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Grade and Cohen-Macaulayness for DG-modules

Yuancheng Ning, Xiaoyan Yang

Abstract

We establish an inequality relating the projective dimension of a DG-module in $\mathrm{D}^\mathrm{b}_\mathrm{f}(A)$ to its grade and introduce the concept of perfect DG-modules as a natural generalization of perfect modules. It is proved that a DG-module $M$ over a local Cohen-Macaulay DG-ring with constant amplitude is Cohen-Macaulay if and only if $M$ is perfect and $\mathrm{amp}M \leq \mathrm{amp}\mathrm{R}Γ_{\bar{\mathfrak{m}}}(M)$. An affirmative answer is provided to Conjecture 2.11 of Yoshida [J. Pure Appl. Algebra 123 (1998) 313--326]. We also study the grade of DG-modules with finite injective dimension and examine the preservation of Cohen-Macaulayness under tensor products.

Grade and Cohen-Macaulayness for DG-modules

Abstract

We establish an inequality relating the projective dimension of a DG-module in to its grade and introduce the concept of perfect DG-modules as a natural generalization of perfect modules. It is proved that a DG-module over a local Cohen-Macaulay DG-ring with constant amplitude is Cohen-Macaulay if and only if is perfect and . An affirmative answer is provided to Conjecture 2.11 of Yoshida [J. Pure Appl. Algebra 123 (1998) 313--326]. We also study the grade of DG-modules with finite injective dimension and examine the preservation of Cohen-Macaulayness under tensor products.
Paper Structure (4 sections, 14 theorems, 15 equations)

This paper contains 4 sections, 14 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Perfect and local Cohen-Macaulay DG-modules
  4. Grade of DG-modules

Key Result

Lemma 2.1

(Mi19Y25ya20). $(1)$ Let $M\in\mathrm{D}^+(A),N\in\mathrm{D}^\mathrm{b}_\mathrm{f}(A)$ with $\mathrm{projdim}_AN<\infty$, then $(2)$ Let $M,N\in\mathrm{D}^{-}_{\mathrm{f}}(A)$ with $\mathrm{projdim}_AM<\infty$, one has an inequality $(3)$ Let $M\in\mathrm{D}^-_\mathrm{f}(A)$ with $\mathrm{projdim}_AM<\infty$, then $\mathrm{projdim}_AM=\mathrm{depth}A-\mathrm{depth}_AM$. $(4)$ Let $N\in\mathrm{D}

Theorems & Definitions (30)

  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • ...and 20 more