6.1.1 Continuous étale cohomology of Jannsen

Let $X$ be a scheme over a field $k$ and let $\operatorname {Ab}(X_{\unicode{x00E9}{\rm t}})^{\mathbb {N}}$ be the abelian category of inverse systems of abelian étale sheaves on the small étale site $X_{\unicode{x00E9}{\rm t}}$ of $X$. This category has enough injectives (see [Reference JannsenJan88]) and we may consider the left exact functor

Jannsen then defines the continuous étale cohomology groups

\[ H^i_{\rm cont}(X,(F_r)):=\operatorname{R}^i (\varprojlim\circ \Gamma)((F_r)) . \]

These groups are closely related to the corresponding étale cohomology groups via the following canonical short exact sequence (see [Reference JannsenJan88, § 1.6]):

(6.1)\begin{equation} 0\longrightarrow \operatorname{R}^1 \varprojlim\ H^{i-1}(X_{{\unicode{x00E9}{\rm t}}},F_r)\longrightarrow H^i_{\rm cont}(X_{{\unicode{x00E9}{\rm t}}},(F_r))\longrightarrow \varprojlim H^i(X_{{\unicode{x00E9}{\rm t}}},F_r)\longrightarrow 0 , \end{equation}

where $\varprojlim$ denotes the inverse limit functor over $r$.

By [Reference JannsenJan88, (3.27)], we have the following Kummer exact sequence in $\operatorname {Ab}(X_{\unicode{x00E9}{\rm t}})^{\mathbb {N}}$:

(6.2)\begin{equation} 0\longrightarrow (\mu_{\ell^r})_r \longrightarrow (\mathbb{G}_m,\times \ell)_r \stackrel{{\times \ell^r}}{\longrightarrow} (\mathbb{G}_m,\operatorname{id})_r \longrightarrow 0 , \end{equation}

where $\ell$ is a prime invertible in $k$. Taking cohomology, the boundary map of the corresponding long exact sequence yields maps

(6.3)\begin{equation} \epsilon:H^0(X,\mathbb{G}_m)\longrightarrow H^1_{\rm cont}(X,\mathbb{Z}_\ell(1))\quad\text{and}\quad c_1:\operatorname{Pic}(X\!)\longrightarrow H^2_{\rm cont}(X,\mathbb{Z}_\ell(1)), \end{equation}

where $H^i_{\rm cont}(X,\mathbb {Z}_\ell (n)):=H^i_{\rm cont}(X,(\mu _{\ell ^r}^{\otimes n})_r)$.

6.1.2 Pro-étale cohomology of Bhatt and Scholze

For a scheme $X$ we denote by $X_{\text{pro}{\unicode{x00E9}} {\rm t}}$ the pro-étale site of $X$ formed by weakly étale maps of schemes $U\to X$ (with $U$ of not too big cardinality), see [Reference Bhatt and ScholzeBS15, Definition 4.1.1 and Remark 4.1.2]. As every étale map is weakly étale, there is a natural map of associated topoi:

(6.4)\begin{equation} \nu:\operatorname{Shv}(X_{\text{pro}{\unicode{x00E9}} {\rm t}})\longrightarrow \operatorname{Shv}(X_{\unicode{x00E9}{\rm t}}). \end{equation}

The pullback $\nu ^\ast :D^+(X_{\unicode{x00E9}{\rm t}})\to D^+(X_{\text{pro}{\unicode{x00E9}} {\rm t}})$ on bounded below derived categories is fully faithful and the adjunction $\operatorname {id}\to \operatorname {R} \nu _\ast \nu ^\ast$ is an isomorphism, see [Reference Bhatt and ScholzeBS15, Proposition 5.2.6]. For a sheaf $F\in \operatorname {Ab}(X_{\text{pro}{\unicode{x00E9}} {\rm t}})$ of abelian groups on $X_{\text{pro}{\unicode{x00E9}} {\rm t}}$, one defines

\[ H^i(X_{\text{pro}{\unicode{x00E9}} {\rm t}},F):=\operatorname{R}^i \Gamma(X_{\text{pro}{\unicode{x00E9}} {\rm t}} , F), \]

where $\operatorname {R}^i \Gamma$ denotes the $i$th right derived functor of the global section functor .

If the transition maps in the inverse system $(F_r)\in \operatorname {Ab}(X_{\unicode{x00E9}{\rm t}})^{\mathbb {N}}$ are surjective, then there is a canonical isomorphism

(6.5)\begin{equation} H^i(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\lim \nu^\ast F_r) \simeq H^i_{\rm cont}(X_{{\unicode{x00E9}{\rm t}}},(F_r)) , \end{equation}

see [Reference Bhatt and ScholzeBS15, § 5.6].

6.1.3 Constructible complexes in the pro-étale topology

We present in this section some parts of the six functor formalism on constructible complexes of Bhatt and Scholze in the special case of algebraic schemes, i.e. separated schemes of finite type over a field, which suffices for our purposes. In Remark 6.3 we add some comments on the more general setting from [Reference Bhatt and ScholzeBS15].

Let $X$ be an algebraic scheme over a field $k$ and recall $\nu$ from (6.4). For a prime $\ell$ invertible in $k$, let

(6.6)\begin{equation} \widehat{\mathbb{Z}}_\ell(n):=\lim \nu^\ast \mu_{\ell^r}^{\otimes n} \in \operatorname{Ab}(X_{\text{pro}{\unicode{x00E9}} {\rm t}}) \end{equation}

and write $\widehat {\mathbb {Z}}_\ell :=\widehat {\mathbb {Z}}_\ell (0)$. Note that $\widehat {\mathbb {Z}}_\ell$ is a sheaf of rings on $X_{\text{pro}{\unicode{x00E9}} {\rm t}}$ and $\widehat {\mathbb {Z}}_\ell (n)$ are $\widehat {\mathbb {Z}}_\ell$-modules, which are, in fact, locally free (e.g. they are free on the pro-étale covering $X_{\overline {k}}\to X$). We may then consider the derived category $D(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$ of the abelian category $\operatorname {Mod}(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$ of sheaves of $\widehat {\mathbb {Z}}_\ell$-modules on $X_{\text{pro}{\unicode{x00E9}} {\rm t}}$. A complex $K\in D(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$ is constructible, if it is complete, i.e. $K\stackrel {\simeq }{\to } \operatorname {R} \lim (K\otimes ^{\mathbb {L}}_{\widehat {\mathbb {Z}}_\ell } \mathbb {Z}/\ell ^r)$, and $K\otimes ^{\mathbb {L}}_{\widehat {\mathbb {Z}}_\ell } \mathbb {Z}/\ell ^r\simeq \nu ^\ast K_r$ for a constructible complex $K_r\in D(X_{\unicode{x00E9}{\rm t}},\mathbb {Z}/\ell ^r)$, see [Reference Bhatt and ScholzeBS15, Definition 6.5.1]. The full subcategory spanned by constructible complexes is denoted by $D_{\rm cons}(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )\subset D(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$. Constructible complexes are bounded, see [Reference Bhatt and ScholzeBS15, Lemma 6.5.3].

For a morphism $f:X\to Y$ of algebraic schemes, $\operatorname {R} f_\ast$ respects constructibility and is right adjoint to $f^\ast _{\rm comp}:D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )\to D_{\rm cons}(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$, which is given by pullback followed by (derived) completion, see [Reference Bhatt and ScholzeBS15, Lemma 6.7.2]. There is also a functor $\operatorname {R} f_!:D_{\rm cons}(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )\to D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$ (see [Reference Bhatt and ScholzeBS15, Definition 6.7.6]) with a right adjoint $f^!: D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )\to D_{\rm cons}( X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$, see [Reference Bhatt and ScholzeBS15, Lemma 6.7.19]. If $f$ is proper, $\operatorname {R} f_!=\operatorname {R} f_\ast$ (by definition).

To explain the construction of $f^!$ in [Reference Bhatt and ScholzeBS15], note that the pullback

(6.7)\begin{equation} \nu^\ast: D_{\rm cons}(X_{\unicode{x00E9}{\rm t}},\mathbb{Z}/\ell^r)\stackrel{{\simeq}}{\longrightarrow} D_{\rm cons}(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\mathbb{Z}/\ell^r) \end{equation}

is an equivalence (see the paragraph after [Reference Bhatt and ScholzeBS15, Definition 6.5.1]). Using this, we freely identify complexes on the two sides with each other. For instance, we freely identify $\mu _{\ell ^r}^{\otimes n}$ on $Y_{\unicode{x00E9}{\rm t}}$ with its pullback $\nu ^\ast \mu _{\ell ^r}^{\otimes n}$ to $Y_{\text{pro}{\unicode{x00E9}} {\rm t}}$. Now let $K\in D_{\rm cons}( Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$ with truncation $K_r=K\otimes ^{\mathbb {L}}_{ \mathbb {Z}_\ell }\mathbb {Z}/\ell ^r$ and let $f_r^!:D_{\rm cons}(Y_{\unicode{x00E9}{\rm t}}, \mathbb {Z}/\ell ^r )\to D_{\rm cons}( X_{\unicode{x00E9}{\rm t}}, \mathbb {Z}/\ell ^r)$ be the exceptional pullback on the étale site, induced by $f$, cf. [Reference Artin, Grothendieck and VerdierSGA4.3, Exposé XVIII]. As any constructible complex of sheaves of $\mathbb {Z}/\ell ^r$-modules on $X_{\text{pro}{\unicode{x00E9}} {\rm t}}$ is also a constructible complex of $\widehat {\mathbb {Z}}_\ell$-modules on $X_{\text{pro}{\unicode{x00E9}} {\rm t}}$, we may by (6.7) identify $f_r^!K_r$ with an object in $D_{\rm cons}( X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$. By [Reference Bhatt and ScholzeBS15, Lemma 6.7.18], the natural reduction maps $K_r\to K_m$ for $m\leqslant r$ make $(f_r^!K_r)$ into a projective system and so, following Bhatt and Scholze (see [Reference Bhatt and ScholzeBS15, Lemma 6.7.19]), one may define

(6.8)\begin{equation} f^!K:=\operatorname{R} \lim f_r^! K_r \in D_{\rm cons}( X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat{\mathbb{Z}}_\ell). \end{equation}

This construction implies that many properties known from the étale site carry over to the pro-étale site.

Proof. Item (i) follows from [Reference Bhatt and ScholzeBS15, Remark 6.5.10]. Item (ii) follows from this together with the fact that for any $K\in D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$,

\[ f_{\rm comp}^\ast K \stackrel{{\simeq}}{\longrightarrow} \operatorname{R} \lim f_r^\ast K_r=\operatorname{R} \lim f_r^!K_r= f^!K \]

because $f_r^!=f_r^\ast$ since $f$ is étale; see [Reference Artin, Grothendieck and VerdierSGA4.3, XVIII, Proposition 3.1.8(iii)].

Let $K\in D_{\rm cons}(Z_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$ with truncations $K_r=K\otimes _{\widehat {\mathbb {Z}}_\ell }^{\mathbb {L}} \mathbb {Z}/\ell ^r \in D_{\rm cons}(Y_{\unicode{x00E9}{\rm t}},\mathbb {Z}/\ell ^r)$. By (6.7) and the construction of $f^!$ from (6.8), there is a natural map

\[ f^!g^!K=\operatorname{R} \lim f_r^!(\operatorname{R} \lim g_r^! K_r)=\operatorname{R} \lim (f_r^!\circ g_r^!(K_r))\stackrel{{\simeq}}{\longrightarrow} \operatorname{R} \lim ((g_r\circ f_r)^!K_r)=(g\circ f)^! K \]

induced by the natural isomorphism $f_r^!\circ g_r^!\stackrel {\sim }{\to } (g_r\circ f_r)^!$, given by adjunction and $\operatorname {R} (g_r)_!\operatorname {R} (f_r)_!\stackrel {\simeq }{\to } \operatorname {R} (g_r\circ f_r)_!$. This proves item (iii).

Let $K\in D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}} ,\widehat {\mathbb {Z}}_\ell )$ with truncations $K_r=K\otimes _{\widehat {\mathbb {Z}}_\ell }^{\mathbb {L}} \mathbb {Z}/\ell ^r \in D_{\rm cons}(Y_{\unicode{x00E9}{\rm t}},\mathbb {Z}/\ell ^r)$. Assume that $f$ is smooth of pure relative dimension $d$. By Poincaré duality on the étale site, there are canonical identifications $f_r^!= f_r^\ast (d)[2d]$, see [Reference Artin, Grothendieck and VerdierSGA4.3, XVIII, Théorème 3.2.5] (cf. [Reference VerdierVer67, § 4.4]). We thus obtain a canonical isomorphism:

\[ f_{\rm comp}^\ast K(d)[2d]\stackrel{{\simeq}}{\longrightarrow} \operatorname{R} \lim f_r^\ast K_r(d)[2d]=\operatorname{R} \lim f_r^! K_r= f^!K. \]

This holds functorially in $K$ and so we get an isomorphism $f_{\rm comp}^\ast (d)[2d]\stackrel {\simeq }{\to } f^!$, which proves item (iv).

By item (ii), $j_{\rm comp}^\ast \simeq j^!$ and so the commutativity of the diagram in item (v) follows from the fact that the isomorphism in item (iv) is compatible with respect to compositions of smooth maps. The latter follows by construction of $f^!$ from the analogous result for constructible complexes on the étale site and, hence, from [Reference Artin, Grothendieck and VerdierSGA4.3, XVIII, diagram above Théorème 3.2.5]. This concludes the proof of the lemma.

Let $f:X\to Y$ be a morphism between algebraic $k$-schemes. By adjunction, there are natural transformations

(6.9)\begin{align} \operatorname{Tr}_f: \operatorname{R} f_! f^!\longrightarrow \operatorname{id} \quad \text{and}\quad \theta_f: \operatorname{id}\longrightarrow \operatorname{R} f_\ast f^\ast_{\rm comp} \end{align}

between functors on $D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$. For $K\in D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$, the maps are defined by asking (see [Reference Bhatt and ScholzeBS15, Lemmas 6.7.2 and 6.7.19]) that the following diagrams commute:

(6.10)

where $K_r=K\otimes ^{\mathbb {L}}_{\widehat {\mathbb {Z}}_\ell }\mathbb {Z}/\ell ^r\in D_{\rm cons}(Y_{\unicode{x00E9}{\rm t}}, \mathbb {Z}/\ell ^r)$, $f_r^\bullet$, $f_{r\bullet }$ denote the corresponding functors on $D_{\rm cons}(Y_{\unicode{x00E9}{\rm t}}, \mathbb {Z}/\ell ^r)$ (respectively, $D_{\rm cons}(X_{\unicode{x00E9}{\rm t}}, \mathbb {Z}/\ell ^r)$) induced by $f$, and where as before we identify $K_r$ with $\nu ^\ast K_r$, using the equivalence (6.7).

For $K\in D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$, $\theta _f$ induces pullback maps:

(6.11)\begin{equation} f^\ast:\operatorname{R}^i \Gamma(Y_{\text{pro}{\unicode{x00E9}} {\rm t}}, K)\longrightarrow \operatorname{R}^i \Gamma(X_{\text{pro}{\unicode{x00E9}} {\rm t}}, f_{\rm comp}^\ast K). \end{equation}

If $f$ is proper, then $\operatorname {R} f_!=\operatorname {R} f_\ast$ and so $\operatorname {Tr}_f$ induces a pushforward map:

(6.12)\begin{equation} f_\ast: \operatorname{R}^i \Gamma(X_{\text{pro}{\unicode{x00E9}} {\rm t}}, f^! K)\longrightarrow \operatorname{R}^i \Gamma(Y_{\text{pro}{\unicode{x00E9}} {\rm t}}, K). \end{equation}

Lemma 6.2 Let $K\in D_{\rm cons}(Y_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat {\mathbb {Z}}_\ell )$ with truncations $K_r=K\otimes ^{\mathbb {L}}_{\widehat {\mathbb {Z}}_\ell }\mathbb {Z}/\ell ^r\in D_{\rm cons}( Y_{\unicode{x00E9}{\rm t}}, \mathbb {Z}/\ell ^r )$. Then for any morphism $f:X\to Y$ between algebraic $k$-schemes, the following diagram commutes with exact rows.

If $f$ is proper, then the following commutes with exact rows as well.

6.1.4 Properties (P1)–(P6), (P7.1), and (P7.2)

Let $k$ be a field and let $\ell$ be a prime that is invertible in $k$. Let $\mathcal {V}$ be the category whose objects are separated schemes of finite type over $k$ and where the morphisms are open immersions of schemes of the same dimension. For $X\in \mathcal {V}$ of dimension $d$ with structure morphism $\pi _X:X\to \operatorname {Spec} k$, we define

(6.13)\begin{gather} H^i(X,\mu_{\ell^r}^{\otimes n}):= \operatorname{R}^{i-2d} \Gamma(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\pi_X^! \mu_{\ell^r}^{\otimes n-d}) \in \operatorname{Mod}_{\mathbb{Z}_\ell}, \end{gather}

(6.14)\begin{gather} H^i(X,\mathbb{Z}_{\ell}(n)):=\operatorname{R}^{i-2d} \Gamma(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\pi_X^!\widehat{\mathbb{Z}}_\ell(n-d)) \in \operatorname{Mod}_{\mathbb{Z}_\ell} , \end{gather}

(6.15)\begin{gather} H^i(X,\mathbb{Q}_\ell/\mathbb{Z}_{\ell}(n)):=\lim_{\substack{\longrightarrow \\ r}}H^i(X,\mu_{\ell^r}^{\otimes n}) \quad\text{and}\quad H^i(X,\mathbb{Q}_{\ell}(n)):=H^i(X,\mathbb{Z}_{\ell}(n))\otimes_{\mathbb{Z}_\ell} \mathbb{Q}_{\ell}, \end{gather}

where $\widehat {\mathbb {Z}}_\ell (n)$ denotes the sheaf on $X_{\text{pro}{\unicode{x00E9}} {\rm t}}$ defined in (6.6). By item (ii) in Lemma 6.1 and (6.11), the cohomology groups in (6.13)–(6.15) are contravariantly functorial with respect to morphisms in $\mathcal {V}$ (in fact, with respect to arbitrary étale maps $U\rightarrow X$ with $\dim U=\dim X$).

Lemma 6.5 Assume that $X\in \mathcal {V}$ is equi-dimensional and smooth over $k$. Then there are canonical isomorphisms

\[ H^i(X,\mu_{\ell^r}^{\otimes n})\simeq \operatorname{R}^{i} \Gamma(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\mu_{\ell^r}^{\otimes n}) \simeq H^i_{\rm cont}(X_{{\unicode{x00E9}{\rm t}}},(\mu_{\ell^r}^{\otimes n},\operatorname{id} )_s)\simeq H^i(X_{{\unicode{x00E9}{\rm t}}},\mu_{\ell^r}^{\otimes n}), \]

and

\[ H^i(X,\mathbb{Z}_{\ell}(n))\simeq \operatorname{R}^{i} \Gamma(X_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat{\mathbb{Z}}_\ell(n)) \simeq H^i_{\rm cont}(X,(\mu_{\ell^r}^{\otimes n})_r) , \]

which are compatible with respect to pullbacks along open immersions.

The main result of this section is the following.

In the terminology of Definitions 4.2, 4.4, and 4.5, the proposition states that $H^\ast (-,A(n))$ is a twisted Borel–Moore cohomology theory on $\mathcal {V}$ that is $\ell$-adic if $k$ is perfect and it is adapted to algebraic equivalence if $k$ is algebraically closed, whereas it is adapted to rational equivalence if $k$ is the perfect closure of a finitely generated field.

Proof of Proposition 6.6 Item (P4) is clear (by definition). As the direct limit functor as well as $\otimes _{\mathbb {Z}_\ell }\mathbb {Q}_\ell$ is exact, it suffices to prove the remaining properties for $A=\mathbb {Z}/\ell ^r$ and $A=\mathbb {Z}_\ell$.

Step 1: item (P1). Let $X,Y\in \mathcal {V}$ and let $f:X\to Y$ be a proper morphism of schemes with $c=\dim Y-\dim X$. The existence of the pushforward $f_\ast :H^{i-2c}(X,A(n-c))\to H^i(Y,A(n))$ follows from (6.12) and item (iii) in Lemma 6.1. Functoriality in $f$ (i.e. $(f\circ g)_\ast =f_\ast g_\ast$) follows from the functoriality of the trace map (which by (6.9) may either be deduced from the corresponding statement on the étale site, or directly from item (iii) in Lemma 6.1). Compatibility of $f_\ast$ with pullbacks along open immersions may by Lemma 6.2 be checked in the case where $A=\mathbb {Z}/\ell ^r$ on the étale site of $X$, which is well-known (and holds, in fact, for arbitrary étale maps in place of open immersions), see [Reference Bloch and OgusBO74, (1.2.2) and § 2.1]. This proves (P1).

Step 2: item (P2). Let $X\in \mathcal {V}$ and let $i:Z\hookrightarrow X$ be a closed immersion with complement $j:U\hookrightarrow X$. Let $c=\dim X-\dim Z$. By [Reference Bhatt and ScholzeBS15, Lemma 6.1.16], there is an exact triangle

\[ \operatorname{R} i_\ast i^! \pi_X^! \stackrel{{\operatorname{Tr}_i}}{\longrightarrow} \pi_X^! \stackrel{{\theta_j}}{\longrightarrow} \operatorname{R} j_\ast j_{\rm comp}^\ast \pi_X^!, \]

where we used $j^\ast \simeq j_{\rm comp}^\ast$ (see Lemma 6.1). By Lemma 6.1, $j_{\rm comp}^\ast \simeq j^!$, $\pi _Z^!\stackrel {\simeq }{\to } i^! \pi _X^!$, and $\pi _{U}^!\stackrel {\simeq }{\to } j_{\rm comp}^\ast \pi _X^!$. Hence, the above triangle identifies to an exact triangle

\[ \operatorname{R} i_\ast \pi_Z^!\longrightarrow \pi_X^!\longrightarrow \operatorname{R} j_\ast \pi_U^! . \]

Applying $\operatorname {R} \Gamma (X_{\text{pro}{\unicode{x00E9}} {\rm t}},-)$, the corresponding long exact sequence yields the Gysin sequence claimed in (P2). The map $\iota _\ast$ in the Gysin sequence from item (P2) coincides, by construction, with the proper pushforward with respect to the inclusion $Z\hookrightarrow X$. Functoriality with respect to open immersions follows by Lemma 6.2 from the case $A=\mathbb {Z}/\ell ^r$ on the étale site which is well-known (and holds, in fact, for arbitrary étale maps), see, e.g., [Reference Bloch and OgusBO74, (1.1.2), Lemma 1.4, and (2.1)]. Similarly, functoriality with respect to proper pushforwards follows by Lemma 6.2 from the case of finite coefficients $A=\mathbb {Z}/\ell ^r$ on the étale site, which is well-known, see, e.g., [Reference Bloch and OgusBO74, (1.2.4) and (2.1)]. This proves (P2).

Step 3: item (P3). By the topological invariance of the pro-étale topos (see [Reference Bhatt and ScholzeBS15, Lemma 5.4.2]), we may replace $X$ be the base change to the perfect closure $k^{per}$ of $k$, and $x$ by the unique point in $X_{k^{per}}$ that lies over it via the natural map $X_{k^{per}}\to X$ (where we use that the latter is a universal homeomorphism). After this reduction step, we may assume that $k$ is perfect.

Note that $H^i(x,A(n))$ in (4.2) is defined as a direct limit where it suffices to run only through the cohomology of regular (hence, smooth, because $k$ is perfect) schemes, so that the vanishing $H^i(x,A(n))=0$ for $i<0$ as well as the canonical isomorphism $H^0(x,A(0))\simeq A$ which is functorial in $A$ follows from Lemma 6.5. The fundamental class $[x]\in H^0(x,\mathbb {Z}_\ell (0))$ corresponds via the canonical isomorphism $H^0(x,\mathbb {Z}_\ell (0))\simeq \mathbb {Z}_\ell$ to $1\in \mathbb {Z}_\ell$. More precisely, let $U\subset \overline {\{x\}}$ be dense and smooth over $k$. By Lemma 6.5, the canonical isomorphism in item (iv) of Lemma 6.1 induces a canonical isomorphism

\begin{align*} H^0(U,\mathbb{Z}_\ell(0))\simeq H^0(U_{\text{pro}{\unicode{x00E9}} {\rm t}},\widehat{\mathbb{Z}}_\ell(0)), \end{align*}

which is compatible with respect to restrictions to open subsets. The class of $H^0(U,\mathbb {Z}_\ell (0))$ induced by the unit section of the pro-étale sheaf $\widehat {\mathbb {Z}}_\ell (0)$ yields a canonical fundamental class $[U]\in H^0(U,\mathbb {Z}_\ell (0))$ with $H^0(U,\mathbb {Z}_\ell (0))=[U]\mathbb {Z}_\ell$. This class is compatible with respect to restrictions to open subsets (see item (v) in Lemma 6.1), hence induces a canonical class $[x]\in H^0(x,\mathbb {Z}_\ell (0))$ in the limit. This proves (P3).

Step 4: item (P5). There is a canonical short exact sequence

\[ 0\longrightarrow \widehat{\mathbb{Z}}_\ell(n) \stackrel{{\times \ell^r}}{\longrightarrow} \widehat{\mathbb{Z}}_\ell(n)\longrightarrow \mu_{\ell^r}^{\otimes n}\longrightarrow 0 \]

of sheaves on $(\operatorname {Spec} k)_{\text{pro}{\unicode{x00E9}} {\rm t}}$. Applying $\pi _X^!$, we arrive at the exact triangle

(6.16)\begin{equation} \pi_X^! \widehat{\mathbb{Z}}_\ell(n) \stackrel{{\times \ell^r}}{\longrightarrow} \pi_X^! \widehat{\mathbb{Z}}_\ell(n)\longrightarrow \pi_X^! \mu_{\ell^r}^{\otimes n} . \end{equation}

The Bockstein sequence in (P5) is (up to some shifts) the long exact sequence associated to this triangle after applying $\operatorname {R} \Gamma (X_{\text{pro}{\unicode{x00E9}} {\rm t}},-)$. Functoriality of the exceptional pullback $\pi _X^!$ shows that the Bockstein sequence is functorial with respect to pullbacks along morphisms in $\mathcal {V}$ and with respect to proper pushforwards from (P1). This proves (P5).

Step 5: item (P6) for $k$ perfect. Let $X\in \mathcal {V}$ and let $x\in X^{(1)}$. In the direct limit (4.2) that defines $H^i(x,A(n))$, we may restrict ourselves to regular (hence, smooth, as $k$ is perfect) dense open subsets $V_x\subset \overline {\{x\}}$, so that Lemma 6.5 identifies $H^i(V_x,A(n))$ canonically with continuous étale cohomology. The map $\epsilon :\kappa (x)^\ast \to H^1(x,\mathbb {Z}_\ell (1))$ is then induced by the Kummer sequence in continuous étale cohomology, see (6.3). Surjectivity of the reduction $\overline {\epsilon }:\kappa (x)^\ast \to H^1(x,\mu _{\ell ^r}\!)$ follows from the Kummer sequence in étale cohomology and Grothendieck's Hilbert theorem 90, which implies that

\[ \lim_{\substack{\longrightarrow \\ F_0X\subset U\subset X}} H^1_{\unicode{x00E9}{\rm t}}(U,\mathbb{G}_m)\simeq \lim_{\substack{\longrightarrow \\ F_0X\subset U\subset X}} \operatorname{Pic}(U)=0 . \]

Let $X\in \mathcal {V}$ be integral of dimension $d$ with generic point $\eta$ and let $x\in X^{(1)}$ be a regular point with closure $i:D\hookrightarrow X$. We claim that the composition ${\partial \circ \epsilon : \kappa (\eta )^\ast \to H^0(x ,\mathbb {Z}_{\ell }(0))=[x] \mathbb {Z}_{\ell }}$ satisfies

\[ \partial\circ \epsilon(f)=[x](- \nu_x(f)), \]

where $\nu _x$ denotes the valuation on $\kappa (\eta )$ induced by $x$. It suffices to check this after reduction modulo $\ell ^r$ for $r\geqslant 1$. It follows from [Reference DeligneSGA4, p. 147, (cycle), Lemme 2.3.6] that the fundamental class $[x]\in H^0(x,\mu _{\ell ^r}^{\otimes 0})$ that we defined above via Poincaré duality (i.e. via item (iv) in Lemma 6.1) is induced by the cycle class

\[ \operatorname{cl}_{\ell^r}(D)\in H^2_D(X_{\unicode{x00E9}{\rm t}},\mu_{\ell^r})=H^2(D_{\unicode{x00E9}{\rm t}},i_r^!\mu_{\ell^r})\simeq H^{2-2d}(D_{\unicode{x00E9}{\rm t}},\pi_D^!\mu_{\ell^r}^{\otimes 1-d}) \]

from [Reference DeligneSGA4, p. 138, (cycle), Définition 2.1.2]. The claim in question follows therefore from the anticommutativity of the diagram in [Reference DeligneSGA4, p. 138, (cycle), (2.1.3)]. This concludes the proof of (P6).

By Lemma 5.8 and Corollary 5.10 (which apply because we have proven (P1)–(P6) already), the proper pushforward map from (P1) together with $H^0(x,\mathbb {Z}_\ell (0))=[x]\mathbb {Z}_\ell$ from (P3) yields for any $X\in \mathcal {V}$ a canonical map

\[ \iota_\ast : \bigoplus_{x\in X^{(1)}}[x]\mathbb{Z}_\ell \longrightarrow H^2(F_1X,\mathbb{Z}_\ell(1))\simeq H^2(X,\mathbb{Z}_\ell(1)) , \]

as claimed in (4.3). If $X$ is a smooth variety, then there is a canonical isomorphism $H^2(X,\mathbb {Z}_\ell (1)) \simeq H^2_{\rm cont}(X,\mathbb {Z}_\ell (1))$ (see Lemma 6.5) and so we may compare $\iota _\ast$ to the first Chern class map $c_1$ from (6.3), as follows.

Proof. The first assertion is [Reference JannsenJan88, Lemma 3.26]. For the second assertion, note that the Kummer sequence (6.2) yields an exact sequence

\[ H^1(X_{{\unicode{x00E9}{\rm t}}}, (\mathbb{G}_m,\times \ell)_r )\longrightarrow \operatorname{Pic}(X\!)\stackrel{{ c_1}}{\longrightarrow} H^2_{\rm cont}(X,\mathbb{Z}_\ell(1)). \]

Functoriality of the extension in (6.1) shows that the image of the first map is given by subgroup of $\ell$-divisible elements of $\operatorname {Pic}(X\!)$ (cf. [Reference JannsenJan88, Remark 6.15]), which concludes the proof of the lemma.

Step 6: item (P7.1) for $k$ algebraically closed. Let us now assume that $k$ is algebraically closed and let $X$ be a regular (hence, smooth, as $k=\overline {k}$) variety over $k$. Note that (P7.1) is well-known in the case where $X$ is smooth projective, and we deduce the general case from this statement in what follows. We denote by $\operatorname {NS}(X\!)=\operatorname {Pic}(X\!)/\sim _{\operatorname {alg}}$ the group of divisors modulo algebraic equivalence on $X$. As $k$ is algebraically closed, the subgroup of algebraically trivial divisors in $\operatorname {Pic}(X\!)$ is $\ell$-divisible. By Lemma 6.7, the first Chern class map from (6.3) descends to a map $c_1:\operatorname {NS}(X\!)\otimes \mathbb {Z}_\ell \to H^2(X,\mathbb {Z}_\ell (1))$ and, again by Lemma 6.7, it suffices to show that this is injective. Let $\overline {X}$ be a projective normal compactification of $X$ and let $\tau :\overline {X}'\to \overline {X}$ be an alteration (i.e. a projective generically finite morphism with regular source) of degree prime to $\ell$, which exists by [Reference Illusie, Laszlo and OrgogozoILO14, Exposé X, Théorème 2.1]. Since $k$ is algebraically closed, $\overline {X}'$ is smooth. Putting $X':=\tau ^{-1}(X\!)$, we get the following commutative diagram.

We claim that it suffices to show that the horizontal arrow in the middle is injective. To see this, let $\alpha \in \operatorname {NS}( X)\otimes \mathbb {Z}_\ell$. Then $c_1(\tau ^\ast \alpha )=\tau ^\ast c_1(\alpha )$ and so injectivity of the horizontal arrow in the middle implies $c_1(\alpha )\neq 0$ unless $\tau ^\ast \alpha =0$, which, in turn, implies $\tau _\ast \tau ^\ast \alpha =\deg \tau \cdot \alpha =0$ and so $\alpha =0$ because $\deg \tau$ is coprime to $\ell$.

By the localization sequence, the kernel of $\operatorname {NS}(\overline {X}') \otimes \mathbb {Z}_\ell \to \operatorname {NS}(X')\otimes \mathbb {Z}_\ell$ is generated by classes of divisors supported on $\overline {X}'\setminus X$. Similarly, the Gysin sequence (see (P2)) shows that the kernel of the restriction map $H^2(\overline {X}',\mathbb {Z}_\ell (1)) \to H^2(X',\mathbb {Z}_\ell (1))$ is generated by the cycle classes of these divisors. As the first horizontal map in the above diagram is injective by [Reference MilneMil80, p. 216, V.3.28], whereas the restriction map $\operatorname {NS}(\overline {X}')\to \operatorname {NS}( X')$ is surjective (see [Reference FultonFul98, Proposition 1.8]), this shows that the horizontal arrow in the middle of the above diagram is injective, as desired. This proves (P7.1).

Step 7: item (P7.2) for $k$ the perfect closure of a finitely generated field. We use the following well-known lemma.

Now let $k$ be the perfect closure of a finitely generated field $k_0\subset k$, and let $X$ be a regular (hence smooth) variety over $k$. By Lemma 6.7, it suffices to show that the map

(6.17)\begin{equation} c_1\otimes \mathbb{Z}_\ell:\operatorname{Pic}(X\!)\otimes_\mathbb{Z} \mathbb{Z}_\ell\longrightarrow H^2(X,\mathbb{Z}_\ell(1)) \end{equation}

induced by $c_1$ from (6.3) is injective, where we note that the right-hand side identifies to continuous étale cohomology by Lemma 6.5. Using the existence of prime to $\ell$ alterations, the same argument as in step 6 reduces us to the case where $X$ is smooth projective over $k$. At this point the argument is similar to [Reference JannsenJan88, Remark 6.15].

As $X$ is defined over some finitely generated field, we may assume (up to enlarging $k_0$) that $X=X_0\times _{k_0}k$ for some smooth $k_0$-variety $X_0$.

Assume for the moment that $X_0$ is geometrically integral. By Grothendieck's theorem, the Picard functor on $X_0$ is then represented by the Picard scheme $\operatorname {\textbf {Pic}}_{X_0/k}$, see e.g. [Reference KleimanKle05, Theorem 9.4.8]. In particular, $\operatorname {Pic}(X_0)$ is given by the group of $k_0$-rational points of $\operatorname {\textbf {Pic}}_{X_0/k}$. The quotient of $\operatorname {\textbf {Pic}}_{X_0/k}$ by the identity component is always a finitely generated group scheme (the Néron–Severi group). Moreover, the identity component is an abelian variety over $k_0$ and because $k_0$ is finitely generated, its group of $k_0$-rational points is finitely generated by Néron's Mordell–Weil theorem [Reference NéronNér52]. It follows that $\operatorname {Pic} (X_0)$ is a finitely generated abelian group.

In general, $X_0$ will split into a finite union of geometrically integral smooth projective varieties after a finite extension of the base field. The above argument together with a pull and push argument then shows that, in general, $\operatorname {Pic}(X_0)$ contains an $n$-torsion subgroup $T$ for some $n\geqslant 1$ such that $Q:=\operatorname {Pic}(X_0)/T$ is a finitely generated abelian group. We consider the short exact sequence $0\to T\to \operatorname {Pic}(X_0)\to Q\to 0$. As $Q$ is finitely generated and $T$ is $n$-torsion, this sequence remains exact if we apply either the $\ell$-adic completion functor or $\otimes _\mathbb {Z} \mathbb {Z}_\ell$. Comparing the two resulting short exact sequences, we find that

(6.18)\begin{equation} \lim_{\substack{\longleftarrow\\ r}} (\operatorname{Pic}(X_0)/\ell^r) \stackrel{{\simeq}}{\longrightarrow} \operatorname{Pic}(X_0)\otimes_\mathbb{Z} \mathbb{Z}_\ell. \end{equation}

The usual Kummer sequence on the étale site yields compatible injections $\operatorname {Pic}(X_0)/\ell ^r\hookrightarrow H^2(X_0,\mu _{\ell ^r}^{\otimes 1})$, i.e. an injection of projective systems. Applying the inverse limit functor, this yields by (6.18) an injection $\operatorname {Pic}(X_0)\otimes _\mathbb {Z}\mathbb {Z}_\ell \hookrightarrow \lim H^2(X_0,\mu _{\ell ^r}^{\otimes 1})$. By (6.1) and the construction of $c_1$ in (6.3), this injection factors through

\[ c_1\otimes\mathbb{Z}_\ell:\operatorname{Pic}(X_0)\otimes_\mathbb{Z}\mathbb{Z}_\ell\longrightarrow H^2(X_0,\mathbb{Z}_\ell(1)) \]

and so the latter must be injective as well. It follows that (6.17) is injective, because $c_1$ is functorial with respect to pullbacks, and the canonical pullback maps yield isomorphisms $\operatorname {Pic}(X_0)\otimes _\mathbb {Z} \mathbb {Z}_\ell \simeq \operatorname {Pic}(X\!)\otimes _\mathbb {Z} \mathbb {Z}_\ell$ (see Lemma 6.8) and $H^2(X_0,\mathbb {Z}_\ell (1))\simeq H^2(X,\mathbb {Z}_\ell (1))$ (see [Reference Bhatt and ScholzeBS15, Lemma 5.4.2]), because $k/k_0$ is purely inseparable by assumption. This concludes the proof of (P7.2) and, hence, finishes the proof of the proposition.