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topos theory

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## Definition

Let $X$ be a paracompact Hausdorff space. A sheaf $F$ of groups over $X$ is fine if for every two disjoint closed subsets $A,B\subset X$, $A\cap B = \emptyset$, there is an endomorphism of the sheaf of groups $F\to F$ which restricts to the identity in a neighborhood of $A$ and to the $0$ endomorphism in a neighborhood of $B$. 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 $X$ is a sheaf of $\mathcal{A}$-modules, where $\mathcal{A}$ is a sheaf of rings such that, for every open cover $U_i$ of $X$, there is a partition of unity $1 = \sum f_i$ (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 $X$ is a sheaf of $\mathcal{A}$-modules, where $\mathcal{A}$ is a sheaf of rings such that, for every open cover $U_i$ of $X$, there is a partition of unity $1 = \sum f_i$ (where the sum is locally finite) subordinate to this covering.

A technical point: I infer from context that, for Voisin, being subordinate to $U_i$ means that, for each $U_i$, there is an open set $V_i$ such that $X = U_i \cup V_i$ and $f|_{V_i}=0$. This is slightly stronger than requiring that $f|_{X \setminus U_i} =0$. 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 $T_3$. A partition of unity subordinate to the covering means as usual that for each $i$ there is $j$ such that $supp f_i \subset U_j$. Thanks for the other correction.

category: sheaf theory

Last revised on March 6, 2013 at 19:45:01. See the history of this page for a list of all contributions to it.