# 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 $Y$, [1].

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) [2], [4].

Coherent analytic sheaves were introduced in connection with problems in the theory of analytic functions on domains in $\mathbf C^n$ (see [3], [5]). 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.

See also Vector bundle, analytic; Duality in the theory of analytic spaces.

#### References

 [1] S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) MR0175897 Zbl 0205.50401 [2] C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian) MR0463470 Zbl 0334.32001 [3] H. Cartan, "Idéaux et modules de fonctions analytiques de variables complexes" Bull. Soc. Math. France , 78 (1950) pp. 28–64 MR0036848 Zbl 0038.23703 [4] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601 [5] K. Oka, "Sur les fonctions analytiques de plusieurs variables (VII. Sur quelques notions arithmétiques)" Bull. Soc. Math. France , 78 (1950) pp. 1–27 MR0035831 Zbl 0036.05202