Relative higher homology and representation theory

Higher homological algebra, basically done in the framework of anncluster 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 AuslanderSolberg, of the higher homological algebra. To this end, we consider an additive sub-bifunctor F of ExtnM(-, -) as the basis of our relative theory. This, in turn, specifies a collection ofnexact sequences in M, which allows us to delve into the relative higher homological algebra. Our results include a proof of the relative nAuslander-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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.