We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer n M such that for all finitely generated Λ-modules N , if Ext i Λ (M, N ) = 0 for all i ≫ 0, then Ext i Λ (M, N ) = 0 for all i ≥ n M . Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.

Abstract. Let R be a local ring with maximal ideal m admitting a non-zero element a ∈ m for which the ideal (0 : a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M , with particular attention to the case when m 4 = 0. Let e denote the minimal number of generators of m. If R is Gorenstein with m 4 = 0 and e ≥ 3, we show that P R M (t) is rational with denominator H R (−t) = 1 − et + et 2 − t 3 , for each finitely generated R-module M . In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.

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).

We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative Noetherian local ring R and a reflexive R-module M such that Ext i R (M, R) = 0 for all i > 0, but Ext i R (M * , R) = 0 for all i > 0. (2000): 13D07. Mathematics Subject Classification

Let (R, m, k) be a commutative noetherian local ring with dualizing complex D R , normalized by Ext depth(R) R (k, D R )k. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative) k-algebras of finite rank, we conjecture that if Ext n R (D R , R) = 0 for all n > 0, then R is Gorenstein, and prove this in several significant cases.