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Injective Sheaves of Abelian Groups: A Counterexample

Published online by Cambridge University Press:  20 November 2018

B. Banaschewski
B. Banaschewski*
Affiliation:
McMaster University, Hamilton, Ontario
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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 V ⊆ U a group homomorphism, denoted s ⟿ s|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:AU → BU. In the following, AbShX will be the category with these A as objects and these h:A → B as maps (= morphisms).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1518 - 1521
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Dobbs, D. E., On characterizing injective sheaves, Can. J. Math. 24 (1977), 1031–1039.Google Scholar
2. Herrlich, H., Topologische Reflexionen und Coreflexionen, LNM 78 (Springer-Verlag, Berlin-Heidelberg-New York, 1968).CrossRefGoogle Scholar
3. Mitchell, B., Theory of categories (Academic Press, New York and London, 1976).Google Scholar
4. Pierce, R. S., Modules over commutative regular rings, Mem. AMS 70 (American Mathematical Society, Providence, Rhode Island, 1967).Google Scholar