Injective Sheaves of Abelian Groups: A Counterexample Academic Article uri icon

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abstract

  • It has been claimed that a sheaf of abelian groups on a Hausdorff space in which the compact open sets form a basis is injective in the category of all such sheaves whenever its group of global elements is divisible (Dobbs [1]). The purpose of this note is to present an optimal counterexample to this by showing, more generally, that on any nondiscrete T0-space there exists a sheaf of the type in question which is not injective.Recall that a sheaf A of abelian groups on a space X assigns to each open set U in X an abelian group AU and to each pair U, V of open sets in X such that VU a group homomorphism, denoted ss|V, satisfying the familiar sheaf conditions ([3, p. 246]) which make A a special type of contravariant functor from the category given by the inclusion relation between the open sets of X into the category Ab of abelian groups, and that a map between sheaves A and B of abelian groups is a natural transformation h:A → B, with component homomorphisms hu:AUBU. In the following, AbShX will be the category with these A as objects and these h:A → B as maps (= morphisms).

publication date

  • December 1, 1980