coherent analytic sheaf

Let M be a complex manifold and be an analytic sheaf. For zM, denote by z the stalk of at z. By 𝒪 denote the sheaf of germs of analytic functionsMathworldPlanetmath. For a section f and a point zM denote by fz the germ of f at z.

is said to be locally finitely generatedMathworldPlanetmath if for every zM, there is a neighbourhood U of z, a finite number of sections f1,,fkΓ(U,) such that for each wU, w is a finitely generated module (as an 𝒪w-module).

Let U be a neighbourhood in M and Suppose that f1,,fk are sections in Γ(U,). Let (f1,,fk) be the subsheaf of 𝒪k over U consisting of the germs


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


is called a coherent analytic sheaf if is locally finitely generated and if for every open subset UM, and f1,,fkΓ(U,), the sheaf (f1,,fk) is locally finitely generated.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven 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