# Locally constant integer sheaf over riemann surface higher cohomology vanishes?

The reason I am asking this question is the following. Consider $$D$$ any divisor over Riemann surface $$X$$. Denote any $$U\subset X$$ biholomorphic to an open disk of $$C$$. Denote $$O_D=\{f\in M(U)|(f)+D\geq 0\}$$ where $$(f)$$ is the divisor of $$f$$ and $$M(X)$$ is meromorphic functions over $$U$$.

In Forster Lectures on Riemann Surfaces Exercise 16.3, it asks to show that $$H^1(U,O_D)=0$$ for any $$D$$ divisor which in turn implies open disk covering $$X$$ will form Leray covering against $$O_D$$ sheaf.

Now consider the sheaf exact sequence $$0\to Z\to O\xrightarrow{exp}O^\star\to 0$$ where $$O^\star$$ is the sheaf non-vanishing holomorphic functions. Take cohomology against $$U$$. One obtains $$0\to Z\to O(U)\to O^\star(U)\to H^1(U,Z)\to H^1(U,O)\to H^1(U,O^\star)\to H^2(U,Z)\to\dots$$

Now $$H^1(U,O)=0$$ by Dolbeault Lemma/Thm and $$H^1(U,Z)=0$$ by $$H^1(U,C)=0$$.(Actually, I think $$H^1$$ for any simply connected manifold vanishes for $$Z,C$$ sheafs as the proof involves only partition of unity, basic arithmetic and exponentiation operations.) I want to see whether $$H^1(U,O^\star)$$ vanishing.(i.e. I am asking whether open disks form Leray covering for $$O^\star$$.)

$$\textbf{Q:}$$ Does $$H^2(U,Z)=0$$ for $$U$$ disk where $$Z$$ is locally constant integer sheaf? I am asking whether $$U$$ is a Leray covering of $$O^\star$$.

$$\textbf{Q':}$$ Is there a characterization for what kind of sheaf over disk has trivial cohomology? Is this a topological characterization? I guess it is not as for smooth function sheaf, I need to assume smooth structure to deduce partition of unity which will show triviality of $$H^1$$ for sheaf of smooth functions.

• If $X$ is any simplicial complex then sheaf cohomology with constant coefficients is isomorphic to simplicial (or singular) cohomology (with the same coefficients). Hence, it vanishes above the dimension of the complex. – Moishe Kohan Jan 7 at 3:35
• @MoisheCohen Then it will indeed be the case. – user45765 Jan 7 at 3:36
• In topology, cohomology with coefficients in a constant sheaf is known as Chech cohomology. See for instance, Eilenberg and Steenrod "Algebraic Topology" for a proof of the isomorphism theorem I mentioned as well as for the proof that contractible spaces are acyclic (have zero reduced cohomology). – Moishe Kohan Jan 7 at 3:40
• @MoisheCohen Where do I find cech cohomology in algebraic topology reference, it is not in standard textbook? I have seen cech cohomology in the context of algebraic geometry for sheaf cohomology computation only. – user45765 Jan 7 at 3:41
• See Eilenberg and Steenrod. Or Bredon's book "Sheaf Theory", although the latter will cover much more than you need. Bredon covers sheaf cohomology from the general topology perspective. – Moishe Kohan Jan 7 at 3:43

While everything Moishe said in the comments is true, it's worth noting that you are taking cohomology of a sheaf over a contractible space, since you're interested here only in $$H^i(U, \mathbb{Z})$$, where $$U$$ is biholomorphic to a disk. Since $$U$$ is contractible, it has the cohomology of a point, and so all cohomologies vanish above degree $$0$$.