# Čech complex of rigid $K$-space - Closed image of boundary maps

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?

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

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.)

• 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.
– KKD
Nov 26, 2020 at 11:38
• 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?
– KKD
Jun 1, 2021 at 10:44