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

TL;DR

This paper introduces a new model structure on modules over Ding-Chen rings, characterizing fibrant modules via acyclic complexes of FP-injective modules and providing a novel perspective on the stable module category.

Contribution

It develops a new method for combining cotorsion pairs to create abelian model structures, specifically applied to Ding-Chen rings, and characterizes fibrant modules through acyclic complexes of FP-injective modules.

Findings

Fibrant objects are generated by weakly Ding injective modules.

Modules as cycles of acyclic complexes of FP-injective modules.

New description of the stable module category for Ding-Chen rings.

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.