Growth in the minimal injective resolution of a local ring

Details Growth in the minimal injective resolution of a local ring Growth in the minimal injective resolution of a local ring

2008-12-26

arxiv.org/abs/0812.4672v2

Mathematics

Commutative Algebra

Final version, to appear in J. London Math. Soc.; 21 pp

Scientific paper

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.

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