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fine sheaf

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Definition

Let XX be a paracompact Hausdorff space. A sheaf FF of groups over XX is fine if for every two disjoint closed subsets A,BXA,B\subset X, AB=A\cap B = \emptyset, there is an endomorphism of the sheaf of groups FFF\to F which restricts to the identity in a neighborhood of AA and to the 00 endomorphism in a neighborhood of BB. Every fine sheaf is soft.

A slightly different definition is given in Voisin, in Hodge Theory and Complex Algebraic Geometry I (Definition 4.35):

A fine sheaf \mathcal{F} over XX is a sheaf of 𝒜\mathcal{A}-modules, where 𝒜\mathcal{A} is a sheaf of rings such that, for every open cover U iU_i of XX, there is a partition of unity 1=f i1 = \sum f_i (where the sum is locally finite) subordinate to this covering.

Another definition is given by Godement in Topologie algébrique et théorie des faisceaux (the paragraph after Theorem II.3.7.2): a sheaf FF is fine if the internal hom Hom(F,F)Hom(F,F) is a soft sheaf.

category: sheaf theory

Last revised on March 6, 2020 at 13:15:06. See the history of this page for a list of all contributions to it.