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Let $(X,\mathcal{O}_X)$ be a rigid $K$-space with a finite affinoid covering $(U_i)_{i\in I}$ such that any intersection of the $U_i$ is affinoid too.

Equipping the direct products with the maximum norm, the Čech complex associated to the covering is a complex of Banach $K$-vector spaces:

$0 \rightarrow \prod_{i\in I} \mathcal{O}_X(U_i) \rightarrow \prod_{i,j\in I}\mathcal{O}_X(U_i \cap U_j) \rightarrow \ldots$

Is it then in general true that the image of each boundary map is closed? If not are there necessary/sufficent conditions to get this property?

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    $\begingroup$ I am also very interested in this question. I asked a more general version here: mathoverflow.net/questions/338149/… $\endgroup$ Oct 8, 2019 at 13:33
  • $\begingroup$ I'd like to add that this true for $X$ beeing proper, but I'm interested in the case where proper is not needed. $\endgroup$
    – KKD
    Oct 9, 2019 at 9:01
  • $\begingroup$ Yes, in fact it suffices that $X$ is proper over some affinoid space, not necessarily $\mathrm{Sp}(K)$. $\endgroup$ Oct 9, 2019 at 10:51

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This is false in general. Let $X$ be a smooth separated quasicompact rigid space over $\mathbf{Q}_p$. Let $\mathcal{O}^+$ be the sheaf of power-bounded functions. Then the following three statements are actually equivalent:

1) The torsion subgroup of $H^1(X,\mathcal{O}^+)$ is killed by a power of p.

2) For any covering $\mathcal{U}$ of $X$ by finitely many affinoids, the torsion subgroup of $\check{H}^{1}_{\mathcal{U}}(X,\mathcal{O}^+)$ is killed by a power of $p$.

3) For some covering $\mathcal{U}$ of $X$ by finitely many affinoids, the torsion subgroup of $\check{H}^{1}_{\mathcal{U}}(X,\mathcal{O}^+)$ is killed by a power of $p$.

This is a straightforward exercise in playing with the Cech-to-derived functor spectral sequence, using critically a theorem of Bartenwerfer that $H^1(U,\mathcal{O}^+)$ is killed by a power of $p$ for any smooth affinoid $U$.

On the other hand, for a fixed covering $\mathcal{U}$ of $X$ by finitely many affinoids and some fixed $n$, consider the following statements.

4) The torsion subgroup of $\check{H}^{n}_{\mathcal{U}}(X,\mathcal{O}^+)$ is killed by a power of $p$.

5) The image of the differential $d^{n-1}_{\mathcal{U}}$ is closed.

Then 5) implies 4), essentially by the open mapping theorem. (It might actually be true that 4) and 5) are equivalent, but I didn't think it through carefully.)

So to find an example where 5) fails (for $n=1$), we just need to find some smooth separated quasicompact $X$ where 1) fails, i.e. where $H^1(X,\mathcal{O}^+)$ contains torsion of arbitrarily high $p$-power order. Such spaces do exist, although not obviously so: de Jong found an example of such an $X$ last year (at my request).

Here is a sketch of de Jong's construction. Take an elliptic curve $E / \mathbf{Z}_p$ with good reduction. Choose a line bundle $L$ on $E$ which is non-torsion in the Picard group but which is trivial modulo $p$. Let $Y \to E$ be the total space of $L$, and let $\mathfrak{Y}$ be the formal completion of $Y$ along $(p)$. Finally, let $X$ be the rigid generic fiber of $\mathfrak{Y}$. Then $H^1(X,\mathcal{O}^+)$ contains torsion of arbitrarily high $p$-power order. (Idea: Reduce to the fact that $H^1(Y,\mathcal{O})=\oplus_{n \geq 0} H^1(E, L^{\otimes n})$ contains torsion of arbitrary $p$-power. Details omitted.)

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  • $\begingroup$ Recenlty somebody told me something similar. Thats why I am back to your answer. So I got the impression that if the space is maybe not to special it could be true. Are there not to strong conditions on $X$ which guarantee the strictness resp. finite p-torsion? I think of $X$ being an admissible open subset in the rigid analytification of a partial flag variety with the property above. $\endgroup$
    – KKD
    Nov 26, 2020 at 11:38
  • $\begingroup$ After thinking again about your answer, I got unsure about your notation and then about your statements. First of all to which Čech complex complex belongs $d^{n-1}_{\mathcal{U}}$ to? How is the open mapping theorem used to say something about the torsion part? But in general you argue with the sheaf of power bounded functions, whereas I consider the structure sheaf. What is the link here? $\endgroup$
    – KKD
    Jun 1, 2021 at 10:44

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