the trivial action of $\mathrm {I}$. This gives $\mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ the structure of a $\mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$-module via $\mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell }) \times \mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell }) \to \mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell }) : (E,F) \mapsto U(E)\otimes F$. This bi-functor distributes over colimits, thus by the adjoint functor theorem, we obtain an enrichment of $\mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ over $\mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$. Note that $\mathcal {D}^{\mathrm {I}}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$ is also enriched over itself.

It is important to note that the $\mathrm {I}$-invariants functor commutes with base change in the following sense. Let $f : \mathrm {Spec}\, k \longrightarrow X_s$ be a geometric point. The morphism $f$ defines a geometric point of $(X_s)_{\unicode{x00E9} {\textrm {t}}}^{\sim }$ and, thus, induces a geometric morphism of toposes

\[ \eta_{\unicode{x00E9} {\textrm {t}}}^{\sim} \longrightarrow (X/S)^{\nu}_{\unicode{x00E9} {\textrm {t}}}. \]

We thus have an inverse image functor

\[ f^{*} : \mathcal{D}_{\rm c}^{\mathrm{I}}(X_s,\mathbb{Q}_{\ell}) \longrightarrow\mathcal{D}_c(\eta,\mathbb{Q}_{\ell})=\mathcal{D}_{\rm c}(\mathrm{I}, \mathbb{Q}_{\ell}). \]

where $\mathcal {D}_c(\mathrm {I}, \mathbb {Q}_{\ell })$ is the $\infty$-category of finite-dimensional complexes of $\ell$-adic representations of $\mathrm {I}$. As usual, the square of $\infty$-functors

comes equipped with a natural transformation

\[ f^{*}((-)^{\mathrm{I}}) \Rightarrow (f^{*}(-))^{\mathrm{I}}. \]

It can be checked that this natural transformation is always an equivalence. In particular, for any geometric point $x$ in $X_s$, we have a natural equivalence of $\ell$-adic complexes $(E)^{\mathrm {I}}_x \simeq (E_x)^{\mathrm{I}}$, for any $E \in \mathcal {D}_c^{\mathrm{I}}(X_s,\mathbb {Q}_{\ell })$.

The dualizing complex $\omega$ of the scheme $X_s$ can be used to obtain a dualizing object in $\mathcal {D}_{\rm c}^{\mathrm{I}}(X_s,\mathbb {Q}_{\ell })$ as follows. We consider $\omega$ as an object in $\mathcal {D}_c^{\mathrm{I}}(X_s,\mathbb {Q}_{\ell })$ endowed with the trivial $\mathrm {I}$-action. We then have an equivalence of $\infty$-categories

\[ \mathbb{D}_I : \mathcal{D}_c^{\mathrm{I}}(X_s,\mathbb{Q}_{\ell}) \longrightarrow \mathcal{D}_c^{\mathrm{I}}(X_s,\mathbb{Q}_{\ell})^{\rm op} \]

sending $E$ to $\mathbb {R}\underline {Hom}(E,\omega )$, where $\mathbb {R}\underline {Hom}$ denotes the natural enrichment of $\mathcal {D}_c^{\mathrm {I}}(X_s,\mathbb {Q}_{\ell })$ over itself. We obviously have a canonical biduality equivalence $\mathbb {D}_{\mathrm {I}}^{2} \simeq \textrm {id}$. The duality functor $\mathbb {D}_I$ is compatible with the usual Grothendieck duality functor $\mathbb {D}$ for the scheme $X_s$ up to a shift, as explained by the following lemma.

Lemma 3.2.3 For any object $E\in \mathcal {D}_{\rm c}^{\mathrm{I}}(X_s,\mathbb {Q}_{\ell })$, there is a functorial equivalence in $\mathcal {D}_{\rm c}(X_s,\mathbb {Q}_{\ell })$

\[ d : \mathbb{D}(E^{\mathrm{I}})[-1]\simeq (\mathbb{D}_I(E))^{\mathrm{I}}. \]

Proof. Taking $\mathrm {I}$-invariants is a lax-monoidal $\infty$-functor, so we have a natural map $E^{\mathrm{I}} \otimes \mathbb {D}_I(E)^{\mathrm{I}} \longrightarrow (E\otimes \mathbb {D}_I(E))^{\mathrm{I}},$ that can be composed with the evaluation morphism $E\otimes \mathbb {D}_I(E) \longrightarrow \omega$ to obtain $E^{\mathrm{I}} \otimes \mathbb {D}_I(E)^{\mathrm{I}} \longrightarrow \omega ^{\mathrm{I}}$. As the action of $\mathrm {I}$ on $\omega$ is trivial, we have a canonical equivalence $\omega ^{\mathrm{I}}\simeq \omega \otimes \mathbb {Q}_{\ell }^{\mathrm{I}} \simeq \omega \oplus \omega [-1]$. By projection on the second factor we obtain a pairing $E^{\mathrm{I}}\otimes \mathbb {D}_I(E)^{\mathrm{I}} \longrightarrow \omega [-1]$ and, thus, a map

\[ \mathbb{D}_I(E)^{\mathrm{I}} \longrightarrow \mathbb{D}(E^{\mathrm{I}})[-1]. \]

We claim that the above morphism is an equivalence in $\mathcal {D}_c(X_s,\mathbb {Q}_{\ell })$. For this it is enough to check that the above morphism is a stalkwise equivalence. Now, the stalk of the above morphism at a geometric point $x$ of $X_s$ can be written as

\[ (E(x)^{\vee})^{\mathrm{I}} \longrightarrow (E(x)^{\mathrm{I}})^{\vee}[-1], \]

where $E(x):=H^{*}_x(X,E) \in D_c(\mathbb {Q}_{\ell })$ is the local cohomology of $E$ at $x$, and $(-)^{\vee }$ is now the standard linear duality over $\mathbb {Q}_{\ell }$. The result now follows from the following well-known duality for $\mathbb {Q}_{\ell }$-representations of $\mathrm {I}$: for any finite-dimensional $\mathbb {Q}_{\ell }$-representation $V$ of $\mathrm {I}$, the fundamental class in $H^{1}(\mathrm {I},\mathbb {Q}_{\ell })\simeq \mathbb {Q}_{\ell }$, induces a canonical isomorphism of Galois cohomologies

\[ H^{*}(\mathrm{I},V^{\vee}) \simeq H^{1-*}(\mathrm{I},V)^{\vee}. \]

3.3 Invariant vanishing cycles and dg-categories

From [Reference Deligne and KatzSGA7-II, Exp. XIII] and [Reference Blanc, Robalo, Toën and VezzosiBRTV18, 4.1], the vanishing cycles construction provides an $\infty$-functor

\[ \phi : \mathcal{D}_{\rm c}(X,\mathbb{Q}_{\ell}) \longrightarrow \mathcal{D}_{\rm c}^{\mathrm{I}}(X_s,\mathbb{Q}_{\ell}). \]

Applied to the constant sheaf $\mathbb {Q}_{\ell }$, in this way we obtain an object denoted by $\nu _{X/S}$ (or simply $\nu _X$ if $S$ is clear) in $\mathcal {D}_{\rm c}^{\mathrm {I}}(X_s,\mathbb {Q}_{\ell })$.

Definition 3.3.1 The $\mathrm {I}$-invariant vanishing cycles of $X$ relative to $S$ (or $\mathrm {I}$-vanishing cycles, for short) is the object

\[ \nu_X^{\mathrm{I}}:= (\nu_X)^{\mathrm{I}} \in \mathcal{D}_{\rm c}(X_s,\mathbb{Q}_{\ell}). \]

There are several possible descriptions of invariant vanishing cycles. First of all, by its very definition, $\nu _X^{\mathrm{I}}$ is related to the $\mathrm {I}$-invariant nearby cycles $\psi _X^{\mathrm {I}}:= (\psi _X)^{\mathrm {I}}$ by means of an exact triangle in $D_c(X_s,\mathbb {Q}_{\ell })$:

(2)

Another description, in terms of local cohomology, is the following. We let $U=X_K$ be the open complement of $X_s$ inside $X$, and $j_X : U \hookrightarrow X$ and $i_X : X_s \hookrightarrow X$ the corresponding immersions. Then, the $\mathrm {I}$-vanishing cycles enters in an exact triangle in $D_c(X_s,\mathbb {Q}_{\ell })$:

(3)

Triangle (

) follows from the octahedral axiom applied to the triangles (

) and

\[ \mathbb{Q}_{\ell} \longrightarrow \mathbb{Q}_{\ell}^{\mathrm{I}} \simeq \mathbb{Q}_{\ell} \oplus \mathbb{Q}_{\ell}[-1] \longrightarrow \mathbb{Q}_{\ell}[-1], \]

also taking into account the triangle

\[ i_X^{!} \mathbb{Q}_{\ell} \longrightarrow \mathbb{Q}_{\ell} \longrightarrow i_X^{*}(j_X)_*j_X^{*}\mathbb{Q}_{\ell} \simeq \psi_X^{\mathrm{I}}. \]

We obtain one more description of $\nu _X^{\mathrm{I}}$ (or, rather, of $\nu _X^{\mathrm{I}}(\beta ):=\nu _X^{\mathrm{I}}\otimes \mathbb {Q}_{\ell }(\beta )$) using the $\ell$-adic realization of the dg-category of singularities studied in [Reference Blanc, Robalo, Toën and VezzosiBRTV18], at least when $X$ is a regular scheme with smooth generic fiber. Let $\mathsf {Sing}(X_s)= \mathsf {Coh}^{b} (X_s)/\mathsf {Perf} (X_s)$ be the dg-category of singularities of the scheme $X_s$. This dg-category is naturally linear over the dg-categories $\mathsf {Perf}(X_s)$ and $\mathsf {Perf}(X)$ and, thus, we can take its $\ell$-adic realization $r_{\ell, X}(\mathsf {Sing}(X_s))$ over $X$ (see Remark 2.2.2) which is a $\mathbb {Q}_{\ell, X}(\beta )$-module in $\mathcal {D}_{\rm ic}(X,\mathbb {Q}_{\ell })$ supported on $X_s$, hence can be identified with a $\mathbb {Q}_{\ell, X_s}(\beta )$-module in $\mathcal {D}_{\rm ic}(X_s,\mathbb {Q}_{\ell })$. When $X$ is a regular scheme and $X_K$ is smooth over $K$, we have from [Reference Blanc, Robalo, Toën and VezzosiBRTV18] a canonical equivalenceFootnote ⁸ in $\mathcal {D}_{\rm c}(X_s,\mathbb {Q}_{\ell }(\beta ))$,

(4)\begin{equation} \nu_X^{\mathrm{I}}(\beta)[1] \simeq r_{\ell, X}(\mathsf{Sing}(X_s)), \end{equation}

where $\nu _X^{\mathrm{I}}(\beta )$ stands for $\nu _X^{\mathrm{I}}\otimes \mathbb {Q}_{\ell }(\beta )$.

We conclude this section with another description of $\nu _X^{\mathrm{I}}(\beta )$ (see equivalence (6)), this time in terms of sheaves of singularities. We need a preliminary result.

Proof. First, separated finite-type morphisms of noetherian schemes are compactifiable (by Nagata's theorem), thus we can assume that $p: X \to S$ is proper.

Part (1) follows immediately from [Reference Blanc, Robalo, Toën and VezzosiBRTV18, Proposition 3.9 and formula (3.7.13)].

To prove part (2), we first produce a map $\alpha : r_{\ell }(\mathsf {Coh}^{b} (X)) \to \omega _X (\beta )$. In the notation of [Reference Blanc, Robalo, Toën and VezzosiBRTV18, § 3] (note that notation in [Reference Blanc, Robalo, Toën and VezzosiBRTV18] for $\mathsf {M}_X^{T}$ is $\mathcal {M}_X^{\vee }(T)$), we first construct a map $\alpha ^{\textrm {mot}}: \mathcal {M}^{\vee }_X(\mathsf {Coh}^{b} (X)) \to p^{!}(\mathbf {B}\mathbb {U}_S)=: \omega ^{\textrm {mot}}_X$ in $\mathrm {SH}(X)$, whose étale $\ell$-adic realization will be $\alpha$. As $p$ is proper, $\alpha ^{\textrm {mot}}$ is the same thing, by adjunction, as a map $p_*(\mathcal {M}^{\vee }_X(\mathsf {Coh}^{b} (X))) \to \mathbf {B}\mathbb {U}_S$ in $\mathrm {SH}(S)$. Now, $p_*(\mathcal {M}^{\vee }_X(\mathsf {Coh}^{b} (X)))$ is just $\mathcal {M}^{\vee }_S(\mathsf {Coh}^{b} (X)))$, where $\mathsf {Coh}^{b} (X)$ is viewed as a dg-category over $S$, via $p$. If $Y$ is smooth over $S$, we have by [Reference PreygelPre11, Proposition B.4.1], an equivalence of $S$-dg-categories

(5)\begin{equation} \mathsf{Coh}^{b}(X)\otimes_S \mathsf{Coh}^{b}(Y) \simeq \mathsf{Coh}^{b}(X\times_S Y). \end{equation}

Through this identification, $\mathcal {M}_S^{\vee }(\mathsf {Coh}^{b}(X))\in \mathrm {SH}(S)$ is the $\infty$-functor $Y\mapsto \mathrm {KH}(\mathsf {Coh}^{b}(X\times _S Y))$, and $\mathrm {KH}(\mathsf {Coh}^{b}(X\times _S Y))$ is equivalent to the $\mathrm {G}$-theory spectrum $\mathrm {G}(X\times _S Y)$ of $X\times _S Y$, by $\mathbb {A}^{1}$-invariance of G-theory. As $S$ is regular, $\mathbf {B}\mathbb {U}_S \simeq \mathrm {G}_S:= \mathrm {G}(-/S)$ canonically in $\mathrm {SH}(S)$, and we can take the map $\mathcal {M}^{\vee }_S(\mathsf {Coh}^{b} (X))) \to \mathbf {B}\mathbb {U}_S \simeq \mathrm {G}_S$ to be the push-forward $p_*$ on G-theories $\mathrm {G}(X\times _S -) \to G(-/S)$. This gives us a map $\alpha ^{\textrm {mot}}: \mathcal {M}^{\vee }_X(\mathsf {Coh}^{b} (X)) \to p^{!}(\mathbf {B}\mathbb {U}_S)=: \omega ^{\textrm {mot}}_X$. Now observe that by [Reference Blanc, Robalo, Toën and VezzosiBRTV18, formula (3.7.13)], the étale $\ell$-adic realization of $p^{!}(\mathbf {B}\mathbb {U}_S)$ is canonically equivalent to $p^{!}(\mathbb {Q}_{\ell } (\beta )) \simeq \omega _X (\beta )$ (because étale $\ell$-adic realization commutes with six operations [Reference Blanc, Robalo, Toën and VezzosiBRTV18, Remark 3.23]). Therefore, we obtain our map $\alpha : r_{\ell }(\mathsf {Coh}^{b} (X)) \to \omega _X (\beta )$. Checking that $\alpha$ is an equivalence is a local statement, that is, it is enough to show that if $j: V=\mathrm {Spec}\, A \hookrightarrow X$ is an open affine subscheme, then $j^{*}(\alpha )$ is an equivalence. Now, $j^{*}r_{\ell }(\mathsf {Coh}^{b} (X)) \simeq r_{\ell, V}(j^{*}\mathsf {Coh}^{b} (X)) \simeq r_{\ell, V}(\mathsf {Coh}^{b} (V))$ (where $r_{\ell, V}$ denotes the $\ell$-adic realization over $V$), and $j^{*}\omega _X \simeq j^{!}\omega _X \simeq \omega _V$, so $j^{*}\alpha$ identifies with a map $r_{\ell, V}(\mathsf {Coh}^{b} (V))\to \omega _V(\beta )$. As $V$ is affine and of finite type over $S$, we can choose a closed immersion $i:V\hookrightarrow V'$, with $V'$ affine and smooth (hence, regular) over $S$. Let $h: V'\setminus V \hookrightarrow V'$ be the complementary open immersion. As $V'$ and $V'\setminus V$ are regular, by Quillen localization and the properties of the non-commutative realization functor $\mathcal {M}^{\vee }$ (see [Reference Blanc, Robalo, Toën and VezzosiBRTV18]), we obtain a cofiber sequence

\[ \mathcal{M}^{\vee}_{V'}(\mathsf{Coh}^{b} (V)/V') \to \mathbf{B}\mathbb{U}_{X'} \to h_*\mathbf{B}\mathbb{U}_{V'\setminus V}, \]

where the notation $\mathsf {Coh}^{b} (V)/V'$ means that $\mathsf {Coh}^{b} (V)$ is viewed as a dg-category over $V'$, via $i$. In other words, $\mathcal {M}^{\vee }_{V'}(\mathsf {Coh}^{b} (V)/V')\simeq i_*\mathcal {M}^{\vee }_{V}(\mathsf {Coh}^{b} (V))$. If we apply $i^{*}$ to this cofiber sequence, and compare what we obtain to the application of $i^{*}$ to the standard localization sequence

\[ i_*i^{!} \mathbf{B}\mathbb{U}_{V'} \to \mathbf{B}\mathbb{U}_{V'}\to h_*h^{*}\mathbf{B}\mathbb{U}_{V'}=h_*\mathbf{B}\mathbb{U}_{V'\setminus V}, \]

we finally obtain, after étale $\ell$-adic realization, that $\omega _V(\beta ) \simeq i^{!}\mathbb {Q}_{\ell } (\beta ) \simeq r_{\ell, V}(\mathsf {Coh}^{b} (V))$. This implies that $j^{*}\alpha$ is also an equivalence.

By parts (1) and (2), the map in part (3) is finally obtained by applying $r_{\ell, X}$ to the inclusion $\mathsf {Perf} (X) \to \mathsf {Coh}^{b} (X)$.

Definition 3.3.4 Let $X/S$ be a separated and finite type $S$-scheme. The sheaf of singularities of $X$ is defined to be the cofiber of the 2-periodic $\ell$-adic fundamental class morphism $\eta _X$ (Lemma 3.3.2(3)):

\[ \omega_X^{o} := \textrm{Cofib}(\eta_X :\mathbb{Q}_{\ell}(\beta) \longrightarrow \omega_X(\beta)). \]

Let us come back to $X$, a regular scheme, proper over $S$. We have a canonical equivalence in $\mathcal {D}_c(X_s,\mathbb {Q}_{\ell }(\beta ))$:

(6)\begin{equation} \nu_X^{\mathrm{I}}(\beta)[1] \simeq \omega_{X_s}^{o}. \end{equation}

This is a reformulation of the equivalence (4), in view of Lemma 3.3.2.

Remark 3.3.7 When $X$ is no longer regular, but still proper and lciFootnote ¹⁰ over $S$, there is nonetheless a natural morphism $\nu _X^{\mathrm{I}}(\beta )[1] \longrightarrow \omega _{X_s}^{\rm op}$, constructed as follows. Consider again the triangle (3)

On $X$, we do have the 2-periodic $\ell$-adic fundamental class $\eta _X : \mathbb {Q}_{\ell }(\beta ) \longrightarrow \omega _X(\beta )$, and by taking its $!$-pullback by $i_X^{!}$, we obtain a morphism $i_X^{!}(\mathbb {Q}_{\ell })(\beta ) \longrightarrow \omega _{X_s}(\beta )$. This produces a sequence of morphisms

The resulting composite morphism $\mathbb {Q}_{\ell }(\beta ) \rightarrow \omega _{X_s}(\beta )$ is the 2-periodic $\ell$-adic fundamental class of $X_s$. Moreover, by construction, the composition

is canonically the zero map, and this induces the natural morphism

\[ \alpha_X : \nu_X^{\mathrm{I}}(\beta)[1] \longrightarrow \omega_{X_s}^{o} \]

we were looking for. Summing up, the morphism $\alpha _X$ always exists for any proper, lci scheme $X$ over $S$, and is an equivalence whenever $X$ is regular.

3.4 A Künneth theorem for invariant vanishing cycles

In this section, we consider two separated, finite-type $S$-schemes $X$ and $Y$, such that $X_K$ and $Y_K$ are smooth over $K$, and both $X$ and $Y$ are regular and connected. For simplicityFootnote ¹¹ we also assume that $X$ and $Y$ are flat over $S$. We set $Z:=X\times _S Y$, and consider this as a scheme over $S$. We have $Z_s \simeq X_s \times _s Y_s$, and the $\infty$-category $D_c^{\mathrm{I}}(Z_s,\mathbb {Q}_{\ell })$ comes equipped with pull-back functors

\[ p^{*} : D_c^{\mathrm{I}}(X_s,\mathbb{Q}_{\ell}) \longrightarrow D_c^{\mathrm{I}}(Z_s,\mathbb{Q}_{\ell}) \longleftarrow D_c^{\mathrm{I}}(Y_s,\mathbb{Q}_{\ell}) :q^{*}. \]

By taking their tensor product, we obtain an external product functor

\[ \boxtimes:=p^{*}(-)\otimes q^{*}(-) : D_c^{\mathrm{I}}(X_s,\mathbb{Q}_{\ell}) \times D_c(Y_s,\mathbb{Q}_{\ell}) \longrightarrow D_c^{\mathrm{I}}(Z_s,\mathbb{Q}_{\ell}). \]

For two objects $E \in D_c^{\mathrm{I}}(X_s,\mathbb {Q}_{\ell })$ and $F \in D_c^{\mathrm{I}}(Y_s,\mathbb {Q}_{\ell })$, we can define the Künneth morphism in $D_c(Z_s,\mathbb {Q}_{\ell })$:

\[ \mathsf{k} : (E\boxtimes F)^{\mathrm{I}}[-1] \longrightarrow E^{\mathrm{I}} \boxtimes F^{\mathrm{I}} \]

as follows. As Grothendieck duality on $Z_s$ is compatible with external products, to define $\mathsf {k}$ it is enough to define its dual

\[ \mathbb{D}(E^{\mathrm{I}}) \boxtimes \mathbb{D}(F^{\mathrm{I}}) \longrightarrow \mathbb{D}((E\boxtimes F)^{\mathrm{I}})[1]. \]

By Lemma 3.2.3, the datum of such a morphism is equivalent to that of a morphism

\[ \mathbb{D}_I(E)^{\mathrm{I}} \boxtimes \mathbb{D}_I(F)^{\mathrm{I}} \longrightarrow \mathbb{D}_I(E\boxtimes F)^{\mathrm{I}}. \]

We now define $\mathsf {k}$ as the map induced by the composite

where $\mu _{(-)^{\mathrm{I}}}$ is the lax-monoidal structure on $(-)^{\mathrm{I}}$, and $\mu _{D_I}$ is that on $D_I$.Footnote ¹²

Definition 3.4.1 With the above notation, the $\mathrm {I}$-invariant convolution of the two objects $E \in D_c^{\mathrm{I}}(X_s,\mathbb {Q}_{\ell })$ and $F \in D_c^{\mathrm{I}}(Y_s,\mathbb {Q}_{\ell })$ is defined to be the cone of the Künneth morphism, and denoted by $(E\circledast F)^{\mathrm{I}}$. By definition, it sits in a triangle

The main result of this section is the following proposition, relating the $\mathrm {I}$-invariant convolution of vanishing cycles on $X$ and $Y$ to the dualizing complex of $Z$. It can also be considered as a computation of the $\ell$-adic realization of the $\infty$-category $\mathsf {Sing}(Z)$ of singularities of $Z$.

Note that, as $X$ and $Y$ are generically smooth over $S$, so is $Z$, and thus the 2-periodic $\ell$-adic fundamental class map $\eta _Z : \mathbb {Q}_{\ell }(\beta ) \longrightarrow \omega _Z(\beta )$ of Lemma 3.3.2 is an equivalence over the generic fiber. Therefore, $\omega _Z^{o}$ is supported on $Z_s$, so that it can (and will) be considered canonically as an object in $\mathcal {D}_c(Z_s,\mathbb {Q}_{\ell })$.

Theorem 3.4.2 With the above notation and assumptions, there is a canonical equivalence

\[ \omega_Z^{o} \simeq (\nu_X \circledast \nu_Y)^{\mathrm{I}}(\beta) \]

in $D_c(Z_s,\mathbb {Q}_{\ell }(\beta ))$.

Proof. The proof of this theorem combines various exact triangles with an application of Gabber's Künneth formula for nearby cycles.

To start with, the vanishing cycles $\nu _Z$ of $Z$ sits in an exact triangle in $\mathcal {D}_c^{\mathrm{I}}(Z_s,\mathbb {Q}_{\ell })$:

where $\psi _Z=\psi _Z(\mathbb {Q}_{\ell })$ is the complex of nearby cycles of $Z$ over $S$. According to [Reference Beilinson, Bernstein, Gel'fand and GindikinBB93, Lemma 5.1.1] or [Reference IllusieIll94, 4.7], we have a natural equivalence in $\mathcal {D}_c^{\mathrm{I}}(Z_s,\mathbb {Q}_{\ell })$, induced by the external product

\[ \psi_Z \simeq \psi_X \boxtimes \psi_Y. \]

The object $\nu _Z$ then becomes the cone of the tensor product of the two morphisms in $\mathcal {D}_c^{\mathrm{I}}(Z_s,\mathbb {Q}_{\ell })$:

\[ \mathbb{Q}_{\ell} \longrightarrow p^{*}(\psi_X), \quad \mathbb{Q}_{\ell} \longrightarrow q^{*}(\psi_Y), \]

where $p$ and $q$ are the two projections from $Z$ down to $X$ and $Y$, respectively. Now, cones of tensor products are computed via the following well-known lemma.

Lemma 3.4.3 Let $\mathcal {C}$ be a stable symmetric monoidal $\infty$-category, and

\[ u : x \rightarrow y, \quad v : x' \rightarrow y' \]

two morphisms. Let $C(u)$ be the cone of $u$, $C(v)$ be the cone of $v$, and $C(u\otimes v)$ the cone of the tensor product $u\otimes v : x \otimes x' \rightarrow y \otimes y'$. Then, there exists a natural exact triangle

Proof of Lemma 3.4.3. Factor $u\otimes v$ as

, and apply the octahedral axiom to the triangles

to obtain a triangle

together with the compatibility $\theta = f[1]\circ d'$. Now observe that $\theta \circ (u \otimes \mathrm {id}_{C(v)})=0$, and apply the octahedral axiom to the triangles

to conclude.

Lemma 3.4.3 implies the existence of a natural exact triangle in $\mathcal {D}_c^{\mathrm{I}}(Z_s,\mathbb {Q}_{\ell })$

which, by taking $\mathrm {I}$-invariants, yields an exact triangle in $\mathcal {D}_c(Z_s,\mathbb {Q}_{\ell })$:

Lemma 3.4.4 Let $k$ be an algebraically closed field, $s:= \mathrm {Spec} \, k$, $p_X: X \to s$, $p_Y: Y \to s$ be proper morphisms of schemes, and $p_1: Z:= X \times _s Y \to X$, $p_2: Z:= X \times _s Y \to Y$ the natural projections. If $\omega _Z$, $\omega _X$, $\omega _Y$ denote the $\mathbb {Q}_{\ell }$-adic dualizing complexes of $Z$, $X$, and $Y$, respectively, there is a canonical equivalence

\[ a: p_1^{*}\omega_X \otimes p_2^{*}\omega_Y \longrightarrow \omega_Z. \]

Proof of Lemma 3.4.4 We first exhibit the map $a$. We denote simply by $\underline {\mathrm {Hom}}_T(-, -)$ the derived internal hom in $\mathcal {D}_{\rm c}(T,\mathbb {Q}_{\ell })$ (so that $D:= \underline {\mathrm {Hom}}_Z(-, \omega _Z)$ is the $\mathbb {Q}_{\ell }$-adic duality on $Z$). By adjunction, giving a map $a$ is the same thing as giving a map $\omega _X \to (p_1)_*\underline {\mathrm {Hom}}_Z(p_2^{*} \omega _Y, p_2^{!} \omega _Y )$. As $p_X$ is proper, by [Reference Artin, Grothendieck and VerdierSGA4-III, Exp XVIII, 3.1.12], we have a canonical equivalence

\[ (p_1)_*\underline{\mathrm{Hom}}_Z(p_2^{*} \omega_Y, p_2^{!} \omega_Y ) \longrightarrow p_X^{!} (p_Y)_* \underline{\mathrm{Hom}}_Y(\omega_Y, \omega_Y). \]

Therefore, we are left to define a map

\[ \omega_X \simeq p_X^{!} \mathbb{Q}_{\ell} \to p_X^{!} (p_Y)_* \underline{\mathrm{Hom}}_Y(\omega_Y, \omega_Y), \]

and we take $p_X^{!}(\alpha )$ for this map, where $\alpha$ is the adjoint to the canonical map $\mathbb {Q}_{\ell } \simeq p_Y^{*}\mathbb {Q}_{\ell } \to \underline {\mathrm {Hom}}_Y(\omega _Y, \omega _Y)$. One can then prove that $a$ is an equivalence, by checking it stalkwise.

By Lemma 3.4.4, the dualizing complex $\omega _{Z_s}= (Z_s\to s)^{!} \mathbb {Q}_{\ell }$ of $Z_s \simeq X_s \times _s Y_s$ is canonically equivalent to $\omega _{X_s} \boxtimes \omega _{Y_s}$, and, through this equivalence, the virtual fundamental class of $Z_s$

\[ \eta_{Z_s} : \mathbb{Q}_{\ell}(\beta) \longrightarrow \omega_{Z_s}(\beta) \simeq \omega_{X_s} \boxtimes \omega_{Y_s} (\beta) \]

is simply given by the external tensor product of the virtual fundamental classes of $X_s$ and $Y_s$. By Lemma 3.4.3 we, thus, obtain a second exact triangle in $\mathcal {D}_c(Z_s,\mathbb {Q}_{\ell }(\beta ))$:

There is a morphism from the triangle (T1) to the triangle (T2) which is defined using the natural morphism

\[ \alpha_Z : \nu_Z^{\mathrm{I}}(\beta)[1] \longrightarrow \omega_{Z_s}^{o} \]

introduced in Remark 3.3.7. In fact, $Z$ is proper and lci over $S$ (because $X/S$ is flat and lci,Footnote ¹³ and being lci is stable under flat base change and composition), and the map $\alpha _Z$ is defined for any proper, lci scheme $Z$ over $S$, being an equivalence when $Z$ is regular with smooth generic fiber. Using the compatible maps $\alpha _X$, $\alpha _Y$ and $\alpha _Z$ we obtain the following commutative square.

This produces a morphism from triangle $(T1)$ (tensored by $\mathbb {Q}_{\ell }(\beta )[1]$) to triangle $(T2)$ as follows.

(7)

As $X$ and $Y$ are regular with smooth generic fibers, the maps $\alpha _X$ and $\alpha _Y$ are equivalences, therefore the leftmost vertical morphism is also an equivalence. Thus, the right-hand square is a cartesian square.

Now, the rightmost vertical morphism can be written, again using the equivalences $\alpha _X$ and $\alpha _Y$, as

\[ (\nu_X \boxtimes \nu_Y)^{\mathrm{I}}(\beta)[1] \longrightarrow (\nu_X^{\mathrm{I}}(\beta)[1]) \boxtimes_{\mathbb{Q}_{\ell}(\beta)} (\nu_Y^{\mathrm{I}}(\beta)[1]) \simeq (\nu_X^{\mathrm{I}} \boxtimes \nu_Y^{\mathrm{I}})[2](\beta). \]

This morphism is the Künneth map $\mathsf {k}$ of Definition 3.4.1 tensored by $\mathbb {Q}_{\ell }[2](\beta )\simeq \mathbb {Q}_{\ell }(\beta )$ and, thus, its cone is $(\nu _X \circledast \nu _Y)^{\mathrm{I}}(\beta )$. To finish the proof of the proposition, it then remains to show that the cone of the middle vertical morphism in (7)

\[ \alpha_Z: \nu_Z^{\mathrm{I}}(\beta)[1] \longrightarrow \omega_{Z_s}^{o} \]

can be canonically identified with $\omega _Z^{o}$.

For this, we recall the exact triangle (3) tensored by $\mathbb {Q}_{\ell } (\beta )$:

Using the fundamental class $\eta _Z : \mathbb {Q}_{\ell }(\beta ) \rightarrow \omega _Z(\beta )$, we obtain a morphism of triangles as follows.

The right-hand square is, thus, cartesian, so that the cone of the vertical morphism on the right is canonically identified with the cone of the vertical morphism in the middle. By definition, this cone is $i_Z^{!}(\omega _Z^{o})$. As $Z_{K}$ is smooth over $K$, the $\ell$-adic complex $\omega _Z^{o}$ is supported on $Z_s$ and, thus, $i_Z^{!}(\omega _Z^{o})$ is canonically equivalent to $\omega _Z^{o}$, and we conclude.