Poincaré series of modules over compressed Gorenstein local rings

Abstract

Given two positive integers e and s we consider Gorenstein Artinian local rings R whose maximal ideal m satisfies m s = 0 = m s+1 and rank R/m (m/m 2 ) = e. 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 = 3, we prove that the Poincaré 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 Poincaré series exist, due to Bøgvad.2010 Mathematics Subject Classification. 13D02 (primary), 13A02, 13D07, 13H10 (secondary).

The structure of Gorenstein-linear resolutions of Artinian algebras

“…The rings A of the present paper satisfy the hypothesis of [14]; on the other hand, [14] also applies to non-graded rings and rings with odd socle degree. The paper [14] is about the Betti numbers in a resolution of M by free A-modules. The present paper is about the differentials in a resolution of A by free S-modules.…”

“…The connection is that, traditionally, one has learned about the Poincaré series of the A-module M by studying the S-resolution of A. So the present paper supplies information that [14] might have used if the information had been available. Furthermore, [14] pointed us in the direction of Fröberg's paper [7].…”

“…We think of the present paper as a companion to the recent paper of Rossi and Ş ega [14] which proves that the Poincaré series of a finitely generated module M over a compressed local Artinian ring A is rational, provided the socle degree of A is not 3. The rings A of the present paper satisfy the hypothesis of [14]; on the other hand, [14] also applies to non-graded rings and rings with odd socle degree.…”

“…So the present paper supplies information that [14] might have used if the information had been available. Furthermore, [14] pointed us in the direction of Fröberg's paper [7]. Fröberg may have had some insight into the skeleton of B.…”

The explicit minimal resolution constructed from a Macaulay inverse system

“…In some sense, this question asks for the generic resolution of standard-graded Artinian Gorenstein algebras with odd socle degree. Marilina Rossi and Liana Ş ega have recently written a remarkable paper [18] in which they prove that the Poincaré series of every finitely generated module over every local Artinian Gorenstein compressed algebra is rational provided the socle degree is not 3. is especially remarkable because so many of the usual tools for proving that Poincaré series are rational are not available to them. In particular, they do not know the minimal R-resolution of R/I and they do not know if the minimal R-resolution of R/I is an associative DG-algebra.…”