Grothendieck groups arising from contravariantly finite subcategories

Abstract

The subject of the paper is the study of the relative homological properties of a given additive category C in relation to a given contravariantly finite subcategory X in C under the assumption that any X-epic has a kernel in C. We introduce the notion of the Grothendieck group relative to the pair (C, X) and also that of the Cartan map cx relative to (C, X) and we show that the cokernel of cx is isomorphic to the corresponding Grothendieck group of the stable category C/Jx We also show that if the right x-dimension of C is finite, then cx is an isomorphism. In case C is a finite dimensional k-additive Krull-Schmidt category, we introduce the notion of the x-dimension vector of an object of C. We give criteria for when an indecomposable object is determined, up to isomorphism, by its x-dimension vector.