International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 1, Pages 89-98

Abelian group in a topos of sheaves: torsion and essential extensions

Kiran R. Bhutani

Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA

Received 24 September 1987

Copyright © 1989 Kiran R. Bhutani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the properties of torsion groups and their essential extensions in the category AbShL of Abellan groups in a topos of sheaves on a locale. We show that every torsion group is a direct sum of its p-primary components and for a torsion group A, the group [A,B] is reduced for any Bε AbShL.. We give an example to show that in AbShL the torsion subgroup of an injective group need not be injective. Further we prove that if the locale is Boolean or finite then essential extensions of torsion groups are torsion. Finally we show that for a first countable hausdorff space X essential extensions of torsion groups in AbSh0(X) are torsion iff X is discrete.