Cech cohomology of holomorphic sections of a line bundle on finite good covers

Let $X$ be a Riemann surface and $E\to X$ a holomorphic line bundle (Riemann surfaces and holomorphic atlases, Holomorphic line bundles and meromorphic sections on a Riemann surface). For every open $U\subseteq X$, let ${\mathcal{O}}_X(E)(U)$ be the $\mathbb{C}$-vector space of holomorphic sections of $E{\mathrm{\mid }}_U$, with the usual restriction maps. Compatible local sections glue uniquely as sections of a bundle, so these groups form a sheaf of $\mathbb{C}$-vector spaces (A sheaf on a topological space, Presheaves and sheaves of groups, rings, and modules, Sections, restrictions, and global sections of a presheaf). Holomorphic local coefficients are smooth, so ${\mathcal{O}}_X(E)$ is a subsheaf of the sheaf of smooth sections (Subsheaves, Smooth sections, local sections, and support, Smoothness of a section is equivalent to smooth local components, Holomorphic functions are real analytic and smooth in their two real coordinates). When $E=\mathcal{O}(D)$, its global sections identify with $L(D)$ as in The holomorphic line bundle associated to a divisor; the general noncompact construction of $\mathcal{O}(D)$ uses ${\mathrm{A}\mathrm{C}}_{\omega }$, while its compact finite-cover construction is choice-free.

A finite good cover of $X$ is a finite indexed open cover $\mathfrak{U}=(U_0,\dots ,U_n)$ by holomorphic chart domains, each biholomorphic to a disc, such that every nonempty finite intersection of its members is also biholomorphic to a disc. This definition applies to a supplied finite good cover; it does not assert that every Riemann surface admits one.

For a supplied finite good cover $\mathfrak{U}$, define the ordered Čech cochain complex $C^{\bullet }(\mathfrak{U},{\mathcal{O}}_X(E))$ and its differential $\delta$ as in Ordered Čech cochain complex of a cover. Its degree-$p$ cocycles and coboundaries are $Z^p=\ker {\delta }^p$ and $B^p=\mathrm{im}{\delta }^{p-1}$, and the fixed-cover Čech cohomology is ${\check{H}}^p(\mathfrak{U},{\mathcal{O}}_X(E)):=Z^p\mathrm{/}{B}^p$ as in Fixed-cover Čech cohomology. In degree zero, restriction identifies ${\check{H}}^0(\mathfrak{U},{\mathcal{O}}_X(E))$ with $\mathrm{\Gamma }(X,E)$ (Čech H0 equals global sections).

If a finite good cover $\mathfrak{V}=(V_j)$ refines $\mathfrak{U}=(U_i)$ by a refinement function $c$ with $V_j\subseteq U_{c(j)}$, restriction defines a cochain map and hence a map on fixed-cover Čech cohomology (Refinement map of ordered open covers). The induced map on cohomology is independent of the chosen refinement function (Refinement choices induce the same Čech map). For a supplied pair of covers and refinement function, these Čech definitions use no Choice principle; the separate ${\mathrm{A}\mathrm{C}}_{\omega }$ input above pertains only to constructing the general noncompact divisor bundle.