Growth in the minimal injective resolution of a local ring

Abstract

Abstract. Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module Ext i R (k, R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that it is non-decreasing and grows exponentially if R is Golod, a non-trivial fiber product, or Teter, or if it has radical cube zero.

Applications and homological properties of local rings with decomposable maximal ideals

“…The next result summarizes some of the points of Fact 2.6. It is almost certainly well-known to the experts, in light of [5,17], Corollary 2.7. The following conditions on the local ring R are equivalent.…”

“…The rings of Theorems A and B have decomposable maximal ideals, which means that they are realized as non-trivial fiber products; see Fact 2.1 below. Such rings have very rigid homological properties, as one sees, e.g., in [5,16,17,21,23]. For instance, they satisfy (UAC), they have at most the two trivial semidualizing modules, they are G-regular, and they satisfy the Auslander-Reiten Conjecture.…”

Rings that are Homologically of Minimal Multiplicity

“…For rings of minimal multiplicity, it is straightforward to answer this question: reduce to the case where m 2 = 0 and show that β R i (ω R ) = (r 2 − 1)r i−1 for all i 1; here r is the Cohen-Macaulay type of R. (See Example 2.4 below.) This question has been answered in the affirmative for other classes of rings by Jorgensen and Leuschke [15] and Christensen, Striuli and Veliche [8]. These classes include the classes of Golod rings, rings with codimension at most 3, rings that are one link from a complete intersection, rings with m 3 = 0, Teter rings, and nontrivial fiber product rings.…”

Chains of semidualizing modules

“…There are several works which were motivated by the following questions of Huneke about the Bass numbers µ i R (R) = rank k (Ext i R (k, R)). For instance, see [3], [5], [12] and [15]. However, each of the following questions is still open in general.…”