View source for Coherent analytic sheaf

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A coherent sheaf of $\mathcal O$ modules on an [[Analytic space|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 $Y$, [[#References|[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) [[#References|[2]]], [[#References|[4]]].

Coherent analytic sheaves were introduced in connection with problems in the theory of analytic functions on domains in $\mathbf C^n$ (see [[#References|[3]]], [[#References|[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|Kodaira theorem]]; [[Ample vector bundle|Ample vector bundle]]; [[Stein space|Stein space]]) as well as criteria for its finiteness and separability (see [[Finiteness theorems|Finiteness theorems]] in the theory of analytic spaces) play an important role in this theory.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) {{MR|0175897}} {{ZBL|0205.50401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian) {{MR|0463470}} {{ZBL|0334.32001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Cartan, "Idéaux et modules de fonctions analytiques de variables complexes" ''Bull. Soc. Math. France'' , '''78''' (1950) pp. 28–64 {{MR|0036848}} {{ZBL|0038.23703}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Oka, "Sur les fonctions analytiques de plusieurs variables (VII. Sur quelques notions arithmétiques)" ''Bull. Soc. Math. France'' , '''78''' (1950) pp. 1–27 {{MR|0035831}} {{ZBL|0036.05202}} </TD></TR></table>



====Comments====
See also [[Coherent sheaf|Coherent sheaf]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Coherent analytic sheaves" , Springer (1984) (Translated from German) {{MR|0755331}} {{ZBL|0537.32001}} </TD></TR></table>