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Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension

Junpeng Wang, Sergio Estrada

Abstract

Let $R$ be a ring with Gwgldim$(R)<\infty$. We obtain a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{GProj})\simeq \mathrm{K}(R\text{-}\mathrm{GInj})$ which restricts to a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{Proj})$ $\simeq \mathrm{K}(R\text{-}\mathrm{Inj})$. This class of rings includes, among others, (left) Gorenstein rings, Ding-Chen rings and the more general Gorenstein $n$-coherent rings ($n\in \mathbb{N}\cup \{\infty\}, n\geq 2$). As application, we establish some triangle-equivalences of Grothendieck duality over Ding-Chen rings and Gorenstein $n$-coherent rings.

Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension

Abstract

Let be a ring with Gwgldim. We obtain a triangle-equivalence which restricts to a triangle-equivalence . This class of rings includes, among others, (left) Gorenstein rings, Ding-Chen rings and the more general Gorenstein -coherent rings (). As application, we establish some triangle-equivalences of Grothendieck duality over Ding-Chen rings and Gorenstein -coherent rings.
Paper Structure (13 sections, 42 theorems, 85 equations)

This paper contains 13 sections, 42 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cotorsion pairs.
  4. Duality pairs and relative Gorenstein rings
  5. Some basic notions of complexes.
  6. Gorenstein homological modules and dimensions
  7. dg-Gorenstein homological complexes
  8. Gorenstein derived categories
  9. Realizing relative derived categories as homotopy categories.
  10. Applying to rings with finite Gorenstein weak global dimension
  11. Homotopy equivalence on totally acyclic complexes
  12. Homotopy equivalence with respect to duality pairs
  13. Grothendieck duality

Key Result

Theorem 1.1

(=Chen2010). Let $R$ be a left Gorenstein ring. Then there is a triangle-equivalence $\hbox{\rm K}(R\text{-}\hbox{\rm GProj})\simeq \hbox{\rm K}(R\text{-}\hbox{\rm GInj})$ which restricts to a triangle-equivalence $\hbox{\rm K}(R\text{-}\hbox{\rm Proj})\simeq \hbox{\rm K}(R\text{-}\hbox{\rm Inj}).$

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 91 more