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 $\mathfrak{m}$ satisfies $\mathfrak{m}^s\ne 0=\mathfrak{m}^{s+1}$. 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 $s\ne 3$, 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 $s=3$ 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