Cohomology of p-adic Stein spaces

Abstract

We compute p-adic étale and pro-étale cohomologies of Drinfeld half-spaces. In the pro-étale case, the main input is a comparison theorem for p-adic Stein spaces; the cohomology groups involved here are much bigger than in the case of étale cohomology of algebraic varieties or proper analytic spaces considered in all previous works. In the étale case, the classical p-adic comparison theorems allow us to pass to a computation of integral differential forms cohomologies which can be done because the standard formal models of Drinfeld half-spaces are pro-ordinary and their differential forms are acyclic.

Alternative proof of Corollary 6.25

We present in this appendix an alternative proof of Corollary 6.25 (hence also of Proposition 6.23 which easily follows from it) that does not use the ordinarity of the truncated log-schemes $Y_s$.

Let $X/k^0$ be a fine log-scheme of Cartier type. Recall that we have the subsheaves

$$\begin{aligned} Z_{\infty }^j=\cap _{n\ge 0} Z_n^j, \quad B_{\infty }^j=\cup _{n\ge 0} B_n^j \end{aligned}$$

of $\Omega ^j=\Omega ^j_{X/k^0}$ (in what follows we will omit the subscripts in differentials if understood). Via the maps $C: Z_{n+1}^j\rightarrow Z_n^j$ (with kernels $B_n^j$), $Z_{\infty }^j$ is the sheaf of forms $\omega$ such that $dC^n \omega =0$ for all n. This sheaf is acted upon by the Cartier operator C, and we recover

$$\begin{aligned} B_{\infty }^j=\cup _{n\ge 0} (Z_{\infty }^j)^{C^n=0}, \quad \Omega ^j_{\log }=(Z_{\infty }^j)^{C=1}. \end{aligned}$$

The following result is proved in [40, 2.5.3] for classically smooth schemes. It holds most likely in much greater generality than the one stated below, but this will be sufficient for our purposes.

Lemma A.1

Assume that $X/k^0$ is semi-stable (with the induced log structure) and that k is algebraically closed. Then the natural map of étale sheaves

$$\begin{aligned} B_{\infty }^j\oplus (\Omega ^j_{\log }\otimes _{{\mathbf {F}}_p} k)\rightarrow Z_{\infty }^j \end{aligned}$$

is an isomorphism.

Proof

It suffices to show that, for X as above and affine, $B_{\infty }^j(X)\oplus (\Omega ^j_{\log }(X)\otimes _{{\mathbf {F}}_p} k)\rightarrow Z_{\infty }^j(X)$ is an isomorphism. Take an open dense subset $j: U\hookrightarrow X$ which is smooth over k. Then $\Omega ^i_{X}$ is a subsheaf of $j_*\Omega ^i_U$ and so $Z^i_{\infty ,X}$ is a subsheaf of $j_*Z^i_{\infty , U}$, giving an inclusion $Z^i_{\infty , X}(X)\subset Z^i_{\infty , U}(U)$. By a result of Raynaud [40, Prop. 2.5.2], $Z^i_{\infty , U}(U)$ is a union of finite dimensional k-vector spaces stable under C. We deduce that $Z^i_{\infty , X}(X)$ is also such a union.

The result follows now from the following basic result of semi-linear algebra (this is where the hypothesis that k is algebraically closed is crucial): if E is a finite dimensional k-vector space stable under C, then $E=E_{\mathrm{nilp}}\oplus E_{\mathrm{inv}}$, where $E_{\mathrm{nilp}}=\cup _{n} E^{C^n=0}$, $E_{\mathrm{inv}}=\cap _{n} C^n(E)$, and the natural map $E^{C=1}\otimes _{{\mathbf {F}}_p} k\rightarrow E_{\mathrm{inv}}$ is an isomorphism. $\square$

Proof of Corollary 6.25

(1) We prove this in several steps. We start with the case $i=0$ (the most delicate). By Lemma A.1,

$$\begin{aligned} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, Z_{\infty }^j)\simeq H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, B_{\infty }^j)\oplus H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log }\otimes _{{\mathbf {F}}_p} k). \end{aligned}$$

We need the following intermediate result:

Lemma A.2

We have $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, B_{\infty }^j)=0$.

Proof

We note that $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, B_n^j)=0$ for all n: because we have $B^j_n\simeq B^j_{n+1}/B^j_1$ this follows from Lemma 6.13. This however does not allow us to deduce formally our lemma because $B_{\infty }^j=\cup _{n} B_n^j$ and $\overline{Y}$ is not quasi-compact. Instead, we argue as follows: the formation of the sheaves $B_{\infty }^j$ being functorial, we have a natural map $\alpha : H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, B_{\infty }^j)\rightarrow \prod _{C\in F^0} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{C}, B_{\infty }^j)$. It suffices to prove that $\alpha$ is injective and that $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{T}, B_{\infty }^j)=0$. To prove the injectivity of $\alpha$, it suffices to embed both the domain and target of $\alpha$ in $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j)$ and $\prod _{C} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{C}, \Omega ^j)$, and to use the injectivity of the natural map $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j)\rightarrow \prod _{C} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{C}, \Omega ^j)$. Next, since $\overline{T}$ is quasi-compact,

$$\begin{aligned} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{T}, B_{\infty }^j)=\varinjlim _{n} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{T}, B_n^j)=0, \end{aligned}$$

the second equality being a consequence of Proposition 6.3 and Lemma 6.13 (as above, in the case of $\overline{Y}$). $\square$

Consider now the sequence of natural maps

$$\begin{aligned} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log }){\widehat{\otimes }}_{{{\mathbf {F}}}_p}\overline{k}{\mathop {\rightarrow }\limits ^{\sim }} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log }\otimes _{{\mathbf {F}}_p} \overline{k}){\mathop {\rightarrow }\limits ^{\sim }} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, Z_{\infty }^j) \rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j). \end{aligned}$$

The first map is clearly a topological isomorphism, the second one is a topological isomorphism by Lemma A.2. Hence it remains to show that the last map is a topological isomorphism as well. Or that all the natural maps $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y},Z^j_{n})\rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y},\Omega ^j)$, $n\ge 1$, are topological isomorphisms. But this was done in the proof of Lemma 6.20. This gives the desired result for $i=0$.

We prove next the result for $i>0$, i.e., that $H^i_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log })\widehat{\otimes }_{{\mathbf {F}}_p}\overline{k}=0$ for $i\ge 1$. We start with showing that $H^i_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log })=0$. The exact sequence

yield the exact sequence

and $H^i_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log })=0$ for $i>1$. It suffices therefore to prove that $1-C^{-1}$ is surjective on $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j)$. For this, write $A_s=H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, \Omega ^j_{\log })$. As we have already seen, we have an isomorphism

$$\begin{aligned} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j)\simeq \varprojlim _s (A_s\otimes _{{\mathbf {F}}_p} \overline{k})=H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log }){\widehat{\otimes }}_{{{\mathbf {F}}}_p}\overline{k}. \end{aligned}$$

We have $C^{-1}=\varprojlim _{s}(1\otimes \varphi )$ ($\varphi$ being the absolute Frobenius on $\overline{k}$; note that $C-1=0$ on $A_s$). To conclude that $1-C^{-1}$ is surjective on $H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j)$, it suffices to pass to the limit in the exact sequences

$$\begin{aligned} 0\rightarrow A_s\rightarrow A_s\otimes _{{\mathbf {F}}_p} \overline{k}{\mathop {\longrightarrow }\limits ^{1-\varphi }} A_s\otimes _{{\mathbf {F}}_p} \overline{k}\rightarrow 0, \end{aligned}$$

whose exactness is ensured by the Artin-Schreier sequence for $\overline{k}$ and the fact that $(A_s)_{s}$ is Mittag-Leffler.

This shows that $H^i_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log })=0$ for $i>0$. Choosing a basis $(e_{\lambda })_{\lambda \in I}$ of $\overline{k}$ over ${\mathbf {F}}_p$ we obtain an embedding

$$\begin{aligned} H^i_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log }){\widehat{\otimes }}_{{{\mathbf {F}}}_p}\overline{k}\subset \prod _{\lambda \in I} H^i_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log })=0, \end{aligned}$$

which finishes the proof of (1).

(2) We prove the claim for $W_n$ by induction on n, the case $n=1$ being part (1). We pass from n to $n+1$ using the strictly exact sequences

$$\begin{aligned}&0\rightarrow H^0(\overline{Y}, \Omega ^j){\mathop {\rightarrow }\limits ^{V^n}} H^0(\overline{Y}, W_{n+1}\Omega ^j)\rightarrow H^0(\overline{Y}, W_n\Omega ^j)\rightarrow 0,\\&0\rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log }){\widehat{\otimes }}_{{{\mathbf {F}}}_p}\overline{k}{\mathop {\rightarrow }\limits ^{V^n}}H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y},W_{n+1}\Omega ^j_{\log }){\widehat{\otimes }}_{ {{\mathbf {Z}}} /p^{n+1}}W_{n+1}(\overline{k})\\&\quad \quad \quad \rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y},W_{n}\Omega ^j_{\log }){\widehat{\otimes }}_{ {{\mathbf {Z}}} /p^n}W_n(\overline{k})\rightarrow 0, \end{aligned}$$

as well as the natural map between them. The first sequence is exact by Lemma 6.20. To show that the second sequence is exact, consider, as above, the exact sequences

$$\begin{aligned} 0\rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, \Omega ^j_{\log }){\mathop {\rightarrow }\limits ^{V^n}} H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, W_{n+1}\Omega ^j_{\log })\rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, W_n\Omega ^j_{\log })\rightarrow H^1_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, \Omega ^j_{\log }). \end{aligned}$$

Tensoring over ${\mathbf {Z}}$ by $W_{n+1}(\overline{k})$, we can rewrite them as

$$\begin{aligned} 0\rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, \Omega ^j_{\log })\otimes _{{\mathbf {F}}_p} \overline{k}&\rightarrow H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, W_{n+1}\Omega ^j_{\log })\otimes _{{\mathbf {Z}}/p^{n+1}} W_{n+1}(\overline{k})\\ \rightarrow&H^0_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, W_n\Omega ^j_{\log })\otimes _{{\mathbf {Z}}/p^n} W_n(\overline{k}) \rightarrow H^1_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}_s^{\circ }, \Omega ^j_{\log })\otimes _{{\mathbf {F}}_p} \overline{k}. \end{aligned}$$

To lighten the notation, write them simply as $0\rightarrow A_s\rightarrow B_s\rightarrow C_s\rightarrow D_s$. Using that $(A_s)_s, (C_s)_s$ are finite $W_n(\overline{k})$-modules and that $\varprojlim _s D_s= H^1_{\acute{\mathrm{e}}\mathrm {t}}(\overline{Y}, \Omega ^j_{\log }){\widehat{\otimes }}_{{{\mathbf {F}}}_p}\overline{k}=0$ (as follows from (1)), we obtain the exact sequence

$$\begin{aligned} 0\rightarrow \varprojlim _s A_s\rightarrow \varprojlim _s B_s\rightarrow \varprojlim _s C_s\rightarrow 0, \end{aligned}$$

which finishes the proof of (2) for $W_n$, $n\ge 1$. Passing to the limit over n gives us the proof for W. $\square$