# Growth in the minimal injective resolution of a local ring

@article{Christensen2010GrowthIT,
title={Growth in the minimal injective resolution of a local ring},
author={Lars Winther Christensen and Janet Striuli and Oana Veliche},
journal={Journal of the London Mathematical Society},
year={2010},
volume={81}
}
• Published 26 December 2008
• Mathematics
• Journal of the London Mathematical Society
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 ExtiR(k, R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that…
Local rings with quasi-decomposable maximal ideal
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2018
Abstract Let (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 is a
Rings that are Homologically of Minimal Multiplicity
• Mathematics
• 2009
Let R be a local Cohen–Macaulay ring with canonical module ω R . We investigate the following question of Huneke: If the sequence of Betti numbers has polynomial growth, must R be Gorenstein? This
Multiplicities and enumeration of semidualizing modules
• Mathematics
• 2010
A finitely generated module C over a commutative noetherian ring R is semidualizing if HomR(C, C) ∼= R and ExtR(C, C) = 0 for all i > 1. For certain local Cohen-Macaulay rings (R, m), we verify the
Multiplicities of Semidualizing Modules
• Mathematics
• 2010
A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom R(C, C) ≅ R and for all i ≥ 1. For certain local Cohen–Macaulay rings (R, 𝔪), we verify the equality of
Homological Invariants of Powers of Fiber Products
• Mathematics
Acta Mathematica Vietnamica
• 2019
Let R and S be polynomial rings of positive dimensions over a field k. Let I ⊆ R, J ⊆ S be non-zero homogeneous ideals none of which contains a linear form. Denote by F the fiber product of I and J
Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber product rings
• Mathematics
• 2022
. In this paper we explore consequences of the vanishing of Ext for ﬁnitely generated modules over a quasi-ﬁber product ring R ; that is, R is a local ring such that R/ ( x ) is a non-trivial ﬁber
Almost Gorenstein rings arising from fiber products
• Mathematics
• 2020
Abstract The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the
On reducing homological dimensions over Noetherian rings
• Mathematics
• 2020
Let $\Lambda$ be a left and right noetherian ring. First, for $m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given $\Lambda$-module to be $n$-torsionfree and have
On the initial Betti numbers
Let $R$ be a Cohen-Macaulay local ring possessing a canonical module. We compare the initial and terminal Betti numbers of modules in a series of nontrivial cases. We pay special attention to the

## References

SHOWING 1-10 OF 49 REFERENCES
Homology over local homomorphisms
• Mathematics
• 2003
The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism φ: R → S.
Injective dimension in Noetherian rings
Introduction. Among Noetherian rings quasi-Frobenius rings are those which are self injective [10, Theorem 18]. This paper is concerned primarily with Noetherian rings whose self injective dimension
On the growth of the Betti sequence of the canonical module
• Mathematics
• 2006
We study the growth of the Betti sequence of the canonical module of a Cohen–Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein.
Two applications of change of rings theorems for Poincaré series
• Mathematics
• 1979
Let (R, m, k) be an artinian Gorenstein ring with dim* m/m2 > 2. If PR denotes the Poincare-series and <f>Ä denotes the Bass-series of R, then <¡>K (PR l)x\l P'x2)' ' with R = A/0: m, see Proposition
Boundedness versus periodicity over commutative local rings
• Mathematics
• 1990
Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They
Rings that are almost Gorenstein
• Mathematics
• 2006
We introduce classes of rings that are close to being Gorenstein and prove they arise naturally as specializations of rings of countable CM type. We study these rings in detail and, inter alia,