Abelian subcategories closed under extensions: K-theory and decompositions

Details Abelian subcategories closed under extensions: K-theory and decompositions Abelian subcategories closed under extensions: K-theory and decompositions

2005-07-15

arxiv.org/abs/math/0507320v2

Communications in Algebra 35 (2007) 807-819

Mathematics

K-Theory and Homology

Improved presentation, fixed typos, to appear in Communications in Algebra, 12 Pages, 1 figure

Scientific paper

A full subcategory of modules over a commutative ring $R$ is wide if it is abelian and closed under extensions. Hovey \cite{wide} gave a classification of wide subcategories of finitely presented modules over regular coherent rings in terms of certain specialisation closed subsets of $\Spec(R)$. We use this classification theorem to study K-theory and Krull-Schmidt type decompositions for wide subcategories. It is shown that the K-group, in the sense of Grothendieck, of a wide subcategory $\W$ of finitely presented modules over a regular coherent ring is isomorphic to that of the thick subcategory of perfect complexes whose homology groups belong to $\W$. We also show that the wide subcategories of finitely generated modules over a noetherian regular ring can be decomposed uniquely into indecomposable ones. This result is then applied to obtain a decomposition for the K-groups of wide subcategories.

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