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fine sheaf

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Topos Theory

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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.

Discussion

David Speyer asks: Voisin, in Hodge Theory and Complex Algebraic Geometry I, definition 4.35 makes a different definition of fine sheaf. I can see that they are related, but I can’t see precisely what the relation is.

According to Voisin:

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.

A technical point: I infer from context that, for Voisin, being subordinate to U iU_i means that, for each U iU_i, there is an open set V iV_i such that X=U iV iX = U_i \cup V_i and f| V i=0f|_{V_i}=0. This is slightly stronger than requiring that f| XU i=0f|_{X \setminus U_i} =0. When working on a regular (T3) space, I believe that, if partitions of unity exist in the weaker sense, than they also exist in the stronger sense.

Zoran: paracompact Hausdorff space is automatically normal (Dieudonne’s theorem) so a fortiori T 3T_3. A partition of unity subordinate to the covering means as usual that for each ii there is jj such that suppf iU jsupp f_i \subset U_j. Thanks for the other correction.

category: sheaf theory

Revised on March 6, 2013 19:45:01 by Zoran Škoda (161.53.130.104)