Proof. The functor

$$ \begin{align*}\mathrm{N}_{\mathrm{dg}}(\mathrm{C}^b(\mathrm{FLCA}))\rightarrow\mathrm{N}_{\mathrm{dg}}(\mathrm{C}^b(\mathrm{LCA}))\rightarrow \mathbf{D}^b(\mathrm{LCA})\end{align*} $$

induces an exact functor $\mathbf {D}^b(\mathrm {FLCA})\rightarrow \mathbf {D}^b(\mathrm {LCA})$ by [Reference Nikolaus and Scholze20, Theorem I.3.3(i)]. It remains to check that this functor is fully faithful. The functors $R\mathrm {Hom}_{\mathrm {LCA}}(-,-)$ and $R\mathrm {Hom}_{\mathrm {FLCA}}(-,-)$ induce the same functor

$$ \begin{align*}\mathrm{D}^b(\mathrm{FLCA})^{\mathrm{op}}\times \mathrm{D}^b(\mathrm{FLCA})\rightarrow \mathrm{D}^b(\mathrm{TA})\end{align*} $$

by [Reference Hoffmann and Spitzweck14, Remark 4.9]. Moreover, for any $X,Y\in \mathrm {D}^b(\mathrm {FLCA})$ , we have [Reference Hoffmann and Spitzweck14, Proposition 4.12(i)]

$$ \begin{align*}\mathrm{disc}(H^0(R\mathrm{Hom}_{\mathrm{LCA}}(X,Y)))\simeq \mathrm{Hom}_{\mathrm{D}^b(\mathrm{LCA})}(X,Y)\end{align*} $$

and similarlyFootnote ¹

$$ \begin{align*}\mathrm{disc}(H^0(R\mathrm{Hom}_{\mathrm{FLCA}}(X,Y)))\simeq \mathrm{Hom}_{\mathrm{D}^b(\mathrm{FLCA})}(X,Y).\end{align*} $$

Therefore, the map

$$ \begin{align*}\mathrm{Hom}_{\mathrm{D}^b(\mathrm{FLCA})}(X,Y)\rightarrow \mathrm{Hom}_{\mathrm{D}^b(\mathrm{LCA})}(X,Y)\end{align*} $$

is an isomorphism of abelian groups, that is, $\mathrm {D}^b(\mathrm {FLCA})\rightarrow \mathrm {D}^b(\mathrm {LCA})$ is fully faithful. Hence,

(4) $$ \begin{align} \mathbf{D}^b(\mathrm{FLCA})\rightarrow \mathbf{D}^b(\mathrm{LCA}) \end{align} $$

is an exact functor of stable $\infty $ -categories which induces a fully faithful functor between the corresponding homotopy categories. It follows that (4) is fully faithful.

Notation 2.4. For any $X\in \mathbf {D}^b(\mathrm {LCA})$ and any $i\in \mathbb Z$ , we consider

$$ \begin{align*}H^{i}(X):=\tau^{\geq 0}\tau^{\leq 0} (X[i]) \in \mathcal{LH}(\mathrm{LCA}).\end{align*} $$

Proof. If $X\rightarrow Y\rightarrow Z$ is a fibre sequence in $\mathbf {D}^b(\mathrm {LCA})$ , such that $X,Z\in \mathbf {D}^b(\mathrm {FLCA})$ , then $Y\in \mathbf {D}^b(\mathrm {FLCA})$ . Indeed, a stable subcategory is closed under extensions. Let $X\in \mathbf {D}^b(\mathrm {LCA})$ , such that $H^{i}(X)\in \mathcal {LH}( \mathrm {FLCA})$ for any i. Note that $H^{i}(X)=0$ for all but finitely many $i\in \mathbb Z$ . Therefore, X has a finite exhaustive filtration with i-graded piece $H^{i}(X)[-i]\in \mathbf {D}^b(\mathrm {FLCA})$ , so that X belongs to $\mathbf {D}^b(\mathrm {FLCA})$ by induction.

The converse follows from the fact that the inclusion functor $\mathbf {D}^b(\mathrm {FLCA})\rightarrow \mathbf {D}^b(\mathrm {LCA})$ is t-exact by Remark 2.3.

Proof. Let

$$ \begin{align*}X= [\cdots\rightarrow X^{i-1}\stackrel{d_X^{i-1}}{\rightarrow} X^{i}\stackrel{d_X^{i}}{\rightarrow} X^{i+1}\rightarrow \cdots ]\end{align*} $$

be an object of $\mathbf {D}^b(\mathrm {LCA})$ , such that $H^{i}(X)\in \mathcal {LH}(\mathrm {LCA})$ belongs to $\mathrm {LCA}$ for any $i\in \mathbb Z$ . We first observe that the differentials $d_X^{i}$ all are strict morphisms. By [Reference Schneiders23, Proposition 1.2.19], the object $H^{i}(X)$ of $\mathcal {LH}(\mathrm {LCA})$ is given by the complex $[0\rightarrow \mathrm {Coim}(d_X^{i-1})\stackrel {\delta }{\rightarrow } \mathrm {Ker}(d_X^{i})\rightarrow 0]$ , where $\mathrm {Ker}(d_X^{i})$ sits in degree $0$ and $\delta $ is a monomorphism. Since $H^{i}(X)\in \mathrm {LCA}$ , the map $\delta $ is strict by [Reference Schneiders23, Proposition 1.2.29], that is, $\delta $ is a closed embedding. Since $\mathrm {Ker}(d_X^{i})\hookrightarrow X^{i}$ is a closed embedding as well, so is the map $\mathrm {Coim}(d_X^{i-1})=X^{i-1}/\mathrm {Ker}(d_X^{i-1})\rightarrow X^{i}$ . Hence, $d_X^{i-1}$ is strict.

We set $Y:=X^D$ so that $Y^{-i}=(X^{i})^D$ and $d_Y^{-i}:Y^{-i}\rightarrow Y^{-i+1}$ is the map $d_Y^{-i}=(d_X^{i-1})^D$ . The differentials $d_X^{*}$ are all strict morphisms, hence, so are their duals $d_Y^{*}$ . We have the following isomorphisms of locally compact abelian groups:

(5) $$ \begin{align} H^{i}(X)^D&\simeq\left(\mathrm{Coker}(\mathrm{Coim}(d_X^{i-1})\stackrel{\delta}{\rightarrow} \mathrm{Ker}(d_X^{i}))\right)^D \end{align} $$

(6) $$ \begin{align} &\qquad\qquad\simeq\mathrm{Ker}\left(\mathrm{Coker}((d_X^{i})^D)\rightarrow \mathrm{Im}((d_X^{i-1})^D)\right) \end{align} $$

(7) $$ \begin{align} &\qquad\simeq\mathrm{Ker}\left(\mathrm{Coker}(d_Y^{-i-1})\rightarrow \mathrm{Im}(d_Y^{-i})\right) \end{align} $$

(8) $$ \begin{align} &\quad \simeq\mathrm{Ker}\left(\mathrm{Coker}(d_Y^{-i-1})\rightarrow Y^{-i+1}\right) \end{align} $$

(9) $$ \begin{align} &\qquad\qquad\quad\simeq\mathrm{Ker}\left(Y^{-i}/d_Y^{-i-1}(Y^{-i-1})\rightarrow Y^{-i+1}\right) \end{align} $$

(10) $$ \begin{align} &\ \ \simeq\mathrm{Ker}(d_Y^{-i})/d_Y^{-i-1}(Y^{-i-1}) \end{align} $$

(11) $$ \begin{align} &\simeq\mathrm{Ker}(d_Y^{-i})/\mathrm{Coim}(d_Y^{-i-1}) \end{align} $$

(12) $$ \begin{align} &\simeq H^{-i}(Y) ,\qquad\qquad\qquad\end{align} $$

where the kernels, cokernels, images and coimages are all computed in $\mathrm {LCA}$ . The isomorphism (5) is valid by [Reference Schneiders23, Proposition 1.2.29] since the map $\delta $ is strict, and (6) holds since Pontryagin duality $(-)^D:\mathrm {LCA}^{\mathrm {op}}\rightarrow \mathrm {LCA}$ is an equivalence of categories with kernels and cokernels. The identification (7) is given by definition of the maps $d_Y^*$ , and (8) holds since $\mathrm {Im}(d_Y^{-i})\rightarrow Y^{-i+1}$ is a monomorphism. We have (9) in view of $\mathrm {Coker}(d_Y^{-i-1})=Y^{-i}/d_Y^{-i-1}(Y^{-i-1})$ , which is valid since $d_Y^{-i-1}(Y^{-i-1})$ is closed in $Y^{-i}$ , as $d_Y^{-i-1}$ is strict. The isomorphism of locally compact abelian groups (10) is clear; (11) holds since $\mathrm {Coim}(d_Y^{-i-1})\rightarrow d_Y^{-i-1}(Y^{-i-1})= \mathrm {Im}(d_Y^{-i-1})$ is an isomorphism in $\mathrm {LCA}$ since $d_Y^{-i-1}$ is strict. Finally, (12) holds by [Reference Schneiders23, Propositions 1.2.19 and 1.2.29] since $\mathrm {Coim}(d_Y^{-i-1})\rightarrow \mathrm {Ker}(d_Y^{-i})$ is strict; indeed, $Y^{-i-1}/\mathrm {Ker}(d_Y^{-i-1})\rightarrow Y^{-i}$ is a closed embedding, hence, so is $Y^{-i-1}/\mathrm {Ker}(d_Y^{-i-1})\rightarrow \mathrm {Ker}(d_Y^{-i})$ .

Proof. The functor

$$ \begin{align*}\mathrm{C}^b(\mathrm{Ab})\stackrel{i}{\rightarrow} \mathrm{C}^b(\mathrm{LCA})\stackrel{\mathrm{disc}}{\rightarrow} \mathrm{C}^b(\mathrm{Ab})\end{align*} $$

is isomorphic to the identity functor of $\mathrm {C}^b(\mathrm {Ab})$ . We obtain a natural transformation

(13) $$ \begin{align} \mathrm{Id}_{\mathbf{D}^b(\mathrm{Ab})}\stackrel{\sim}{\rightarrow} \mathrm{disc}\circ i. \end{align} $$

Similarly, there is a natural transformation

$$ \begin{align*}i\circ \mathrm{disc}\rightarrow \mathrm{Id}_{\mathbf{D}^b(\mathrm{LCA})}.\end{align*} $$

Let $X\in \mathbf {D}^b(\mathrm {Ab})$ , and let $Y\in \mathbf {D}^b(\mathrm {LCA})$ . Let $F\stackrel {\sim }{\rightarrow }X$ be a bounded flat resolution, and let $Y\stackrel {\sim }{\rightarrow } D$ be a strict quasi-isomorphism, where D is a bounded complex of divisible locally compact abelian groups. Then F is a bounded complex of codivisibleFootnote ² discrete groups $F^{i}$ (in particular, such that $F^{i}_{\mathbb {S}^1}=0$ ). Therefore, we have

$$ \begin{align*}R\mathrm{Hom}_{\mathrm{LCA}}(i(X),Y)\simeq \underline{\mathrm{Hom}}^{\bullet}(F,D):=\mathrm{Tot}(\underline{\mathrm{Hom}}(F,D))\end{align*} $$

by [Reference Hoffmann and Spitzweck14, Corollary 4.7], where $\underline {\mathrm {Hom}}(F,D)$ is the double complex of continuous maps endowed with the compact-open topology and $\mathrm {Tot}$ denotes the total complex. We obtain

$$ \begin{align*} \mathrm{disc}\left(R\mathrm{Hom}_{\mathrm{LCA}}(i(X),Y)\right)&\simeq \mathrm{disc}\left(\underline{\mathrm{Hom}}^{\bullet}(F,D)\right)\\ &\simeq \mathrm{Hom}^{\bullet}(F,\mathrm{disc}(D))\\ &\simeq R\mathrm{Hom}(X,\mathrm{disc}(Y)). \end{align*} $$

In view of [Reference Hoffmann and Spitzweck14, Proposition 4.12], we have

$$ \begin{align*} H^0(\mathrm{disc}\left(R\mathrm{Hom}_{\mathrm{LCA}}(i(X),Y[-n])\right))&\simeq \mathrm{disc}(H^0\left(R\mathrm{Hom}_{\mathrm{LCA}}(i(X),Y[-n])\right))\\ &\simeq\mathrm{Hom}_{\mathrm{D}^b(\mathrm{LCA})}(i(X),Y[-n])\\ &\simeq\pi_0(\mathrm{Map}_{\mathbf{D}^b(\mathrm{LCA})}(i(X),\Omega^nY))\\ &\simeq \pi_n( \mathrm{Map}_{\mathbf{D}^b(\mathrm{LCA})}(i(X),Y)) ,\end{align*} $$

where $\Omega (-):=0\times _{(-)}0$ is the loop space functor. Similarly, we have

$$ \begin{align*}H^0(R\mathrm{Hom}(X,\mathrm{disc}(Y[-n])))\simeq \pi_n( \mathrm{Map}_{\mathbf{D}^b(\mathrm{Ab})}(X,\mathrm{disc}(Y))).\end{align*} $$

Hence, the map

$$ \begin{align*}\mathrm{Map}_{\mathbf{D}^b(\mathrm{LCA})}(i(X),Y)\rightarrow \mathrm{Map}_{\mathbf{D}^b(\mathrm{Ab})}(X,\mathrm{disc}(Y))\end{align*} $$

is an equivalence of $\infty $ -groupoids. The result then follows from [Reference Lurie17, Proposition 5.2.2.8] and from the fact that the unit transformation (13) is an equivalence.

Proof. Let $P\stackrel {\sim }{\rightarrow } iX$ and $iY\stackrel {\sim }{\rightarrow } I$ be strict quasi-isomorphisms, where $P\in \mathrm {C}^b(\mathrm {P})$ (respectively, $I\in \mathrm {C}^b(\mathrm {I})$ ). We denote by $P^{\delta }:=\mathrm {disc}(P)$ and $I^{\delta }:=\mathrm {disc}(P)$ the underlying complexes of discrete abelian groups. Then the maps $P^{\delta }\stackrel {\sim }{\rightarrow } X$ and $Y\stackrel {\sim }{\rightarrow } I^{\delta }$ are quasi-isomorphisms in the usual sense. Hence, we have $\mathrm {Hom}^{\bullet }(X,I^{\delta })\simeq R\mathrm {Hom}(X,Y)$ , where $\mathrm {Hom}^{\bullet }$ denotes the total complex of the double complex of morphisms of abelian groups. We denote by $\underline {\mathrm {Hom}}^{\bullet }(P,I)$ the total complex of the double complex of continuous morphisms endowed with the compact-open topology. Then we have morphisms

$$ \begin{align*}R\mathrm{Hom}(X,Y)\simeq \mathrm{Hom}^{\bullet}(X,I^{\delta})\rightarrow \underline{\mathrm{Hom}}^{\bullet}(iX,I)\rightarrow \underline{\mathrm{Hom}}^{\bullet}(P,I)\simeq R\underline{\mathrm{Hom}}(iX,iY).\end{align*} $$

Suppose now that X and Y are perfect complexes of abelian groups. We may suppose that $X^{n}$ is a finitely generated free abelian group for all $n\in \mathbb Z$ , zero for almost all n and similarly for Y. We have a strict quasi-isomorphism

$$ \begin{align*}iY\stackrel{\sim}{\rightarrow} I:=\mathrm{Tot}[Y\otimes\mathbb{R}\rightarrow Y\otimes\mathbb{R}/\mathbb Z],\end{align*} $$

where $[Y\otimes \mathbb {R}\rightarrow Y\otimes \mathbb {R}/\mathbb Z]$ is seen as a double complex of locally compact abelian groups and $\mathrm {Tot}$ is the total complex. Then $iX\in \mathrm {C}^b(\mathrm {P})$ and $I\in \mathrm {C}^b(\mathrm {I})$ , and we have a strict quasi-isomorphism

$$ \begin{align*}i\mathrm{Hom}^{\bullet}(X,Y)\stackrel{\sim}{\rightarrow} \underline{\mathrm{Hom}}^{\bullet}(iX,I).\end{align*} $$

We obtain

$$ \begin{align*}iR\mathrm{Hom}(X,Y)\simeq i\mathrm{Hom}^{\bullet}(X,Y)\stackrel{\sim}{\rightarrow} \underline{\mathrm{Hom}}^{\bullet}(iX,I)\simeq R\underline{\mathrm{Hom}}(iX,iY).\end{align*} $$

Proof. The co-unit map

$$ \begin{align*}i\circ \mathrm{disc} \,R\underline{\mathrm{Hom}}(i(X),\mathbb Z/m)\rightarrow R\underline{\mathrm{Hom}}(i(X),\mathbb Z/m)\end{align*} $$

is an equivalence by assumption. Moreover, we have

$$ \begin{align*}R\mathrm{Hom}(X,\mathbb Z/m) \simeq\mathrm{disc}\, R\underline{\mathrm{Hom}}(i(X),\mathbb Z/m),\end{align*} $$

hence,

$$ \begin{align*}i\,R\mathrm{Hom}(X,\mathbb Z/m)\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(i(X),\mathbb Z/m).\end{align*} $$

We obtain

$$ \begin{align*} X\widehat{\underline{\otimes}}\widehat{\mathbb Z}:=&\ \kern-2pt\left(i(\mathrm{colim}\,R\mathrm{Hom}(X,\mathbb Z/m)\right))^D\\\simeq&\ \kern-2pt\left(\mathrm{colim}\,(i\,R\mathrm{Hom}(X,\mathbb Z/m)\right))^D\\\simeq&\ \mathrm{lim} \left((i\,R\mathrm{Hom}(X,\mathbb Z/m))^D\right)\\\simeq&\ \mathrm{lim} \left(R\underline{\mathrm{Hom}}(i(X),\mathbb Z/m)^D\right)\\\simeq&\ \mathrm{lim} \left(i(X)\underline{\otimes}^{\mathbb L} \mathbb Z/m\right) \end{align*} $$

since the left adjoint functor i commutes with arbitrary colimits and since $(-)^D$ transforms colimits into limits. Hence, we have

$$ \begin{align*} \mathrm{disc}(X\widehat{\underline{\otimes}}\widehat{\mathbb Z}) &\simeq \mathrm{disc}\left(\mathrm{lim} (i(X)\underline{\otimes}^{\mathbb L} \mathbb Z/m) \right)\\ &\simeq \mathrm{lim}\left(\mathrm{disc} (i(X)\underline{\otimes}^{\mathbb L} \mathbb Z/m) \right)\\ &\simeq \mathrm{lim}\left(\mathrm{Cofib}(\mathrm{disc} \circ i(X)\stackrel{m}{\longrightarrow} \mathrm{disc} \circ i(X)) \right)\\ &\simeq \mathrm{lim}\left(X\otimes^{\mathbb L} \mathbb Z/m \right) \end{align*} $$

since the right adjoint functor $\mathrm {disc}$ commutes with arbitrary limits.

Proof. The composite map

$$ \begin{align*}i(R\mathrm{Hom}(X,\mathbb Z/m))\stackrel{\sim}{\rightarrow}i\circ \mathrm{disc} (R\underline{\mathrm{Hom}}(iX,\mathbb Z/m))\rightarrow R\underline{\mathrm{Hom}}(iX,\mathbb Z/m)\end{align*} $$

$$ \begin{align*}\rightarrow R\underline{\mathrm{Hom}}(iX,\mathbb{R}/\mathbb Z)\simeq (iX)^D\end{align*} $$

induces

$$ \begin{align*}i(\mathrm{colim}\,R\mathrm{Hom}(X,\mathbb Z/m))\simeq \mathrm{colim}\,i(R\mathrm{Hom}(X,\mathbb Z/m))\rightarrow (iX)^D.\end{align*} $$

We obtain

$$ \begin{align*}iX\stackrel{\sim}{\rightarrow} (iX)^{DD}\rightarrow \left(i(\mathrm{colim}\,R\mathrm{Hom}(X,\mathbb Z/m))\right)^D=:X\widehat{\underline{\otimes}}\widehat{\mathbb Z}.\end{align*} $$

Notation 2.16. In the next sections, given $X\in \mathbf {D}^b(\mathrm {Ab})$ , we often simply denote by X its image $iX$ in $\mathbf {D}^b(\mathrm {LCA})$ . In particular, for $X,Y\in \mathbf {D}^b(\mathrm {Ab})$ , we denote by $R\mathrm {Hom}(X,Y)\in \mathbf {D}^b(\mathrm {Ab})\subseteq \mathbf {D}^b(\mathrm {LCA})$ the usual $R\mathrm {Hom}$ seen as an object of $\mathbf {D}^b(\mathrm {LCA})$ .

Proof. To prove perfectness of $R\Gamma _{Wh}(Y,\mathbb Z)$ , one first reduces to the smooth and projective case by [Reference Geisser8, Proposition 3.2], in which case, one can conclude with loc. cit. Theorem 4.3 and [Reference Lichtenbaum16].

Proof. To get the first triangle, noting that $i_*$ and $\pi _*$ are exact, it suffices to show that the exact sequence

$$ \begin{align*}0\to \mathbb Z\to \pi_*\mathbb Z\oplus i_*\mathbb Z \to (\pi\circ i')_*\mathbb Z\to 0\end{align*} $$

of étale sheaves on T is exact. But this follows by considering stalks at points $t\in T$ . If $t\not \in Y$ , then the sequence reduces to the isomorphism $\mathbb Z\cong \pi _*\mathbb Z$ , and if $t\in Y$ , then $\mathbb Z\cong i_*\mathbb Z$ and $\pi _*\mathbb Z \cong (\pi \circ i')_*\mathbb Z$ .

The second triangle can be obtained by applying $R\Gamma (W,-)$ to the first triangle after base extension to the algebraic closure.

Proof. Since étale cohomology with coefficients in $\mathbb Z$ does not change if we replace T by $T^{\mathrm {red}}$ , we can assume that T is reduced. We proceed by induction on dimension of T and the number of irreducible components of T. If the number of components is one, then $T^{\mathrm {red}}$ is smooth and proper and the result follows from [Reference Geisser8, Theorem 4.3]. In general, let $T=\cup _{i\in I}S_i$ and set $Y=S_1$ and $T'=\cup _{i\not =1}S_i$ . Then the hypotheses of Lemma 3.4 are satisfied, Y is smooth, $T'$ is a normal crossing scheme with fewer irreducible components and $Y'$ a normal crossing scheme of smaller dimension. Hence, we obtain the first statement on perfectness and the second statement by comparing with the corresponding triangle for Wh-cohomology.

Proof. Using blow-up squares and induction on the dimension, it suffices to prove the statement for smooth and proper schemes T of dimension, at most, $d-1$ . By [Reference Geisser8, Corollary 5.5], the Weil-eh cohomology groups agree with Weil-étale cohomology groups. By conjecture $P_{-1}(T)$ , they are torsion, hence, finite by comparison with étale cohomology groups.

Proof. Let $f:Y\rightarrow \mathrm {Spec}(k)=s$ be the structure morphism. The pushforward map [Reference Geisser9, Corollary 3.2]

$$ \begin{align*}Rf_*{\mathbb Z}^c(0)^{Y}\rightarrow {\mathbb Z}^c(0)^{s}\simeq\mathbb Z[0]\end{align*} $$

induces a trace map

$$ \begin{align*}R\Gamma_{W}(Y,{\mathbb Z}^c(0))\rightarrow R\Gamma_{W}(s,\mathbb Z) \rightarrow \mathbb Z[-1].\end{align*} $$

We consider the map

(14) $$ \begin{align} R\Gamma_{W}(Y,\mathbb Z)\longrightarrow R\mathrm{Hom}(R\Gamma_W(Y,{\mathbb Z}^c(0)), \mathbb Z[-1]) \end{align} $$

induced by the pairing

$$ \begin{align*} R\Gamma_{W}(Y,\mathbb Z)\otimes^{\mathbb L}R\Gamma_W(Y,{\mathbb Z}^c(0))\rightarrow R\Gamma_W(Y,{\mathbb Z}^c(0)) \rightarrow R\Gamma_W(s,{\mathbb Z}^c(0))\rightarrow \mathbb Z[-1], \end{align*} $$

which, in turn, is induced by the obvious pairing $\mathbb Z\otimes ^{\mathbb L}{\mathbb Z}^c(0)\rightarrow {\mathbb Z}^c(0)$ . In order to show that the morphism of perfect complexes (14) is an equivalence, it is enough to show that $(14)\otimes ^{\mathbb L}\mathbb Z/m\mathbb Z$ is an equivalence for any integer m. But $(14)\otimes ^{\mathbb L}\mathbb Z/m\mathbb Z$ may be identified with the canonical map

(15) $$ \begin{align} R\Gamma_{et}(Y,{\mathbb Z}/m{\mathbb Z})\longrightarrow R\mathrm{Hom}(R\Gamma_{et}(Y,{\mathbb Z}^c(0)/m), {\mathbb Q}/{\mathbb Z}[-1]) \end{align} $$

since we have an equivalence of lax symmetric monoidal functors

$$ \begin{align*}R\Gamma_W(Y,-)\otimes^{\mathbb L}_{\mathbb Z}\mathbb Z/m\mathbb Z\simeq R\Gamma_{et}(Y,(-)\otimes^{\mathbb L}_{\mathbb Z}\mathbb Z/m\mathbb Z).\end{align*} $$

But (15) is an equivalence by [Reference Geisser9, Theorem 5.1]. Hence, (14) is an equivalence as well.

Proof. Composing the morphism in $\mathbf {D}^b(\mathrm {Ab})$

$$ \begin{align*}R\Gamma_{Wh}(\mathcal{X}_s,\mathbb Z(n))\rightarrow C_W(\mathcal{X},n)\end{align*} $$

and the morphism in $\mathbf {D}^b(\mathrm {LCA})$

$$ \begin{align*}C_W(\mathcal{X},n)\rightarrow C_W(\mathcal{X},n)\widehat{\underline{\otimes}}\widehat{\mathbb Z}\end{align*} $$

given by Lemma 2.14, we obtain a morphism in $\mathbf {D}^b(\mathrm {LCA})$

(21) $$ \begin{align} R\Gamma_{Wh}(\mathcal{X}_s,\mathbb Z(n))\rightarrow C_W(\mathcal{X},n)\widehat{\underline{\otimes}}\widehat{\mathbb Z}. \end{align} $$

We define $\mathbf {R\Gamma }_{ar}(\mathcal {X},\mathbb Z(n))$ as the fibre of (21), and we obtain the fibre sequence (20) in $\mathbf {D}^b(\mathrm {LCA})$ . Lemma 2.14 gives a map from (19) to (20), hence, a map

$$ \begin{align*}R\Gamma_{W}(\mathcal{X},\mathbb Z(n))\rightarrow \mathbf{R\Gamma}_{ar}(\mathcal{X},\mathbb Z(n)).\end{align*} $$

Then we define $\mathbf {R\Gamma }_{ar}(\mathcal {X}_K,\mathbb Z(n))\in \mathbf {D}^b(\mathrm {LCA})$ as the cofibre of the composite map

$$ \begin{align*}R\Gamma_{W}(\mathcal{X}_s,Ri^{!}\mathbb Z(n))\rightarrow R\Gamma_{W}(\mathcal{X},\mathbb Z(n))\rightarrow \mathbf{R\Gamma}_{ar}(\mathcal{X},\mathbb Z(n)).\end{align*} $$

Proof. By proper base change, the map

$$ \begin{align*}R\Gamma_{et}(\mathcal{X},\mathbb Z/m)\rightarrow R\Gamma_{et}(\mathcal{X}_s,\mathbb Z/m)\end{align*} $$

is an equivalence, hence, $C_W(\mathcal {X},0)\otimes ^{\mathbb L}\mathbb Z/m\simeq 0$ . We obtain $C_W(\mathcal {X},0)\underline {\widehat {\otimes }}\widehat {\mathbb Z}\simeq 0$ . The first equivalence of the proposition follows. The second equivalence is obtained the same way, in view of the fact that

$$ \begin{align*}R\Gamma_{et}(\mathcal{X},\mathbb Z/m)\rightarrow R\Gamma_{et}(\mathcal{X}_s,\mathbb Z/m)\rightarrow R\Gamma_{eh}(\mathcal{X}_s,\mathbb Z/m)\end{align*} $$

is an equivalence, again, by proper base change.

Proof. This follows from Corollary 3.5 and Proposition 4.5.

Notation 4.7. If ${\mathcal X}^{{\mathrm {red}}}_s$ is a simple normal crossing scheme, we denote by $R\Gamma _{ar}({\mathcal X},\mathbb Z)$ and $R\Gamma _{ar}({\mathcal X}_K,\mathbb Z)$ the complexes defined above. In view of Proposition 4.6, we set $R\Gamma _{ar}({\mathcal X},\mathbb Z):=\mathbf {R\Gamma }_{ar}({\mathcal X},\mathbb Z)$ and $R\Gamma _{ar}({\mathcal X}_K,\mathbb Z):=\mathbf {R\Gamma }_{ar}({\mathcal X}_K,\mathbb Z)$ for arbitrary regular ${\mathcal X}$ of dimension, at most, $3$ or when we are assuming resolution of singularities.

Proof. If $R\Gamma _{W}(\mathcal {X}_K,\mathbb Z(d))$ and $R\Gamma _{W}(\mathcal {X}_s,Ri^!\mathbb Z(d))\simeq R\Gamma _W(\mathcal {X}_s,{\mathbb Z}^c(0))[-2d]$ are cohomologically bounded, then the same holds for $R\Gamma _{W}(\mathcal {X},\mathbb Z(d))$ by the localisation triangle

$$ \begin{align*}\cdots\to R\Gamma_{W}(\mathcal{X}_s,Ri^!\mathbb Z(d)) \to R\Gamma_{W}(\mathcal{X},\mathbb Z(d))\to R\Gamma_{W}(\mathcal{X}_K,\mathbb Z(d))\to \cdots.\end{align*} $$

In this case, $R\Gamma _{W}(\mathcal {X},\mathbb Z(d))\stackrel {\sim }{\rightarrow } \tau ^{>a}R\Gamma _{W}(\mathcal {X},\mathbb Z(d))$ is an equivalence for $a<<0$ , hence,

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X},\mathbb Z(d))\simeq R\Gamma_{W}(\mathcal{X},\mathbb Z(d))\widehat{\underline{\otimes}}\widehat{\mathbb Z}.\end{align*} $$

In view of (19) and (20), we obtain a map of fibre sequences

since $(-)\widehat {\underline {\otimes }}\widehat {\mathbb Z}$ is an exact functor. If the cohomology of $R\Gamma _{Wh}(\mathcal {X}_s,\mathbb Z(d))$ consists of finite groups, then the middle vertical map

$$ \begin{align*}R\Gamma_{Wh}(\mathcal{X}_s,\mathbb Z(d))\rightarrow R\Gamma_{Wh}(\mathcal{X}_s,\mathbb Z(d)) \widehat{\underline{\otimes}}\widehat{\mathbb Z}\end{align*} $$

in the above diagram is an equivalence by Proposition 2.11 and the two fibre sequences are equivalent.

Proof. The isomorphism follows because normality of ${\mathcal X}$ implies that $\mathbb {Q}\cong Rj_*j^*\mathbb {Q}$ , hence, $Ri^!\mathbb {Q}\cong 0$ . Since $H^j_{et}(\mathcal {X}_s,Ri^!\mathbb {Q}/\mathbb Z)$ is torsion and discrete, it is both of finite $\mathbb Z$ -rank and of finite $\mathbb {S}^1$ -rank. It remains to see that it is of finite p-rank for any prime number p. But $H^j_{et}(\mathcal {X}_s,Ri^!\mathbb Z/p\mathbb Z)$ is a finite group for any $j\in \mathbb Z$ , because of the fibre sequence

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_s,Ri^!\mathbb Z/p\mathbb Z)\rightarrow R\Gamma_{et}(\mathcal{X},\mathbb Z/p\mathbb Z) \rightarrow R\Gamma_{et}(\mathcal{X}_K,\mathbb Z/p\mathbb Z),\end{align*} $$

and classical finiteness results in étale and Galois cohomology.

Proof. a) Under the hypothesis, the complexes $R\Gamma _{W}(\mathcal {X}_s,\mathbb Z)$ and $R\Gamma _{Wh}(\mathcal {X}_s,\mathbb Z)$ are perfect complexes of abelian groups by Propositions 3.2 and 3.5, respectively, hence, they belong to $\mathbf {D}^b(\mathrm {FLCA})$ by Lemma 2.5. The result for $R\Gamma _{ar}(\mathcal {X},\mathbb Z)$ then follows from Proposition 4.5 (using Notation 4.7), and the result for $R\Gamma _{ar}(\mathcal {X}_K,\mathbb Z)$ follows from Proposition 4.6 and Lemma 4.12.

b) By the proof of Proposition 5.4, $R\Gamma _{ar}(\mathcal {X},\mathbb Z(d))$ is (up to a shift) dual to $R\Gamma _{et}(\mathcal {X}_s,Ri^!\mathbb {Q}/\mathbb Z)$ . Hence, the result follows from Lemmas 2.5 and 4.12. The statement for $R\Gamma _{ar}(\mathcal {X}_K,\mathbb Z(d))$ follows from the statement for $R\Gamma _{ar}(\mathcal {X},\mathbb Z(d))$ together with the perfectness of $R\Gamma _W(\mathcal {X}_s,Ri^!\mathbb Z(d))\simeq R\Gamma _W(\mathcal {X}_s,\mathbb Z^c(0))[-2d]$ by hypothesis.

Proof. Since ${\mathcal X}/\mathcal {O}_K$ has strictly semistable reduction, the morphism ${\mathcal X}\rightarrow \mathrm {Spec}(\mathcal {O}_K)$ is log smooth with respect to the log structures associated with ${\mathcal X}_s$ and s, respectively, where s is the closed point of $\mathrm {Spec}(\mathcal {O}_K)$ , and ${\mathcal X}_s$ is a normal crossing divisor on ${\mathcal X}$ . Therefore, the results of [Reference Sato22] apply. We have isomorphisms

$$ \begin{align*} H^{i}_{et}(\mathcal{X},\mathbb{Q}_p(d))\simeq H^{i}_{et}(\mathcal{X},\mathbb{Q}^S_p(d)) \end{align*} $$

compatible with the map (22), where

$$ \begin{align*}R\Gamma_{et}(\mathcal{X},\mathbb{Q}^S_p(d)):= R\mathrm{lim}R\Gamma_{et}(\mathcal{X},\mathfrak{T}_r(d))\otimes_{\mathbb Z_p}\mathbb Q_p\end{align*} $$

is the complex studied in [Reference Sato21] and [Reference Sato22]. Indeed, this follows from the equivalences

$$ \begin{align*} R\Gamma({\mathcal X}_{et},\mathfrak{T}_r(d))&\simeq R\mathrm{Hom}(R\Gamma_{{\mathcal X}_s}({\mathcal X}_{et},\mathbb Z/p^{r}), \mathbb Z/p^r)[-2d-1]\\ &\simeq R\Gamma({\mathcal X}_{et},\mathbb Z/p^r(d)), \end{align*} $$

given by [Reference Sato21, Theorem 10.1.1] and [Reference Geisser9, Proof of Theorem 7.5] and from the fact that (22) is induced by the dual of the map

$$ \begin{align*}R\Gamma({\mathcal X}_{K,et},\mathbb Z/p^{r})[-1]\rightarrow R\Gamma_{{\mathcal X}_s}({\mathcal X}_{et},\mathbb Z/p^{r}).\end{align*} $$

Hence, we are reduced to show that the map

$$ \begin{align*}H^{i}_{et}(\mathcal{X},\mathbb{Q}^S_p(d))\rightarrow H^{i}_{et}(\mathcal{X}_K,\mathbb{Q}_p(d))\end{align*} $$

is injective. By [Reference Sato22, Proposition 3.4(1)], [Reference Sato22, Section 4.1] and [Reference Sato22, Theorem 5.3], there is a morphism of spectral sequences from

$$ \begin{align*}H^{i}_f(G_K, H^{j}_{et}({\mathcal X}_{\bar{K}},\mathbb Q_p(d)))\Rightarrow H^{i+j}_{et}(\mathcal{X},\mathbb{Q}^S_p(d))\end{align*} $$

to

$$ \begin{align*}H^{i}(G_K, H^{j}_{et}({\mathcal X}_{\bar{K}},\mathbb Q_p(d)))\Rightarrow H^{i+j}_{et}(\mathcal{X}_K,\mathbb{Q}_p(d)),\end{align*} $$

where the first spectral sequence degenerates into isomorphisms

$$ \begin{align*}H^{j}_{et}(\mathcal{X},\mathbb{Q}^S_p(d))\stackrel{\sim}{\rightarrow}H^{1}_f(G_K, H^{j-1}_{et}({\mathcal X}_{\bar{K}},\mathbb Q_p(d))).\end{align*} $$

Since we have [Reference Sato22, Proposition 5.10(1)]

$$ \begin{align*}H^{0}_f(G_K, H^{j}_{et}({\mathcal X}_{\bar{K}},\mathbb Q_p(d)))=H^{0}(G_K, H^{j}_{et}({\mathcal X}_{\bar{K}},\mathbb Q_p(d)))=0\end{align*} $$

for any $j\in \mathbb Z$ , we obtain a commutative square

where the vertical maps are edge morphisms of the corresponding spectral sequences. Here, the left vertical map is an isomorphism and the lower horizontal map is injective. It follows that the upper horizontal map is injective as well.

Proof. By definition, we have $\mathbb Z(d)\cong \mathbb Z^c(0)[-2d]$ , and by [Reference Geisser9, Proposition 7.10 a)] and Gabber’s purity theorem [Reference Fujiwara4, Section 8], we have

$$ \begin{align*}\mathbb Z^c/m(0)[-2d] \cong Rf^!\mathbb Z^c/m(0)[-2d]\cong Rf^!\mathbb Z/m[-2d]\cong \mu_m^{\otimes d}\end{align*} $$

on ${\mathcal X}$ for any m prime to p. Moreover, by the proper base change theorem

$$ \begin{align*}R\Gamma_{et}({\mathcal X},\mu_m^{\otimes d})\simeq R\Gamma_{et}({\mathcal X}_{s},\mu_m^{\otimes d}).\end{align*} $$

Thus, it suffices to show that the cohomology of the right-hand side of

$$ \begin{align*}R\Gamma_{et}(\mathcal{X},\widehat{\mathbb Z}'(d))\simeq R\mathrm{lim}_{p\nmid m} \,R\Gamma_{et}({\mathcal X}_{s},\mu_m^{\otimes d})\end{align*} $$

if finite. By the analog of Proposition 3.4 and induction on the number of irreducible components of ${\mathcal X}_{s}$ , it suffices to show that the cohomology $R\Gamma _{et}(Y,\mathbb Z_l(d))$ of each connected component Y of each ${\mathcal X}^{(i)}_{s}$ is finite for all $l\neq p$ and zero for almost all l. Since Y is smooth and proper, this is known for the extension $\bar Y$ to the algebraic closure by Gabber’s theorem [Reference Gabber5], [Reference Suh24], and this extends to Y by a weight argument because $d>\dim Y$ , hence, the Frobenius does not have eigenvalue one on $R\Gamma _{et}(Y,\mathbb Z_l(d))$ .

Proof. a) We have a long exact sequence in the abelian category $\mathcal {LH}(\mathrm {LCA})$

$$ \begin{align*}\cdots\rightarrow H^{j}_{ar}(\mathcal{X},\mathbb Z)\rightarrow H^{j}_{ar}(\mathcal{X}_{K},\mathbb Z)\rightarrow H^{j+1}_{W}(\mathcal{X}_{s},Ri^{!}\mathbb Z)\rightarrow \cdots,\end{align*} $$

where $H^{j+1}_{W}(\mathcal {X}_{s},Ri^{!}\mathbb Z) \simeq H^{j}_{W}(\mathcal {X}_{s},Ri^{!}\mathbb Q/\mathbb Z)$ is a discrete torsion abelian group (see the proof of Proposition 5.4) and $H^{j}_{ar}(\mathcal {X},\mathbb Z)\simeq H^{j}_{W}(\mathcal {X}_s,\mathbb Z)$ is a discrete finitely generated abelian group by Proposition 3.5. Hence, $H^{j}_{ar}(\mathcal {X}_{K},\mathbb Z)\in \mathcal {LH}(\mathrm {LCA})$ is an extension of a torsion abelian group by a finitely generated abelian group. It follows that $H^{j}_{ar}(\mathcal {X}_{K},\mathbb Z)\in \mathrm {LCA}$ since $\mathrm {LCA}\subset \mathcal {LH}(\mathrm {LCA})$ is stable under extensions [Reference Schneiders23, Proposition 1.2.29(c)].

b) We have a long exact sequence in $\mathcal {LH}(\mathrm {LCA})$

$$ \begin{align*}H^{j}_{W}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d))\rightarrow H^{j}_{ar}(\mathcal{X},\mathbb Z(d))\rightarrow H^{j}_{ar}(\mathcal{X}_{K},\mathbb Z(d))\rightarrow H^{j+1}_{W}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d)),\end{align*} $$

where $H^{j}_{W}(\mathcal {X}_{s},Ri^{!}\mathbb Z(d))$ is a discrete finitely generated abelian group by assumption. Moreover, $H^{j}_{ar}(\mathcal {X},\mathbb Z(d))\in \mathcal {LH}(\mathrm {LCA})$ is the group (see Remark 4.11)

$$ \begin{align*}H^{j}_{et}(\mathcal{X},\widehat{\mathbb Z}(d))\simeq\prod_{l} H^{j}_{et}(\mathcal{X},\mathbb Z_l(d)):= \prod_{l} H^{j}(R\mathrm{lim} (R\Gamma_{et}(\mathcal{X},\mathbb Z(d))\otimes^{\mathbb L}\mathbb Z/l^{\bullet}))\end{align*} $$

which by Lemma 4.16 is the product of a finite group and the finitely generated $\mathbb Z_p$ -module $H^{j}_{et}(\mathcal {X},\mathbb Z_p(d))$ endowed with the p-adic topology. If we can show that the image of the map $H^{j}_{W}(\mathcal {X}_{s},Ri^{!}\mathbb Z(d))\rightarrow H^{j}_{et}(\mathcal {X},\widehat {\mathbb Z}(d))$ is finite, then it will follow that $H^{j}_{ar}(\mathcal {X}_{K},\mathbb Z(d))$ is an extension of a finitely generated abelian group by a profinite abelian group. Since we have an isomorphism of finitely generated $\mathbb Z_p$ -modules

$$ \begin{align*}H^{j}_{W}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d))\otimes_{\mathbb Z}\mathbb Z_p\simeq H^{j}_{et}(\mathcal{X}_{s},Ri^{!}\mathbb Z_p(d)),\end{align*} $$

it is enough to show that the image of the map

$$ \begin{align*}H^{j}_{et}(\mathcal{X}_{s},Ri^{!}\mathbb Z_p(d))\rightarrow H^{j}_{et}(\mathcal{X},\mathbb Z_p(d))\end{align*} $$

is finite, or equivalently, that the map

$$ \begin{align*}H^{j}_{et}(\mathcal{X}_{s},Ri^{!}\mathbb{Q}_p(d))\rightarrow H^{j}_{et}(\mathcal{X},\mathbb{Q}_p(d))\end{align*} $$

is the zero map. This follows from Theorem 4.15 by the localisation sequence.

Proof. Recall from Proposition 4.13 that $R\Gamma _{ar}(\mathcal {X}_K,\mathbb Z(n))$ belongs to $\mathbf {D}^b(\mathrm {FLCA})$ for $n=0,d$ , so that the tensor product

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d))\underline{\otimes}^{\mathbb L} R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z),\end{align*} $$

defined in Section 2, makes sense. Moreover, the equivalence $Ri^{!}\mathbb Z^c(0)^{\mathcal {X}}\simeq \mathbb Z^c(0)^{\mathcal {X}_s}$ of Remark 4.9 and the pushforward map $Rf_*\mathbb Z^c(0)^{\mathcal {X}_s}\rightarrow \mathbb Z^c(0)^s\simeq \mathbb Z[0]$ of [Reference Geisser9, Corollary 3.2] induce trace maps

$$ \begin{align*}R\Gamma_{W}(\mathcal{X}_s,Ri^{!}\mathbb Z(d))\simeq R\Gamma_{W}(\mathcal{X}_s,\mathbb Z^c(0)[-2d])\rightarrow R\Gamma_{W}(s,\mathbb Z[-2d])\rightarrow \mathbb Z[-2d-1]\end{align*} $$

and

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_s,Ri^{!}\mathbb Z/m(d))\rightarrow R\Gamma_{et}(s,\mathbb Z/m[-2d])\rightarrow \mathbb Z/m[-2d-1].\end{align*} $$

We start with the following,

Proposition 5.3. The canonical product map $\mathbb Z\otimes ^{\mathbb L} \mathbb Z(d)\rightarrow \mathbb Z(d)$ in the derived $\infty $ -category of étale sheaves over $\mathcal {X}_{\mathcal {O}_{K^{un}}}$ and $\mathcal {X}_{K^{un}}$ induce perfect pairings

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m)\otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X},\mathbb Z/m(d))\rightarrow R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m(d))\rightarrow \mathbb Z/m[-2d-1]\end{align*} $$

and

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m(d))\otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X},\mathbb Z/m)\rightarrow R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m(d))\rightarrow \mathbb Z/m[-2d-1]\end{align*} $$

for any m.

Proof. Consider the commutative square

Taking the fibres of the vertical arrows induces the product map

(23) $$ \begin{align} R\Gamma_{et}(\mathcal{X}_{s},Ri^{!}\mathbb Z) \otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X},\mathbb Z(d)) \rightarrow R\Gamma_{et}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d)), \end{align} $$

and the product map

(24) $$ \begin{align} R\Gamma_{et}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d)) \otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X},\mathbb Z) \rightarrow R\Gamma_{et}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d)) \end{align} $$

is obtained similarly. By [Reference Geisser9, Theorem 7.5] applied to $\mathcal {F}=\mathbb Z/m$ , the pairing

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m)\otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X},\mathbb Z/m(d))\rightarrow R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m(d))\rightarrow \mathbb Z/m[-2d-1],\end{align*} $$

induced by (23), is perfect. The pairing induced by (24)

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m(d))\otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X},\mathbb Z/m)\rightarrow R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m(d))\rightarrow \mathbb Z/m[-2d-1]\end{align*} $$

is perfect as well, since it reduces, by purity and proper base change, to

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_{s},\mathbb Z^c/m[-2d])\otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X}_s,\mathbb Z/m)\rightarrow R\Gamma_{et}(\mathcal{X}_{s},\mathbb Z^c/m[-2d])\rightarrow \mathbb Z/m[-2d-1]\end{align*} $$

which is perfect by [Reference Geisser9, Theorem 5.1] applied to $\mathcal {F}=\mathbb Z/m$ .

For $n=0$ or $n=d$ , consider the product map

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_{\bar{s}},R\bar{i}^!\mathbb Z(n))\otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X}_{\mathcal{O}_{{K}^{un}}},\mathbb Z(d-n))\rightarrow R\Gamma_{et}(\mathcal{X}_{\bar{s}},R\bar{i}^!\mathbb Z(d)).\end{align*} $$

This product map is induced by the obvious product maps $\mathbb Z\otimes ^{\mathbb L} \mathbb Z(d)\rightarrow \mathbb Z(d)$ in the derived $\infty $ -category of étale sheaves over $\mathcal {X}_{\mathcal {O}_{K^{un}}}$ and $\mathcal {X}_{K^{un}}$ , as in the proof of Proposition 5.3. Applying $R\Gamma (W_{\kappa (s)},-)$ and composing with the trace map, we obtain

(25) $$ \begin{align} R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z(n))\otimes^{\mathbb L} R\Gamma_{W}(\mathcal{X},\mathbb Z(d-n))\rightarrow R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z(d))\rightarrow \mathbb Z[-2d-1]. \end{align} $$

This yields the morphisms

$$ \begin{align*}R\Gamma_{W}(\mathcal{X},\mathbb Z(d))\rightarrow R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z),\mathbb Z[-2d-1]),\end{align*} $$

which, in turn, induce

(26) $$ \begin{align} \tau^{>a}R\Gamma_{W}(\mathcal{X},\mathbb Z(d))\rightarrow R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z),\mathbb Z[-2d-1]) \end{align} $$

for $a\ll 0$ , since the right-hand side is bounded. Composing (26) with the canonical map (see Lemma 2.9)

$$ \begin{align*}R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z),\mathbb Z[-2d-1])\rightarrow R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z),\mathbb Z[-2d-1]),\end{align*} $$

we obtain

(27) $$ \begin{align} \tau^{>a}R\Gamma_{W}(\mathcal{X},\mathbb Z(d))\rightarrow R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z),\mathbb Z[-2d-1]). \end{align} $$

Proposition 5.4. The map (27) factors through an equivalence

(28) $$ \begin{align} R\Gamma_{ar}(\mathcal{X},\mathbb Z(d))\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z),\mathbb Z[-2d-1]) \end{align} $$

in $\mathbf {D}^b(\mathrm {FLCA})$ .

Proof. One has

$$ \begin{align*} R\Gamma_{ar}(\mathcal{X},\mathbb Z(d))&:= \tau^{>a}R\Gamma_{W}(\mathcal{X},\mathbb Z(d))\underline{\widehat{\otimes}}\widehat{\mathbb Z} \\ &\simeq (\mathrm{hocolim}\,R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X},\mathbb Z/m(d)),\mathbb{Q}/\mathbb Z))^D \\ &\simeq(\mathrm{hocolim}\,R\mathrm{Hom}(R\Gamma_{et}(\mathcal{X},\mathbb Z/m(d)),\mathbb{Q}/\mathbb Z))^D \\ &\simeq R\underline{\mathrm{Hom}} (\mathrm{hocolim}\,R\mathrm{Hom}(R\Gamma_{et}(\mathcal{X},\mathbb Z/m(d)),\mathbb{Q}/\mathbb Z[-2d-1]),\mathbb{R}/\mathbb Z[-2d-1])\\ &\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}} (\mathrm{hocolim}\,R\Gamma_{et}(\mathcal{X}_{s},Ri^!\mathbb Z/m),\mathbb{R}/\mathbb Z[-2d-1])\\ &\simeq R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb{Q}/\mathbb Z),\mathbb{R}/\mathbb Z[-2d-1])\\ &\simeq R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z[1]),\mathbb{R}/\mathbb Z[-2d-1])\\ &\simeq R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z),\mathbb Z[-2d-1]) ,\end{align*} $$

where we use Proposition 5.3, the vanishing

$$ \begin{align*} R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z[1]),\mathbb{R})\simeq 0 \end{align*} $$

proven in Lemma 5.5 below and $Ri^!\mathbb {Q}\simeq 0$ .

Lemma 5.5. We have

$$ \begin{align*}R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z[1]),\mathbb{R})\simeq R\underline{\mathrm{Hom}}(\mathbb{R},R\Gamma_{W}(\mathcal{X}_{s},Ri^!\mathbb Z[1]))\simeq 0.\end{align*} $$

Proof. As observed above, we have $R\Gamma _{W}(\mathcal {X}_{s},Ri^!\mathbb Z[1])\simeq R\Gamma _{W}(\mathcal {X}_{s},Ri^!\mathbb {Q}/\mathbb Z)$ . Since $R\underline {\mathrm {Hom}}(\mathbb {R},-)$ and $R\underline {\mathrm {Hom}}(-,\mathbb {R})$ are exact functors, and using the t-structure on $\mathbf {D}^b(\mathrm {FLCA})$ , we may suppose that $R\Gamma _{W}(\mathcal {X}_{s},Ri^!\mathbb {Q}/\mathbb Z)$ is cohomologically concentrated in one degree. Hence, one is reduced to show that

$$ \begin{align*}R\underline{\mathrm{Hom}}(A,\mathbb{R})\simeq R\underline{\mathrm{Hom}}(\mathbb{R},A)\simeq 0\end{align*} $$

for any torsion discrete abelian group of finite ranks A. This follows from [Reference Hoffmann and Spitzweck14, Proposition 4.15 (i) and (vii)].

Corollary 5.6. We have

$$ \begin{align*}R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X},\mathbb Z(d)),\mathbb{R})\simeq R\underline{\mathrm{Hom}}(\mathbb{R},R\Gamma_{ar}(\mathcal{X},\mathbb Z(d)))\simeq 0.\end{align*} $$

Similarly, we have the

Proposition 5.7. The map

(29) $$ \begin{align} R\Gamma_{W}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d))\rightarrow R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X},\mathbb Z),\mathbb Z[-2d-1]), \end{align} $$

induced by (25), factors through an equivalence

(30) $$ \begin{align} R\Gamma_{W}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d)) \stackrel{\sim}{\longrightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X},\mathbb Z) ,\mathbb Z[-2d-1]). \end{align} $$

Proof. Recall from Remark 4.9 that we have

$$ \begin{align*}Ri^!\mathbb Z(d)=Ri^!\mathbb Z^c(0)[-2d]\simeq \mathbb Z^c(0)[-2d].\end{align*} $$

If ${\mathcal X}_s^{{\mathrm {red}}}$ is a simple normal crossing scheme, we may, therefore, identify the map

(31) $$ \begin{align} R\Gamma_{W}(\mathcal{X}_{s},Ri^{!}\mathbb Z(d))[2d] \stackrel{\sim}{\longrightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X},\mathbb Z) ,\mathbb Z[-2d-1]) [2d] \end{align} $$

with the composite morphism

$$ \begin{align*} R\Gamma_{W}(\mathcal{X}_{s},\mathbb Z^c(0)) \stackrel{\sim}{\rightarrow} R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X}_s,\mathbb Z) ,\mathbb Z[-1])\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_s,\mathbb Z) ,\mathbb Z[-1]) ,\end{align*} $$

which is an equivalence of perfect complexes of abelian groups by Proposition 3.5, Theorem 3.8 and Lemma 2.9. If $d\leq 2$ , we may identify (31) with the morphism

$$ \begin{align*} R\Gamma_{W}(\mathcal{X}_{s},\mathbb Z^c(0)) \stackrel{\sim}{\rightarrow} R\mathrm{Hom}(R\Gamma_{Wh}(\mathcal{X}_s,\mathbb Z) ,\mathbb Z[-1])\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{Wh}(\mathcal{X}_s,\mathbb Z) ,\mathbb Z[-1]) ,\end{align*} $$

which is an equivalence of perfect complexes of abelian groups by Proposition 3.2 and [Reference Geisser11, Theorem 4.2] (using the fact that for a curve, étale and eh-cohomology agree).

Note that, if $d\leq 2$ , then the following diagram in $\mathbf {D}^b(\mathrm {LCA})$

commutes. If ${\mathcal X}_s^{{\mathrm {red}}}$ is a simple normal crossing scheme, then the same diagram with $R\Gamma _{Wh}(\mathcal {X}_s,\mathbb Z)$ replaced by $R\Gamma _{W}(\mathcal {X}_s,\mathbb Z)$ commutes as well.

We now combine Propositions 5.4 and 5.7 to prove our result for the generic fibre.

Proposition 5.8. There is an equivalence

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d)) \stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) ,\mathbb Z[-2d]),\end{align*} $$

such that, for any m, there is a commutative square

where the lower horizontal map is induced by duality for the usual étale cohomology of the variety $\mathcal {X}_K$ .

Proof. We start with the commutative diagram:

where the map

$$ \begin{align*}R\Gamma_{et}(\mathcal{X}_{\bar{s}},R\bar{i}^!\mathbb Z)\otimes^{\mathbb L} R\Gamma_{et}(\mathcal{X}_{\bar{s}},R\bar{i}^!\mathbb Z(d))\rightarrow R\Gamma_{et}(\mathcal{X}_{\bar{s}},R\bar{i}^!\mathbb Z(d))\end{align*} $$

is induced by the map $\mathbb Z\otimes ^{\mathbb L}\mathbb Z(d)\rightarrow \mathbb Z(d)$ over $\mathcal {X}_{\mathcal {O}_{{K}^{un}}}$ as follows. Consider the morphism

$$ \begin{align*}\bar{i}_*R\bar{i}^!\mathbb Z\rightarrow\mathbb Z\rightarrow R\mathbf{Hom}_{\mathcal{X}_{\mathcal{O}_{{K}^{un}}}}(\mathbb Z(d),\mathbb Z(d))\end{align*} $$

$$ \begin{align*}\rightarrow R\mathbf{Hom}_{\mathcal{X}_{\mathcal{O}_{{K}^{un}}}}(\bar{i}_*R\bar{i}^!\mathbb Z(d),\mathbb Z(d))\simeq \bar{i}_*R\mathbf{Hom}_{\mathcal{X}_{\bar{s}}}(R\bar{i}^!\mathbb Z(d),R\bar{i}^!\mathbb Z(d)),\end{align*} $$

and apply $\bar {i}^*$ , where $\mathbf {Hom}$ denotes the internal Hom in the category of sheaves on the small étale site of the corresponding scheme. Applying $R\Gamma (W_{\kappa (s)},-)$ to the diagram above, we obtain the following commutative diagram in $\mathbf {D}(\mathrm {Ab})$ , where $tr$ is the trace map:

It gives the following commutative diagram in $\mathbf {D}(\mathrm {Ab})$

We obtain the following commutative diagram

in the derived $\infty $ -category $\mathbf {D}^b(\mathrm {LCA})$ , where the lower-right map is given by Lemma 2.9. By construction of the maps (28) and (30), we obtain the following commutative diagram

hence, the upper square in the commutative square

It follows that the diagram is an equivalence of cofibre sequences in $\mathbf {D}^b(\mathrm {LCA})$ . Tensoring the upper commutative square with $\mathbb Z/m$ gives a square equivalent to the commutative square

where the horizontal maps are induced by the perfect pairings of Proposition 5.3. This yields the commutative square of Proposition 5.8.

It remains to prove that

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z)\rightarrow R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d)) ,\mathbb Z[-2d])\end{align*} $$

is an equivalence.

Lemma 5.9. The map

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z)\rightarrow R\underline{\mathrm{Hom}}(R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) ,\mathbb Z),\mathbb Z)\end{align*} $$

is an equivalence.

Proof. We have

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X},\mathbb Z)\simeq R\underline{\mathrm{Hom}}(R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X},\mathbb Z) ,\mathbb Z),\mathbb Z)\end{align*} $$

by Lemma 2.9, since $R\Gamma _{ar}(\mathcal {X},\mathbb Z)$ is a perfect complex of abelian groups by Propositions 3.2, 3.5 and 4.5. In view of the cofibre sequence

$$ \begin{align*}R\Gamma_{W}(\mathcal{X}_s,Ri^!\mathbb Z)\rightarrow R\Gamma_{ar}(\mathcal{X},\mathbb Z)\rightarrow R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z),\end{align*} $$

one is reduced to check that the map

$$ \begin{align*}R\Gamma_{W}(\mathcal{X}_s,Ri^!\mathbb Z)\rightarrow R\underline{\mathrm{Hom}}(R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_s,Ri^!\mathbb Z) ,\mathbb Z),\mathbb Z)\end{align*} $$

is an equivalence. Recall from the proof of Proposition 5.4 that we have

$$ \begin{align*} &R\underline{\mathrm{Hom}}(R\underline{\mathrm{Hom}}(R\Gamma_{W}(\mathcal{X}_s,Ri^!\mathbb Z) ,\mathbb Z),\mathbb Z)\\ &\quad \simeq R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X},\mathbb Z(d))[2d+1],\mathbb Z)\\ &\quad \simeq R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X},\mathbb Z(d))[2d+1],\mathbb{R}/\mathbb Z[-1])\\ &\quad \simeq R\Gamma_{ar}(\mathcal{X},\mathbb Z(d))^D[-2d-2]\\ &\quad \simeq (\mathrm{hocolim}\,R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X},\mathbb Z/m(d)),\mathbb{Q}/\mathbb Z))^{DD}[-2d-2]\\ &\quad \simeq \mathrm{hocolim}\,R\mathrm{Hom}(R\Gamma_{W}(\mathcal{X},\mathbb Z/m(d)),\mathbb{Q}/\mathbb Z[-2d-1])[-1]\\ &\quad \simeq R\Gamma_{W}(\mathcal{X}_s,Ri^!\mathbb{Q}/\mathbb Z)[-1]\\ &\quad \simeq R\Gamma_{W}(\mathcal{X}_s,Ri^!\mathbb Z), \end{align*} $$

where the second equivalence follows from Corollary 5.6.

Consider the pairing

(32) $$ \begin{align} R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d))\underline{\otimes}^{\mathbb L} R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) \longrightarrow \mathbb Z[-2d] \end{align} $$

induced by the equivalence of Proposition 5.8. The induced map

(33) $$ \begin{align} R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d)) \stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) ,\mathbb Z[-2d]) \end{align} $$

is (tautologically) the equivalence of Proposition 5.8. Moreover, the map

(34) $$ \begin{align} R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z)\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d)) ,\mathbb Z[-2d]) \end{align} $$

induced by (32) is an equivalence as well. Indeed, applying $R\underline {\mathrm {Hom}}(-,\mathbb Z[-2d])$ to (33) and using Lemma 5.9, we obtain the composite equivalence

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z)\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) ,\mathbb Z[-2d]),\mathbb Z[-2d])\end{align*} $$

$$ \begin{align*}\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d)),\mathbb Z[-2d]),\end{align*} $$

which is, up to equivalence, the map (34).

Proof. By Theorem 5.2 and [Reference Hoffmann and Spitzweck14, Remark 4.3(ii)], we have

$$ \begin{align*} R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d)) &\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) ,\mathbb Z[-2d])\\ &\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) , R\underline{\mathrm{Hom}}(\mathbb{R}/\mathbb Z,\mathbb{R}/\mathbb Z[-2d])\\ &\simeq R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z)\underline{\otimes}^{\mathbb L}\mathbb{R}/\mathbb Z ,\mathbb{R}/\mathbb Z[-2d])\\ &:=R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb{R}/\mathbb Z),\mathbb{R}/\mathbb Z[-2d]). \end{align*} $$

Applying the functor $R\underline {\mathrm {Hom}}(-,\mathbb {R}/\mathbb Z[-2d])$ and using Pontryagin duality, we obtain the first equivalence of the corollary.

Similarly, we have

$$ \begin{align*} R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z) &\stackrel{\sim}{\rightarrow} R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d)) ,\mathbb Z[-2d])\\ &\simeq R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb Z(d))\underline{\otimes}^{\mathbb L}\mathbb{R}/\mathbb Z ,\mathbb{R}/\mathbb Z[-2d])\\ &:=R\underline{\mathrm{Hom}}(R\Gamma_{ar}(\mathcal{X}_K,\mathbb{R}/\mathbb Z(d)),\mathbb{R}/\mathbb Z[-2d]). \end{align*} $$

Proof. In view of Theorem 4.17 and Lemma 2.6, the equivalence in $\mathbf {D}^b(\mathrm {FLCA})$

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X}_{K},\mathbb{R}/\mathbb Z)\stackrel{\sim}{\longrightarrow } R\Gamma(\mathcal{X}_{K},\mathbb Z(d))^D[-2d]\end{align*} $$

induces isomorphisms

$$ \begin{align*}H^{i}_{ar}(\mathcal{X}_{K},\mathbb{R}/\mathbb Z)\stackrel{\sim}{\longrightarrow } H^{i}(R\Gamma(\mathcal{X}_{K},\mathbb Z(d))^D[-2d])\simeq H^{2d-i}_{ar}(\mathcal{X}_{K},\mathbb Z(d))^D\end{align*} $$

of locally compact abelian groups. Similarly, the equivalence in $\mathbf {D}^b(\mathrm {FLCA})$

$$ \begin{align*}R\Gamma_{ar}(\mathcal{X}_{K},\mathbb{R}/\mathbb Z(d))\stackrel{\sim}{\longrightarrow } R\Gamma(\mathcal{X}_{K},\mathbb Z)^D[-2d]\end{align*} $$

induces isomorphisms

$$ \begin{align*}H^{i}_{ar}(\mathcal{X}_{K},\mathbb{R}/\mathbb Z(d))\stackrel{\sim}{\longrightarrow } H^{i}(R\Gamma(\mathcal{X}_{K},\mathbb Z)^D[-2d])\simeq H^{2d-i}_{ar}(\mathcal{X}_{K},\mathbb Z)^D\end{align*} $$

of compact abelian groups.