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