Relative higher homology and representation theory

Abstract

Higher homological algebra, basically done in the framework of an n-cluster tilting subcategory $M$ of an abelian category $A$, has been the topic of several recent researches. In this paper, we study a relative version, in the sense of Auslander-Solberg, of the higher homological algebra. To this end, we consider an additive sub-bifunctor F of ${\mathrm{Ext}}_M^n(-,-)$ as the basis of our relative theory. This, in turn, specifies a collection of n-exact sequences in $M$, which allows us to delve into the relative higher homological algebra. Our results include a proof of the relative n-Auslander-Reiten duality formula, as well as an exploration of relative Grothendieck groups, among other results. As an application, we provide necessary and sufficient conditions for $M$ to be of finite type.