Stability of abelian groups in a topos of sheaves

Abstract

Recall that a ring R is said to be left stable if the category R-Mod satisfies the following: for any left R-module A and any injective left R-module B, hom(A, B) = 0 implies that hom(E(A),L?) =0 where E(A) is the injective hull of A. Some examples of such rings are, all commutative rings which are either noetherian or perfect. In particular the ring .Z’ of integers is stable noetherian, that is, the category Ab of abelian groups is stable. For further details on stability the reader is referred to [7-10; 131. In this paper we are interested to investigate the notion of stability and related facts for the category AbSb y! of abelian groups in a topos of sheaves on a locale. We first show that stability is a local property (Proposition l.l), and for the smallest non-boolean locale LX = 3 we show that AbSh 3 is not stable (Example 1.2). We prove a number of propositions which lead us to the proof of Proposition 1.7 that if 0 : 9? + Jdt is an onto map of locales, then AbSh LZ? stable implies AbSh & is stable. For a T, space X, we show in Proposition 1.9 that AbSh X stable implies that X is a Ti space. For a finite g, we prove in Corollary 1 .lO that AbSh 61! is stable iff &Z? is boolean. It is shown in [4] that the ring Z, is noetherian iff .zZ has A.C.C. on its elements. In particular for .J.??= 3, Zg is a noetherian ring. Since an A E AbSh LZ? can be viewed as a module over the ring Zg, our Example 1.2 shows that a commutative noetherian ring in a topos Sh 2 is not necessarily stable.