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
Aof abelian groups on a space Xassigns to each open set Uin Xan abelian group AUand to each pair U, Vof open sets in Xsuch that V⊆ Ua group homomorphism, denoted s⟿ s|V, satisfying the familiar sheaf conditions ([ 3, p. 246]) which make Aa special type of contravariant functor from the category given by the inclusion relation between the open sets of Xinto the category Abof abelian groups, and that a map between sheaves Aand Bof abelian groups is a natural transformation h:A → B, with component homomorphisms hu: AU→ BU. In the following, Ab ShXwill be the category with these Aas objects and these h: A → Bas maps (= morphisms).