Acyclic and totally acyclic objects in the $Q$-shaped derived category

Abstract

This paper concerns homological algebra based on $Q$-shaped diagrams, where $Q$ belongs to a certain class of small categories. We will show that the Gorenstein property of noetherian commutative rings is characterised by the condition that acyclicity coincides with total acyclicity for $Q$-shaped diagrams of suitable classes of modules. Many special cases occur as corollaries, for instance $N$-complexes. This generalises a classic result by Iyengar and Krause to the programme of $Q$-shaped derived categories, which builds on an insight of Iyama and Minamoto. We prove our result using an adjoint triple of functors relating $Q$-shaped diagrams to chain complexes. One of the functors was introduced by Jasso, and the whole triple generalises the compression, cocompression, and expansion functors for differential modules introduced by Avramov, Buchweitz, and Iyengar and by Nkansah. We consider the adjoint triple to be the main contribution of this paper.