Coherent analytic sheaf
A coherent sheaf of modules on an analytic space . A space is said to be coherent if 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 are a locally free sheaf (that is, an analytic sheaf locally isomorphic to the sheaf ) and also the sheaf of ideals of an analytic set , that is, the sheaf of germs of analytic functions equal to on , .
If is a coherent analytic sheaf on a complex-analytic space , then the space of its sections, , is endowed with a natural topology turning it into a Fréchet space when is separable. For , this topology is the same as the topology of uniform convergence of analytic functions on compacta. In this case, becomes a Fréchet sheaf, that is, for arbitrary open sets the restriction mapping is continuous. An analytic homomorphism of coherent sheaves induces a continuous linear mapping . If is a coherent analytic sheaf on and is a submodule of , , then the submodule is closed in for any neighbourhood of . The cohomology spaces also have a natural topology, which is not, in general, separable for (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 (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.
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See also Coherent sheaf.
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Coherent analytic sheaf. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Coherent_analytic_sheaf&oldid=23788