14 research outputs found
Homological Properties of Powers of the Maximal Ideal of a Local Ring
AbstractIt is known that the powers mn of the maximal ideal of a local Noetherian ring share certain homological properties for all sufficiently large integers n. For example, the natural homomorphisms R→R/mn are Golod, respectively, small, for all large n. We give effective bounds on the smallest integers n for which such properties begin to hold
Poincar\'e series of modules over compressed Gorenstein local rings
Given positive integers e and s we consider Gorenstein Artinian local rings R
of embedding dimension e whose maximal ideal satisfies
. We say that R is a compressed
Gorenstein local ring when it has maximal length among such rings. It is known
that generic Gorenstein Artinian algebras are compressed. If , we prove
that the Poincare series of all finitely generated modules over a compressed
Gorenstein local ring are rational, sharing a common denominator. A formula for
the denominator is given. When s is even this formula depends only on the
integers e and s. Note that for examples of compressed Gorenstein local
rings with transcendental Poincare series exist, due to B{\o}gvad.Comment: revised version, to appear in Adv. Mat
Quasi-complete intersection homomorphisms
Extending a notion defined for surjective maps by Blanco, Majadas, and
Rodicio, we introduce and study a class of homomorphisms of commutative
noetherian rings, which strictly contains the class of locally complete
intersection homomorphisms, while sharing many of its remarkable properties.Comment: Final version, to appear in the special issue of Pure and Applied
Mathematics Quarterly dedicated to Andrey Todorov. The material in the first
four sections has been reorganized and slightly expande
Self-tests for freeness over commutative artinian rings
AbstractWe prove that the Auslander–Reiten conjecture holds for commutative standard graded artinian algebras, in two situations: the first is under the assumption that the modules considered are graded and generated in a single degree. The second is under the assumption that the algebra is generic Gorenstein of socle degree 3