nLab fine sheaf



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In higher category theory




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 2002 (Vol I, Def. 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 1958 (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.


  • For any fine sheaf JJ over a paracompact Hausdorff space MM, the sheaf cohomology groups of positive degree jj are all trivial
H j(M,J)=0. H^j(M,J) = 0.
0SJ 0J 1 0\to S\to J_0\to J_1\to \cdots

These resolutions can be used to compute the sheaf cohomology groups of SS in terms of the induced maps of sections

d i *:Γ(M,J i)Γ(M,J i+1). d_i^*: \Gamma(M,J_i)\to\Gamma(M,J_{i+1}).


H q(M,S)(kerd q *)/(imd q1 *) H^q(M,S) \cong (\text{ker}d^*_{q})/(\text{im}d^*_{q-1})

(see Section 3 of Gunning (1966).).


category: sheaf theory

Last revised on October 18, 2023 at 05:51:11. See the history of this page for a list of all contributions to it.