  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-Gorenstien 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, 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.
