coherent analytic sheaf


Let M be a complex manifold and β„± be an analytic sheaf. For z∈M, denote by β„±z the stalk of β„± at z. By π’ͺ denote the sheaf of germs of analytic functionsMathworldPlanetmath. For a section f and a point z∈M denote by fz the germ of f at z.

β„± is said to be locally finitely generatedMathworldPlanetmath if for every z∈M, there is a neighbourhood U of z, a finite number of sections f1,…,fkβˆˆΞ“β’(U,β„±) such that for each w∈U, β„±w is a finitely generated module (as an π’ͺw-module).

Let U be a neighbourhood in M and Suppose that f1,…,fk are sections in Γ⁒(U,β„±). Let ℛ⁒(f1,…,fk) be the subsheaf of π’ͺk over U consisting of the germs

{(g1,…,gk)∈π’ͺzkβˆ£βˆ‘j=1kgj⁒(fj)z=0}.

ℛ⁒(f1,…,fk) is called the sheaf of relations.

Definition.

β„± is called a coherent analytic sheaf if β„± is locally finitely generated and if for every open subset UβŠ‚M, and f1,…,fkβˆˆΞ“β’(U,β„±), the sheaf ℛ⁒(f1,…,fk) is locally finitely generated.

References

  • 1 Lars HΓΆrmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 StevenΒ G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title coherent analytic sheaf
Canonical name CoherentAnalyticSheaf
Date of creation 2013-03-22 17:39:05
Last modified on 2013-03-22 17:39:05
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 4
Author jirka (4157)
Entry type Definition
Classification msc 32C35
Defines locally finitely generated sheaf
Defines sheaf of relations