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Localization and class groups of module categories with exactness defects

Published online by Cambridge University Press:  24 October 2008

D. Holland  and
S. M. J. Wilson
D. Holland
Affiliation:
Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, CanadaL8S 4K1
S. M. J. Wilson
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE
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Abstract

We present a new way of forming a grothendieck group with respect to exact sequences. A ‘defect’ is attached to each non-split sequence and the relation that would normally be derived from a collection of exact sequences is only effective if the (signed) sum of the corresponding defects is zero. The theory of the localization exact sequence and, in particular, of the relative group in this sequence is developed. The (‘locally free’) class group of a module category with exactness defect is defined and an idèlic formula for this is given. The role of torsion and of torsion-free modules is investigated. One aim of the work is to enhance the locally trivial, ‘class group’, invariants obtainable for a module while keeping to a minimum the local obstructions to the definition of such invariants.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 2 , March 1993 , pp. 233 - 251
Copyright
Copyright © Cambridge Philosophical Society 1993

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