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Acyclic complexes of FP-injective modules over Ding-Chen rings

James Gillespie

TL;DR

The work introduces a general method to merge two cotorsion pairs into an abelian model structure and applies it to left $R$-modules over a left coherent ring $R$, with fibrant objects generated by the weakly Ding injective class and characterized as cycles $Z_0X$ of acyclic complexes of $FP$-injective modules. In the Ding-Chen setting, these fibrant cycles coincide with cycles of acyclic FP-injectives, enabling a detailed description of the stable module category in terms of Gorenstein FP-pro-injective modules. The authors develop non-hereditary model constructions for derived categories by lifting non-hereditary cotorsion pairs to chain complexes, and they establish several dualities, including a complete duality between $\mathcal Z$ and Gorenstein flat modules via character modules. The results yield new abelian models and representations of derived and stable categories, extend Iacob’s weakly Ding injective framework, and connect FP-injective theory to broader Gorenstein homological structures, providing practical tools for studying Ding-Chen rings and their associated homotopy theories.

Abstract

We present a new method for combining two cotorsion pairs to obtain an abelian model structure and we apply it to construct and study a new model structure on left $R$-modules over a left coherent ring $R$. Its class of fibrant objects is generated by the weakly Ding injective $R$-modules, a class of modules recently studied by Iacob. We give several characterizations of the fibrant modules, one being that they are the cycle modules of certain acyclic complexes of FP-injective (i.e., absolutely pure) $R$-modules. In the case that $R$ is a Ding-Chen ring, we show that they are precisely the modules appearing as cycles of acyclic complexes of FP-injectives. This leads to a new description of the stable module category of a Ding-Chen ring $R$, by way of modules we call Gorenstein FP-pro-injective. These are modules that appear as a cycle module of a totally acyclic complex of FP-projective-injective modules. As a completely separate application of the new model category method, we show that all complete cotorsion pairs, even non-hereditary ones, lift to abelian models for the derived category of a ring.

Acyclic complexes of FP-injective modules over Ding-Chen rings

TL;DR

The work introduces a general method to merge two cotorsion pairs into an abelian model structure and applies it to left -modules over a left coherent ring , with fibrant objects generated by the weakly Ding injective class and characterized as cycles of acyclic complexes of -injective modules. In the Ding-Chen setting, these fibrant cycles coincide with cycles of acyclic FP-injectives, enabling a detailed description of the stable module category in terms of Gorenstein FP-pro-injective modules. The authors develop non-hereditary model constructions for derived categories by lifting non-hereditary cotorsion pairs to chain complexes, and they establish several dualities, including a complete duality between and Gorenstein flat modules via character modules. The results yield new abelian models and representations of derived and stable categories, extend Iacob’s weakly Ding injective framework, and connect FP-injective theory to broader Gorenstein homological structures, providing practical tools for studying Ding-Chen rings and their associated homotopy theories.

Abstract

We present a new method for combining two cotorsion pairs to obtain an abelian model structure and we apply it to construct and study a new model structure on left -modules over a left coherent ring . Its class of fibrant objects is generated by the weakly Ding injective -modules, a class of modules recently studied by Iacob. We give several characterizations of the fibrant modules, one being that they are the cycle modules of certain acyclic complexes of FP-injective (i.e., absolutely pure) -modules. In the case that is a Ding-Chen ring, we show that they are precisely the modules appearing as cycles of acyclic complexes of FP-injectives. This leads to a new description of the stable module category of a Ding-Chen ring , by way of modules we call Gorenstein FP-pro-injective. These are modules that appear as a cycle module of a totally acyclic complex of FP-projective-injective modules. As a completely separate application of the new model category method, we show that all complete cotorsion pairs, even non-hereditary ones, lift to abelian models for the derived category of a ring.
Paper Structure (15 sections, 24 theorems, 57 equations)

This paper contains 15 sections, 24 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cotorsion pairs and abelian model categories
  4. Covers and envelopes and duality pairs
  5. Hom-acyclicity terminology
  6. Ding-Chen rings
  7. Merging a cotorsion pair with an injective model structure
  8. Projective duals
  9. Existence of non-hereditary models for derived categories
  10. Bifibrant objects and totally acyclic complexes
  11. Weakly Gorenstein $\mathcal{B}$-injective modules
  12. The weakly Gorenstein $\mathcal{B}$-injective model structure
  13. The weakly Ding injective modules and model structure
  14. Acyclic complexes of FP-injectives over Ding-Chen rings
  15. Complete duality with Gorenstein flat modules

Key Result

Theorem 1.1

Let $R$ be a Ding-Chen ring. Then each of the following hold.

Theorems & Definitions (52)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Theorem 3.2: New models from injective ones
  • Remark 3.3
  • proof
  • Theorem 3.4: Hereditary condition and fibrant objects
  • proof
  • Theorem 3.5: New models from projective ones
  • Theorem 3.6: Hereditary condition and cofibrant objects
  • ...and 42 more