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G-levels of perfect complexes

Lars Winther Christensen, Antonia Kekkou, Justin Lyle, Zachary Nason, Andrew J. Soto Levins

Abstract

We prove that a commutative noetherian ring $R$ is Gorenstein of dimension at most $d$ if $d+1$ is an upper bound on the G-levels of perfect $R$-complexes. For $R$ local, we prove a formula for levels, with respect to injective or Gorenstein injective $R$-modules, of $R$-complexes with finitely generated homology; it mimics Bass' classic formula for injective dimension of finitely generated $R$-modules.

G-levels of perfect complexes

Abstract

We prove that a commutative noetherian ring is Gorenstein of dimension at most if is an upper bound on the G-levels of perfect -complexes. For local, we prove a formula for levels, with respect to injective or Gorenstein injective -modules, of -complexes with finitely generated homology; it mimics Bass' classic formula for injective dimension of finitely generated -modules.

Paper Structure

This paper contains 5 sections, 21 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. G-Levels of perfect complexes
  3. G-levels over Gorenstein rings
  4. Gorenstein injective levels
  5. Flat and Gorenstein flat levels

Key Result

Lemma 2.4

Let $M$ be a finitely generated $R$-module. If there exists an integer $b \ge1$ such that $\operatorname{Ext}_{R}^{n}(G,M) = 0$ holds for all $n \ge b$ and all $G$ in $\mathsf{G}(R)$, then $\operatorname{Ext}_{R}^{n}(G,M) = 0$ holds for all $n \ge 1$ and all $G$ in $\mathsf{G}(R)$. $\blacktrianglele

Theorems & Definitions (47)

  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Lemma 2.9
  • proof
  • Theorem 2.10
  • proof
  • Remark 2.11
  • ...and 37 more