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 (2001) gave a classification of the wide subcategories of finitely presented modules over regular coherent rings in terms of certain specialization 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.