Abelian subcategories closed under extensions: K-theory and decompositions

Abstract

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.