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Localization and class groups of module categories with exactness defects. (English) Zbl 0782.11031

From the authors’ 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.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29 Class numbers, class groups, discriminants
18F30 Grothendieck groups (category-theoretic aspects)
11R70 \(K\)-theory of global fields
16D90 Module categories in associative algebras
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References:

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