Let be a paracompact Hausdorff space. A sheaf of groups over is fine if for every two disjoint closed subsets , , there is an endomorphism of the sheaf of groups which restricts to the identity in a neighborhood of and to the endomorphism in a neighborhood of . Every fine sheaf is soft.
A slightly different definition is given in Voisin, in Hodge Theory and Complex Algebraic Geometry I, (definition 4.35):
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.
A fine sheaf over is a sheaf of -modules, where is a sheaf of rings such that, for every open cover of , there is a partition of unity (where the sum is locally finite) subordinate to this covering.
A technical point: I infer from context that, for Voisin, being subordinate to means that, for each , there is an open set such that and . This is slightly stronger than requiring that . 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 . A partition of unity subordinate to the covering means as usual that for each there is such that . Thanks for the other correction.