Injective Sheaves of Abelian Groups: A Counterexample
Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1518-1521
Voir la notice de l'article provenant de la source Cambridge University Press
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 T 0-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, Ab ShX will be the category with these A as objects and these h:A → B as maps (= morphisms).
Banaschewski, B. Injective Sheaves of Abelian Groups: A Counterexample. Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1518-1521. doi: 10.4153/CJM-1980-119-1
@article{10_4153_CJM_1980_119_1,
author = {Banaschewski, B.},
title = {Injective {Sheaves} of {Abelian} {Groups:} {A} {Counterexample}},
journal = {Canadian journal of mathematics},
pages = {1518--1521},
year = {1980},
volume = {32},
number = {6},
doi = {10.4153/CJM-1980-119-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-119-1/}
}
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