Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 113 / 4

A flasque sheaf has surjective restriction maps $\mathcal{F}(V)\to \mathcal{F}(U)$ for every pair of opens $U\subseteq V$. To prove the claim for an injective sheaf of modules, let $j_U:U\hookrightarrow X$ and $j_V:V\hookrightarrow X$. The natural mapis a monomorphism: its stalks are either the identity on ${\mathcal{O}}_{X,x}$, the map from zero to that stalk, or the zero-to-zero map. Here $j_{!}$ is extension by zero for module sheaves.

The extension-by-zero adjunction identifiesThe injective object property extends every morphism from $(j_U{)}_{!}{\mathcal{O}}_U$ to one from $(j_V{)}_{!}{\mathcal{O}}_V$. Under the displayed identification this is exactly surjectivity of $I(V)\to I(U)$. Injective module sheaves are therefore flasque. This argument works on an arbitrary ringed space, without Noetherian or separation assumptions.

For the closed inclusion $f$, the direct image sheaf has stalk ${\mathcal{F}}_x$ at $x\in X$ and zero stalk at points outside $X$: outside the closed set there is an open neighbourhood disjoint from $X$. Hence $f_{\ast }$ is exact, since exact sequences of sheaves are detected on stalks.

Take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Direct image preserves flasqueness, because restrictions are the restrictions on inverse-image opens. Exactness makes $f_{\ast }{\mathcal{I}}^{\bullet }$ a flasque resolution of $f_{\ast }\mathcal{F}$. MoreoverThe complexes computing sheaf cohomology are identical. ThereforeThis is sheaf cohomology under a closed inclusion; the closedness hypothesis is used in the exactness of direct image.

For a point $x\in X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$, with $i:\{x\}\hookrightarrow X$, isRestrictions are identities or the map to zero, so it is a flasque sheaf. The global sections equal $A$ and higher cohomology vanishes:No closed-point assumption is needed for this calculation of skyscraper sheaf cohomology. On an arbitrary topological space, its nonzero stalks are at points of $\overline{\{x\}}$ when $A\ne 0$; only for a closed point is the nonzero stalk confined to $x$.