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On containment of trace ideals in ideals of finite homological dimension

Souvik Dey, Monalisa Dutta

Abstract

Motivated by recent result of Pérez and R.G. on equality of test ideal of module closure operation and trace ideal, and the well-known result by Smith that parameter test ideal cannot be contained in parameter ideals, we study the obstruction of containment of trace ideals in ideals of finite projective (or injective) dimension. One of our results says that the trace ideal of any big Cohen--Macaulay module over a Gorenstein complete local domain cannot be contained in any ideal of finite projective dimension, thereby generalizing Smith's result in this case. As consequences of our results, we give upper bounds on $\mathfrak m$-adic order of trace ideals of certain modules over local Cohen--Macaulay rings. We also prove analogous results for ideal of entries of maps in a free resolution of modules.

On containment of trace ideals in ideals of finite homological dimension

Abstract

Motivated by recent result of Pérez and R.G. on equality of test ideal of module closure operation and trace ideal, and the well-known result by Smith that parameter test ideal cannot be contained in parameter ideals, we study the obstruction of containment of trace ideals in ideals of finite projective (or injective) dimension. One of our results says that the trace ideal of any big Cohen--Macaulay module over a Gorenstein complete local domain cannot be contained in any ideal of finite projective dimension, thereby generalizing Smith's result in this case. As consequences of our results, we give upper bounds on -adic order of trace ideals of certain modules over local Cohen--Macaulay rings. We also prove analogous results for ideal of entries of maps in a free resolution of modules.
Paper Structure (7 sections, 47 theorems, 25 equations)

This paper contains 7 sections, 47 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. when are trace ideals not contained in ideals of finite projective or injective dimension?
  4. connections with ideals of entries in free resolutions
  5. order of trace ideals
  6. order of ideals generated by entries of maps in minimal free resolutions
  7. Some additional results on obstruction of containment of trace ideals

Key Result

Theorem 1.4

(hoc) Let $R$ be a complete local domain of characteristic $p > 0$, and let $N \subseteq M$ be finitely-generated $R$-modules. Then $N^*_M$, the tight closure of $N$ in $M$, is equal to the set of elements $u \in M$ that are in $N^{\operatorname{cl}_B}_M$ for some big Cohen-Macaulay algebra $B$.

Theorems & Definitions (95)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 85 more