Auslander–Gorenstein algebras from Serre-formal algebras via replication

Abstract

We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga–Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras – called the replicated algebras – given by certain subquotients of its repetitive algebra. We calculate the self-injective dimension and dominant dimension of all such replicated algebras and determine which of them are minimal Auslander–Gorenstein, i.e. when the two dimensions are finite and equal to each other. In particular, we show that there exist infinitely many minimal Auslander–Gorenstein algebras in such a series if, and only if, the Serre-formal algebra is twisted fractionally Calabi–Yau. We apply these results to a construction of algebras from Yamagata [29], called SGC extensions, given by iteratively taking the endomorphism ring of the smallest generator-cogenerator. We give a sufficient condition so that the SGC extensions and replicated algebras coincide. Consequently, in such a case, we obtain explicit formulae for the self-injective dimension and dominant dimension of the SGC extension algebras.