Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohen-Macaulay approximations and the $\text{SC}_r$-condition

Richard F. Bartels

TL;DR

The paper links MCM approximations and FID hulls over CM local rings with canonical modules, focusing on generically Gorenstein rings to establish dualities for ideals between omega and R and to derive a key exact sequence 0 -> R -> omega^n -> X^{R/omega} -> 0. It introduces the SC_r-condition as a stable isomorphism class with minimal MCM approximations and proves an inductive criterion: if M is MCM and Omega_R^1(M) satisfies SC_{r-1}, and there exists L in CM^{r-1}(R) with X_L st-cong Omega_R^1(M) and a regular sequence x in Ann(L) such that R/xR is a UFD, then M satisfies SC_r. The results generalize Kato's SC_2/UFD characterization to higher r in Gorenstein complete local rings and relate SC_r to UFD properties after factoring by regular sequences. These contributions illuminate stable module structures and factoriality criteria in CM rings, with implications for singularity theory and CM representation type.

Abstract

We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring $R$ with canonical module, specifically when $R$ is generically Gorenstein. We then generalize a result of Kato, who proved that a Gorenstein complete local ring $R$ satisfies the $\text{SC}_{2}$-condition if and only if $R$ is a UFD. For $r \geq 3$, we prove a criterion for when an MCM $R$-module $M$ satisfies the $\text{SC}_{r}$-condition, assuming that its first syzygy $Ω_{R}^{1}(M)$ satisfies the $\text{SC}_{r-1}$-condition.

Cohen-Macaulay approximations and the $\text{SC}_r$-condition

TL;DR

The paper links MCM approximations and FID hulls over CM local rings with canonical modules, focusing on generically Gorenstein rings to establish dualities for ideals between omega and R and to derive a key exact sequence 0 -> R -> omega^n -> X^{R/omega} -> 0. It introduces the SC_r-condition as a stable isomorphism class with minimal MCM approximations and proves an inductive criterion: if M is MCM and Omega_R^1(M) satisfies SC_{r-1}, and there exists L in CM^{r-1}(R) with X_L st-cong Omega_R^1(M) and a regular sequence x in Ann(L) such that R/xR is a UFD, then M satisfies SC_r. The results generalize Kato's SC_2/UFD characterization to higher r in Gorenstein complete local rings and relate SC_r to UFD properties after factoring by regular sequences. These contributions illuminate stable module structures and factoriality criteria in CM rings, with implications for singularity theory and CM representation type.

Abstract

We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring with canonical module, specifically when is generically Gorenstein. We then generalize a result of Kato, who proved that a Gorenstein complete local ring satisfies the -condition if and only if is a UFD. For , we prove a criterion for when an MCM -module satisfies the -condition, assuming that its first syzygy satisfies the -condition.

Paper Structure

This paper contains 3 sections, 19 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. MCM approximations and FID hulls over generically Gorenstein rings
  3. $\text{SC}_r$-conditions for MCM modules

Key Result

Proposition 2.1

Ding90 Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with canonical module $\omega$. Let $M$ be a finitely-generated $R$-module. An arbitrary MCM approximation of $M$ can be written as follows for some non-negative integer $m$. Likewise, an arbitrary FID hull of $M$ can be written as follows for some non-negative integer $n$.

Theorems & Definitions (36)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • ...and 26 more