p-adic étale cohomology of period domains

Abstract

We compute the p-torsion and p-adic étale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. As in the cases of $\ell$-torsion or $\ell$-adic coefficients, $\ell \ne p$, considered by Orlik, the results involve generalized Steinberg representations. For the p-torsion case, we follow the method used by Orlik in his computations of the $\ell$-torsion étale cohomology using as a key new ingredient the computation of ${\text {Ext}}$ groups between mod p generalized Steinberg representations of p-adic groups. For the p-adic case, we do not use Huber’s definition of étale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use continuous étale cohomology with compact support.

Appendix A: Adic potpourri

We gather here, as a reference, some basic facts concerning pseudo-adic spaces and compactly supported étale cohomology.

1.1 Pseudo-adic spaces

We start with pseudo-adic spaces. Recall that, Huber defines in [36] the category PPA of pre-pseudo-adic spaces, consisting of pairs $X=(\underline{X}, |X|)$, where $\underline{X}$ is an adic space and |X| a subset of $\underline{X}$, morphisms $X\rightarrow Y$ being morphisms of adic spaces $\underline{X}\rightarrow \underline{Y}$ that send |X| into |Y|. A morphism $f: X\rightarrow Y$ induces therefore a morphism of adic spaces $\underline{f}: \underline{X}\rightarrow \underline{Y}$ and a map of topological spaces $|f|: |X|\rightarrow |Y|$ (we endow |X| with the topology induced from $\underline{X}$). We say that f is étale if $\underline{f}$ is étale and if |X| is open in $\underline{f}^{-1}(|Y|)$ (this implies that |f| is an open map). The étale site $X_{{\acute{\mathrm{e}}\mathrm{t}} }$ of X is the category of pre-pseudo-adic spaces Y étale over X with the topology such that a family of morphisms $f_i: Y_i\rightarrow Y$ in this category is a covering if $|Y|=\cup _{i} |f_i|(|Y_i|)$.

We mention the following properties of this construction, which we need, and we refer the reader to Huber’s book [36] for the proofs and details (see especially Sections 1.10, 2.3):

• (1) The category PPA contains (as full subcategory) the category of adic spaces (via $X\mapsto (X, |X|)$) and the étale topoi of X and (X, |X|) are equivalent.

• (2) If X is an adic space and $S\subset T$ are subsets of X, the natural morphism $i: (X,S)\rightarrow (X,T)$ in PPA satisfies $i^*i_*F\simeq F$, for all $F\in \mathrm{Sh}((X,S)_{{\acute{\mathrm{e}}\mathrm{t}} })$, thus $i_*: \mathrm{Sh}((X,S)_{{\acute{\mathrm{e}}\mathrm{t}} })\rightarrow \mathrm{Sh}((X,T)_{{\acute{\mathrm{e}}\mathrm{t}} })$ is fully faithful. Moreover, if S is closed in T, then $i_*$ is exact and identifies $\mathrm{Sh}((X,S)_{{\acute{\mathrm{e}}\mathrm{t}} })$ with the full subcategory of $\mathrm{Sh}((X,T)_{{\acute{\mathrm{e}}\mathrm{t}} })$ consisting of sheaves F whose restriction to $(T-S)_{{\acute{\mathrm{e}}\mathrm{t}} }$ is the final object of $\mathrm{Sh}((T-S)_{{\acute{\mathrm{e}}\mathrm{t}} })$ ([36, Lemma 2.3.11]).

• (3) Let PA be the full subcategory of PPA consisting of pseudo-adic spaces, i.e., those X for which |X| is convex and locally pro-constructible in $\underline{X}$. An object X of PA is called quasi-compact/quasi-separated if |X| is so, and a map $f: X\rightarrow Y$ in PA is called quasi-compact/quasi-separated if |f| is so. If $f: X\rightarrow Y$ is a quasi-compact quasi-separated morphism in PA and if f is adic (i.e., $\underline{f}$ is adic), then $\mathrm {R} ^nf_*$ commutes with pseudo-filtered inductive limits. If $X\in PA$ is quasi-compact quasi-separated, then $H^n_{{\acute{\mathrm{e}}\mathrm{t}} }(X,-)$ commutes with pseudo-filtered inductive limits. ([36, Lemma 2.3.13]).

• (4) If x is a point of an adic space X and if K is the henselization of the residue class field k(x) with respect to the valuation ring $k(x)^+$, there is a natural equivalence of categories $\mathrm{Sh}((X,\{x\})_{{\acute{\mathrm{e}}\mathrm{t}} })\simeq \mathrm{Sh}(\mathrm{Spec}(K)_{{\acute{\mathrm{e}}\mathrm{t}} })$ ([36, Prop. 2.3.10]).

• (5) Let P be one of the properties “open, closed, locally closed”. A P-subspace of $X\in PPA$ is an object $Y\in PPA$ for which $\underline{Y}$ is a P-subspace of $\underline{X}$ and |Y| is a P-subspace of |X|. The notion of P-embedding in PPA is defined in the obvious way. If $i: X\rightarrow Y$ is a locally closed embedding in PPA then i induces an equivalence $\mathrm{Sh}(X_{{\acute{\mathrm{e}}\mathrm{t}} })\simeq \mathrm{Sh}((\underline{Y}, i(|X|))_{{\acute{\mathrm{e}}\mathrm{t}} })$ ([36, Cor. 2.3.8]). In particular if $i: X\rightarrow Y$ is a locally closed embedding of adic spaces then $\mathrm{Sh}(X_{{\acute{\mathrm{e}}\mathrm{t}} })\simeq \mathrm{Sh}((Y, i(|X|))_{{\acute{\mathrm{e}}\mathrm{t}} })$.

• (6) A morphism $f: X\rightarrow Y$ in PPA is finite if $\underline{f}$ is finite and |X| is closed in $\underline{f}^{-1}(|Y|)$. If $f: X\rightarrow Y$ is a finite morphism in PA, then $f_*: \mathrm{Sh}(X_{{\acute{\mathrm{e}}\mathrm{t}} })\rightarrow \mathrm{Sh}(Y_{{\acute{\mathrm{e}}\mathrm{t}} })$ is exact and commutes with any base change in PA $Y'\rightarrow Y$ ([36, Prop. 2.6.3]).

• (7) A geometric point (in the category PPA) is an object $S\in PA$ such that $\underline{S}$ is the adic spectrum of a separably algebraically closed affinoid field ([36, 1.1.5]) and $|S|=\{s\}$, where s is the closed point of $\underline{S}$ ([36, 1.1.6]). For a geometric point S, the functor $\Gamma (S,-)$ induces an equivalence $\mathrm{Sh}(S_{{\acute{\mathrm{e}}\mathrm{t}} })\simeq \mathrm{Sets}$. A geometric point of $X\in PPA$ is a morphism $u: S\rightarrow X$ in PPA, where S is a geometric point. The stalk of $F\in \mathrm{Sh}(X_{{\acute{\mathrm{e}}\mathrm{t}} })$ at S is then $F_{S}=\Gamma (S, u^*F)$. Somewhat more explicitly, $F_{S}\simeq \varinjlim _{(V,v)} F(V)$, the limit being over the cofiltered category $C_S$ of pairs (V, v), where V is étale over X and $v: S\rightarrow V$ is an X-morphism.

  The support of u is by definition $u(|S|)\in |X|$. Two geometric points with the same support yield isomorphic stalk functors. Moreover, each $x\in X$ induces a geometric point $\bar{x}$ of X with support x and the family of functors $\mathrm{Sh}(X_{{\acute{\mathrm{e}}\mathrm{t}} })\rightarrow \mathrm{Sets}, F\rightarrow F_{\bar{x}}$, for $x\in |X|$, is conservative [36, 2.5.5]. If $f: X\rightarrow Y$ is a morphism of analytic pseudo-adic spaces (i.e., $\underline{X}, \underline{Y}$ are analytic adic spaces) and f is of weakly finite type and quasi-separated, then, for any maximal point y of |Y| and any $F\in \mathrm{Sh}(X_{{\acute{\mathrm{e}}\mathrm{t}} })$, we have a natural isomorphism [36, Th. 2.6.2]

  $$\begin{aligned} (\mathrm {R} ^n f_* F)_{\bar{y}}\simeq H^n_{{\acute{\mathrm{e}}\mathrm{t}} }(X\times _Y \bar{y}, F). \end{aligned}$$

• (8) One can define (see [36, Sec. 2.5]), for each geometric point $\xi$ of $X\in PPA$, the strict localization $X(\xi )$ of X at $\xi$. It comes with an X-morphism $\xi \rightarrow X(\xi )$, and the isomorphism class of $X(\xi )$ as X-space depends only on the support of $\xi$. If $X\in PA$ and $\xi ,\xi '$ are geometric points of X, a specialization morphism $u:\xi \rightarrow \xi '$ is an X-morphism in PA $X(\xi )\rightarrow X(\xi ')$. It induces functorial maps $u^*(F): F_{\xi '}\rightarrow F_{\xi }$ for $F\in \mathrm{Sh}(X_{{\acute{\mathrm{e}}\mathrm{t}} })$, via the natural isomorphisms $\Gamma (X(\xi ), F|X(\xi ))\simeq F_{\xi }$ and $\Gamma (X(\xi '), F|X(\xi '))\simeq F_{\xi '}$.

1.2 Compactly supported cohomology

We survey Huber’s compactly supported étale cohomology and introduce continuous compactly supported étale cohomology.

1.2.1 Huber’s compactly supported étale cohomology

Huber defined compactly supported étale cohomology of analytic pseudo-adic spaces in [36, Ch. 5]; in [37] he extended this definition to $\ell$-adic sheaves. We will briefly recall its properties.

Fix a prime $\ell$. Let X be a taut separated pseudo-adic space locally of $^{+}$weakly finite type over C (i.e., over ${\text {Spa}} (C,{\mathscr {O}}_C)$). For $i\ge 0$, we set

$$\begin{aligned} H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X, {\mathbf{Z}} _{\ell }):=H^i\mathrm {R} \Gamma _{\mathrm{c,Hu}}(X_{{\acute{\mathrm{e}}\mathrm{t}} },( {\mathbf{Z}} /\ell ^n)_n),\quad H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X, {\mathbf{Q}}_\ell ) :=H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X, {\mathbf{Z}} _\ell )\otimes {\mathbf{Q}}_\ell , \end{aligned}$$

where the functor $\mathrm {R} \Gamma _{\mathrm{c,Hu}}$ is defined in the following way.

If X is partially proper, then it is the right derived functor of $\Gamma _{\mathrm{c,Hu}}$, i.e., of the left exact functor

$$\begin{aligned} \Gamma _\mathrm{c,Hu}: \mathrm{mod}(X_{{\acute{\mathrm{e}}\mathrm{t}} }- {\mathbf{Z}} _\ell ^{{\scriptscriptstyle \bullet }})\rightarrow \mathrm{mod}( {\mathbf{Z}} _\ell ),\quad (F_n)_n\mapsto \Gamma _c\left( X_{{\acute{\mathrm{e}}\mathrm{t}} },\varprojlim _n F_n\right) . \end{aligned}$$

Here $\mathrm{mod}(X_{{\acute{\mathrm{e}}\mathrm{t}} }- {\mathbf{Z}} ^{{\scriptscriptstyle \bullet }}_\ell )$ is the category of projective systems $(F_n)_n$ of ${\mathbf{Z}} _\ell$-modules on $X_{{\acute{\mathrm{e}}\mathrm{t}} }$ such that $p^nF_n=0$, $n\in {\mathbf{N}}$. Recall that, for an étale sheaf F, $\Gamma _c(X_{{\acute{\mathrm{e}}\mathrm{t}} },F)$ denotes the abelian group of global sections whose support is proper.

In general one sets

$$\begin{aligned} \mathrm {R} \Gamma _{\mathrm{c,Hu}}(X_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n)_n):=\mathrm {R} \Gamma _{\mathrm{c,Hu}}(\overline{X}_{{\acute{\mathrm{e}}\mathrm{t}} },(i_!F_n)_n), \end{aligned}$$

where $i:X\hookrightarrow \overline{X}$ is a locally closed embedding and $\overline{X}$ is partially proper. This definition is, of course, independent of the chosen partially proper compactification. We have

$$\begin{aligned} \Gamma _\mathrm{c,Hu}(X_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n)_n)=\{(s_n)_n\in \varprojlim _n\Gamma (X_{{\acute{\mathrm{e}}\mathrm{t}} },F_n)|\overline{\cup _n\mathrm{supp}(s_n)} \text { is proper}\}. \end{aligned}$$

We list the following properties:

• (1) If X is proper then

  $$\begin{aligned} \mathrm {R} \Gamma _{\mathrm{c,Hu}}(X_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n)_n)\simeq \mathrm {R} \Gamma (X_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n)_n). \end{aligned}$$

• (2) An isomorphism [37, Lemma 2.3] of exact functors from $D^+(\mathrm{mod}(X_{{\acute{\mathrm{e}}\mathrm{t}} }- {\mathbf{Z}} ^{{\scriptscriptstyle \bullet }}_\ell ))$ to $D^+(\mathrm{mod}( {\mathbf{Z}} _\ell ))$:

  $$\begin{aligned} \mathrm {R} \Gamma _\mathrm{c,Hu}=\mathrm {R} \Gamma _{!}\circ \mathrm {R} \pi _*, \end{aligned}$$

  for the discretization functor

  $$\begin{aligned} \pi _*: \mathrm{mod}(X_{{\acute{\mathrm{e}}\mathrm{t}} }- {\mathbf{Z}} ^{{\scriptscriptstyle \bullet }}_\ell )\rightarrow \mathrm{mod}(X_{{\acute{\mathrm{e}}\mathrm{t}} }- {\mathbf{Z}} _\ell ),\quad (F_n)_n\mapsto \varprojlim _nF_n, \end{aligned}$$

  and the functor

  $$\begin{aligned} \Gamma _{!}: \mathrm{mod}(X_{{\acute{\mathrm{e}}\mathrm{t}} }- {\mathbf{Z}} _\ell )\rightarrow \mathrm{mod}( {\mathbf{Z}} _\ell ),\quad F\mapsto \Gamma _{c}(X_{{\acute{\mathrm{e}}\mathrm{t}} },F). \end{aligned}$$

• (3) If X is quasi-compact, there is an exact sequence [37, Cor. 2.4]

  $$\begin{aligned} 0\rightarrow \mathrm {R} ^1\varprojlim _nH^{i-1}_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X,F_n)\rightarrow H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X,(F_n)_n)\rightarrow \varprojlim _nH^i_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X,F_n)\rightarrow 0 \end{aligned}$$

• (4) Let U be a taut open subspace of X, let $Z=X{\setminus } U$, and let $i: Z\hookrightarrow X$ be the inclusion. Assume that X, U are partially proper. Then we have a distinguished triangle

  $$\begin{aligned} \mathrm {R} \Gamma _{\mathrm{c,Hu}}(U_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n|U)_n) \rightarrow \mathrm {R} \Gamma _{\mathrm{c,Hu}}(X_{{\acute{\mathrm{e}}\mathrm{t}} }, (F_n)_n)\rightarrow \mathrm {R} \Gamma _{c}(Z_{{\acute{\mathrm{e}}\mathrm{t}} }, i^*\mathrm {R} \pi _*(F_n)_n) \end{aligned}$$

• (5) Let ${\mathbb U}$ be an open covering of X such that every $U \in {\mathbb U}$ is taut and, for every $U, V\in {\mathbb U}$, there exists a $W \in {\mathbb U}$ such that $U \cup V\subset W$. Then, the map

  $$\begin{aligned} \varinjlim _{U\in {\mathbb U}} H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}} (U,(F_n|U)_n)\rightarrow H^i_{{\acute{\mathrm{e}}\mathrm{t}} , \mathrm{c,Hu}} (X,(F_n)_n),\quad i\ge 0,\end{aligned}$$

  is an isomorphism [37, Prop. 2.1.].

• (6) Let X be adic and partially proper and let G be a locally profinite group acting continuously on X. Then $H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X, {\mathbf{Z}} /{\ell }^n)$, $i\ge 0$, is a smooth G-module [6, Cor. 7.8].

Let us now distinguish two cases.

(i) The case $\ell \ne p$. We can say more in this case.

• (1) If X is as at the beginning of this section and of finite type over C and if $(F_n)_n$ is a quasi-constructible ${\mathbf{Z}} _\ell ^{{\scriptscriptstyle \bullet }}$-module on $X_{{\acute{\mathrm{e}}\mathrm{t}} }$ then the natural map

  $$\begin{aligned} H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X,(F_n)_n)\rightarrow \varprojlim _n H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X,F_n) \end{aligned}$$

  is a bijection. Moreover, the projective system $(H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X,F_n))_n$ is $\ell$-adic and every $H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X,F_n)$ is a finitely generated ${\mathbf{Z}} _{\ell }$-module (hence also $H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X,(F_n)_n)$ is a finitely generated ${\mathbf{Z}} _{\ell }$-module).

• (2) If $X=Y^{{\text {ad}} }$, for a separated scheme Y of finite type over C, and if $(F_n)_n$ is a constructible ${\mathbf{Z}} _\ell ^{{\scriptscriptstyle \bullet }}$-module on $Y_{{\acute{\mathrm{e}}\mathrm{t}} }$, there is a natural isomorphism

  $$\begin{aligned} H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(Y,(F_n)_n){\mathop {\rightarrow }\limits ^{\sim }} H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X,(F_n)_n). \end{aligned}$$

• (3) Let X be adic and partially proper and let G be a locally profinite group, with an open pro-p subgroup, acting continuously on X. Let $(F_n)_n$ be a locally constant overconvergent ${\mathbf{Z}} _{\ell }^{{\scriptscriptstyle \bullet }}$-module equipped with a compatible discrete G-action (see [16, B.1.3] for the definition). Then $H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}(X,(F_n)_n)$, $i\ge 0$, is a smooth G-module [16, Prop. B.2.5], [22, 4.1.19].

(ii) The case $\ell = p$. In this case cohomology with compact support behaves very differently. We will discuss an example.

Example 7.1

Let ${\mathbb A}^1_C$ be the adic affine space of dimension 1; this is a period domain, with $G:={\mathbb G}_{m, {\mathbf{Q}}_p}\times {\mathbb G}_{m, {\mathbf{Q}}_p}$ the relevant reductive group [3, 5.3.1, 4.2.2]. We have the exact sequence

$$\begin{aligned} 0&\rightarrow H^1_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty },i^*_{\infty }\mathrm {R} \pi _*( {\mathbf{Z}} /p^n(1))_n) \rightarrow H^2_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}({\mathbb A}^1_C, {\mathbf{Z}} _p(1)) \rightarrow H^2_{{\acute{\mathrm{e}}\mathrm{t}} }({\mathbb P}^1_C, {\mathbf{Z}} _p(1)) \\&\rightarrow H^2_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty },i^*_{\infty }\mathrm {R} \pi _*( {\mathbf{Z}} /p^n(1))_n), \end{aligned}$$

where $i_{\infty }: x_{\infty }\hookrightarrow {\mathbb P}^1_C$ is the point at infinity. Picking the fundamental neighborhoods of $x_{\infty }$ consisting of closed balls, we compute easily that

$$\begin{aligned} i^*_{\infty }\mathrm {R} ^i\pi _*( {\mathbf{Z}} /p^n(1))_n\simeq {\left\{ \begin{array}{ll} {\mathbf{Z}} _p(1) &{} \text{ if } i=0,\\ \varinjlim _j (\varprojlim _nH^1_{{\acute{\mathrm{e}}\mathrm{t}} }(E(j), {\mathbf{Z}} /p^n(1))) &{} \text{ if } i=1,\\ 0 &{} \text{ if } i\ge 2, \end{array}\right. } \end{aligned}$$

where E(j) is the closed ball centered at $x_{\infty }$ and of radius $p^{-j}$. We used here the fact that $H^i_{{\acute{\mathrm{e}}\mathrm{t}} }(E(j), {\mathbf{Z}} /p^n(1))=0, i\ge 2$. Since $\mathrm{Pic}(E(j))=0$, the Kummer exact sequence implies that

$$\begin{aligned} H^1_{{\acute{\mathrm{e}}\mathrm{t}} }(E(j), {\mathbf{Z}} /p^n(1))\simeq C\{T^{-1}\}^*/C\{T^{-1}\}^{*p^n}. \end{aligned}$$

Hence $H^2_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty },i^*_{\infty }\mathrm {R} \pi _*( {\mathbf{Z}} /p^n(1))_n)=0$. We also claim that

$$\begin{aligned} H^1_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty },i^*_{\infty }\mathrm {R} \pi _*( {\mathbf{Z}} /p^n(1))_n) \simeq \varinjlim _j C\{(p^jT)^{-1}\}^{*{\wedge }}. \end{aligned}$$

Since $x_{\infty }$ is simply a geometric point (thus étale sheaves are acyclic on it), we have (using the local-global spectral sequence relating $H^i_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty }, H^j(K))$ and $H^{i+j}_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty }, K)$ for a complex of sheaves K, as well as the computation above)

$$\begin{aligned} H^1_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty },i^*_{\infty }\mathrm {R} \pi _*( {\mathbf{Z}} /p^n(1))_n)&\simeq H^0_{{\acute{\mathrm{e}}\mathrm{t}} }(x_{\infty }, i^*_{\infty }\mathrm {R} ^1\pi _*( {\mathbf{Z}} /p^n(1))_n) \simeq \varinjlim _j (\varprojlim _n H^1_{{\acute{\mathrm{e}}\mathrm{t}} }(E(j), {\mathbf{Z}} /p^n(1))) \\&\simeq \varinjlim _j (\varprojlim _n C\{T^{-1}\}^*/C\{T^{-1}\}^{*p^n}) \simeq \varinjlim _j C\{(p^jT)^{-1}\}^{*{\wedge }}. \end{aligned}$$

It follows that

$$\begin{aligned} H^2_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}({\mathbb A}^1_C, {\mathbf{Z}} _p(1))\simeq (\varinjlim _n C\{(p^nT)^{-1}\}^{*{\wedge }} )\oplus {\mathbf{Z}} _p. \end{aligned}$$

In the case $\ell \ne p$, the same computation gives ${\mathbf{Z}} _\ell$ as a result since $C\{(p^n T)^{-1}\}^{*{\wedge }}$ is $\ell$-divisible. Note that $C\{T^{-1}\}^*/C^*=1+T^{-1}{\mathfrak m}_C\{T^{-1}\}$ and that its image by the logarithm satisfies

$$\begin{aligned} p T^{-1}{\mathfrak m}_C\{T^{-1}\}\subset \log \big ( 1+T^{-1}{\mathfrak m}_C\{T^{-1}\}\big ) \subset (pT)^{-1}{\mathfrak m}_C\{(pT)^{-1}\}. \end{aligned}$$

One gets the same inclusions for the p-adic completion. Hence the above inductive limit is isomorphic, via the logarithm, to the inductive limit of the $(p^nT)^{-1}{\mathfrak m}_C\{(p^nT)^{-1}\}$ and so

$$\begin{aligned} H^2_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}({\mathbb A}^1_C, {\mathbf{Z}} _p(1))\simeq \big ({\mathscr {O}}_{{\mathbb P}^1,\infty }/C\big )\oplus {\mathbf{Z}} _p. \end{aligned}$$

Hence the $\ell$-adic compactly supported cohomology groups behave very differently in the cases $\ell =p$ and $\ell \ne p$, where $H^2_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}({\mathbb A}^1_C, {\mathbf{Z}} _{\ell }(1))\simeq {\mathbf{Z}} _{\ell }$. Note also that the action of $G( {\mathbf{Q}}_p)$ on $H^2_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c,Hu}}({\mathbb A}^1_C, {\mathbf{Z}} _p(1))$ is not smooth, contrary to the case $\ell \ne p$.

1.2.2 Continuous compactly supported étale cohomology

We will also study a different version of Huber’s compactly supported cohomology: For X as in Sect. 1, we define its (continuous) compactly supported cohomology by:

$$\begin{aligned} \mathrm {R} \Gamma _\mathrm{c}(X_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n)_n):= \mathrm {R} \varprojlim _n\mathrm {R} \Gamma _c(X_{{\acute{\mathrm{e}}\mathrm{t}} },F_n). \end{aligned}$$

We have

$$\begin{aligned} \Gamma _\mathrm{c}(X_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n)_n)=\left\{ (s_n)_n\in \varprojlim _n\Gamma (X_{{\acute{\mathrm{e}}\mathrm{t}} },F_n)|\mathrm{supp}(s_n) \text { is proper}\right\} . \end{aligned}$$

The following properties are obtained directly from the definition and the corresponding properties for the compactly supported cohomology of $F_n$’s.

• (1) There is an exact sequence

  $$\begin{aligned} 0\rightarrow \mathrm {R} ^1\varprojlim _nH^{i-1}_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X,F_n)\rightarrow H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c}}(X,(F_n)_n)\rightarrow \varprojlim _nH^i_{{\acute{\mathrm{e}}\mathrm{t}} ,c}(X,F_n)\rightarrow 0. \end{aligned}$$

• (2) Let U be a taut open subspace of X, let $Z=X{\setminus } U$, and let $i: Z\hookrightarrow X$ be the inclusion. Then we have a distinguished triangle

  $$\begin{aligned} \mathrm {R} \Gamma _\mathrm{c}(U_{{\acute{\mathrm{e}}\mathrm{t}} },(F_n|U)_n) \rightarrow \mathrm {R} \Gamma _\mathrm{c}(X_{{\acute{\mathrm{e}}\mathrm{t}} }, (F_n)_n)\rightarrow \mathrm {R} \Gamma _\mathrm{c}(Z_{{\acute{\mathrm{e}}\mathrm{t}} }, i^*(F_n)_n). \end{aligned}$$

To lighten the notation, for $i\ge 0$, we will set

$$\begin{aligned} H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c}}(X, {\mathbf{Z}} _p):=H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c}}(X,( {\mathbf{Z}} /p^n)_n),\quad H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c}}(X, {\mathbf{Q}}_p):=H^i_{{\acute{\mathrm{e}}\mathrm{t}} ,\mathrm{c}}(X, {\mathbf{Z}} _p)\otimes _{ {\mathbf{Z}} _p} {\mathbf{Q}}_p. \end{aligned}$$