Growth in the minimal injective resolution of a local ring

TL;DR

This paper investigates the growth behavior of Bass numbers in the minimal injective resolution of non-Gorenstein local rings, proving exponential growth under various algebraic conditions.

Contribution

It establishes that Bass numbers form a non-decreasing sequence and grow exponentially for certain classes of local rings, providing new insights into their invariants.

Findings

Bass numbers are non-decreasing in these rings.

Bass numbers grow exponentially if the ring is Golod, a fiber product, Teter, or has radical cube zero.

The growth behavior reveals new properties of the injective resolution structure.

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(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.