Date of Award

5-12-2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Graham Leuschke

Keywords

Auslander's delta invariant;Cohen-Macaulay;Ding's conjecture;generalized graded length;generalized Loewy length;index

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

We characterize different classes of Cohen-Macaulay local rings (R,m, k) with positive Krull di?mension in terms of MCM approximations of finitely-generated R-modules. Assume R has a canonical module. For each finitely-generated R-module M, Auslander’s δ?invariant δR(M) equals the rank of a maximal free direct summand of the minimal MCM approxi?mation XM of M. We have δR(R/m) = 1 if and only if R is a regular local ring. Auslander defined the index of R, denoted index(R), as the infimum of positive integers n such that δR(R/mn ) = 1. When R is Gorenstein, we have index(R) ≤ gℓℓ(R) < ∞, where gℓℓ(R) denotes the generalized Loewy length of R, the smallest positive integer n such that mn ⊆ xR for some system of parameters x for R. We call such a system of parameters a witness to the generalized Loewy length of R. In Chapter 3, we generalize a theorem of Ding, who proved that if R is Gorenstein with infinite residue field k and Cohen-Macaulay associated graded ring grm(R), then gℓℓ(R) = index(R). We prove that if R is a one-dimensional Cohen-Macaulay local ring with finite index and nonzerodivi?sor x of order t with grm(R)-regular initial form x ∗ , then gℓℓ(R) ≤ index(R) +t −1. We use this estimate to derive a formula for the generalized Loewy length of a one-dimensional hypersurface R = kJx, yK/(f). If z is a witness to gℓℓ(R) such that z ∗ is grm(R)-regular, then gℓℓ(R) = ordR(z)+e(R)−1, where e(R) denotes the Hilbert-Samuel multiplicity of R. We compute the generalized Loewy lengths of several families {Rn} ∞ n=1 of one-dimensional hypersurfaces over finite and infinite fields such that gℓℓ(Rn) = index(Rn) for all n ≥ 1 or gℓℓ(Rn) = index(Rn) + 1 for all n ≥ 1. Lastly, we study a graded version of the generalized Loewy length of a Noetherian local ring for Noetherian k-algebras (R,m, k), where k is an arbitrary field and m is the irrelevant ideal of R. This invariant is called the generalized graded length of R and denoted ggl(R). After determining bounds for ggl(R) in terms of gℓℓ(R) and the degrees of generators for R, we compute the generalized graded length of numerical semigroup rings. We also characterize witnesses to the generalized graded length of numerical semigroup rings for semigroups with two generators. In Chapter 4, we study criteria for when an MCM module over a Gorenstein complete local ring R is stably isomorphic to an MCM approximation of a finitely-generated R-module of some fixed positive codimension r. If this condition holds for an MCM R-module M, we say with Kato that M satisfies the SCr-condition. If this condition holds for every MCM R-module, we say that R satisfies the SCr-condition. Only the SC1- and SC2- conditions have been characterized for Gorenstein complete local rings R. Kato proved that R satisfies the SC1-condition if and only if R is a domain, and R satisfies the SC2-condition if and only if R is a UFD. For rings of dimension d ≥ 3 and 3 ≤ r ≤ d, we prove an inductive criterion for when an MCM R-module satisfies the SCr-condition when its first syzygy module Ω1 R (M) satisfies the SCr−1-condition. We use this criterion to prove the equivalence of the SCd- and SCd−1-conditions for Gorenstein complete local rings of dimension d ≥ 3 that remain UFDs when factoring out certain regular sequences of length d −2.

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