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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.

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    $\begingroup$ 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. $\endgroup$ – Moishe Kohan Jan 7 at 3:35
  • $\begingroup$ @MoisheCohen Then it will indeed be the case. $\endgroup$ – user45765 Jan 7 at 3:36
  • $\begingroup$ 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). $\endgroup$ – Moishe Kohan Jan 7 at 3:40
  • $\begingroup$ @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. $\endgroup$ – user45765 Jan 7 at 3:41
  • $\begingroup$ 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. $\endgroup$ – Moishe Kohan Jan 7 at 3:43
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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 $0$.

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