# Coherent analytic sheaf

A coherent sheaf of $\mathcal O$ modules on an analytic space $(X,\mathcal O)$. A space $(X,\mathcal O)$ is said to be coherent if $\mathcal O$ is a coherent sheaf of rings. Any analytic space over an algebraically closed field is coherent. The most important examples of a coherent analytic sheaf on such a space $(X,\mathcal O)$ are a locally free sheaf (that is, an analytic sheaf locally isomorphic to the sheaf $\mathcal O^p$) and also the sheaf of ideals of an analytic set $Y\subset X$, that is, the sheaf of germs of analytic functions equal to $0$ on $Y$, .
If $\mathcal F$ is a coherent analytic sheaf on a complex-analytic space $(X,\mathcal O)$, then the space of its sections, $\Gamma(X,\mathcal F)$, is endowed with a natural topology turning it into a Fréchet space when $X$ is separable. For $\mathcal F=\mathcal O$, this topology is the same as the topology of uniform convergence of analytic functions on compacta. In this case, $\mathcal F$ becomes a Fréchet sheaf, that is, for arbitrary open sets $U\subset V\subset X$ the restriction mapping $\Gamma(V,\mathcal F)\to\Gamma(U,\mathcal F)$ is continuous. An analytic homomorphism of coherent sheaves $\mathcal F\to\mathcal G$ induces a continuous linear mapping $\Gamma(X,\mathcal F)\to\Gamma(X,\mathcal G)$. If $\mathcal F$ is a coherent analytic sheaf on $X$ and $M$ is a submodule of $\mathcal F_x$, $x\in X$, then the submodule $\{s\in\Gamma(U,\mathcal F)\colon s(x)\in M\}$ is closed in $\Gamma(U,\mathcal F)$ for any neighbourhood $U$ of $x$. The cohomology spaces $H^p(X,\mathcal F)$ also have a natural topology, which is not, in general, separable for $p>0$ (they are quotient spaces of Fréchet spaces) , .
Coherent analytic sheaves were introduced in connection with problems in the theory of analytic functions on domains in $\mathbf C^n$ (see , ). Later they and their cohomology became a fundamental tool in the global theory of analytic spaces. Criteria for the vanishing of cohomology with values in a coherent analytic sheaf (cf. Kodaira theorem; Ample vector bundle; Stein space) as well as criteria for its finiteness and separability (see Finiteness theorems in the theory of analytic spaces) play an important role in this theory.