Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 18 / 3 / a / Solution

Order the index set of the open cover $\mathcal{U}=(U_i)$. For the sheaf of abelian groups $\mathcal{F}$, the Čech cochain group isIt is a product, not a finite-support sum. The Čech coboundary isEvery term in ${\delta }^{2}c$ occurs twice with opposite signs, so ${\delta }^2=0$. The Čech cohomology of the cover is ${\check{H}}^p(\mathcal{U},\mathcal{F})=\ker \delta /\mathrm{im}\delta$. Refinements induce maps on these groups; different choices of refinement indices give maps related by a cochain homotopy, hence the same map on cohomology. Taking the direct limit over covers givesOne may use locally finite covers, since a complex manifold is a paracompact space and these covers are cofinal. In degree zero this recovers the group of global sections by the sheaf gluing axiom.