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

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.

💡 Research Summary

The paper introduces a novel method for merging two cotorsion pairs to produce a new abelian model structure, and applies this technique to left coherent rings, especially Ding‑Chen rings. The authors first develop a general construction (Theorems 3.2 and 3.4) that, given an existing abelian model structure (i.e., a Hovey triple (𝒬,𝒲,ℛ)) and a second complete cotorsion pair (𝒜,ℬ), yields a new model structure (𝒜,𝒲,ℬ). The trivial class 𝒲 is kept unchanged, while the cofibrant (or fibrant) objects are taken from the second cotorsion pair. This “cotorsion‑pair merging” provides a systematic way to create model structures with prescribed cofibrant or fibrant classes without altering the homotopy theory.

Specializing to a left coherent ring R, the authors consider the complete cotorsion pair (FP‑projective, FP‑injective) where FP‑injective modules are the absolutely pure modules. They define a class of “weakly Ding‑injective” modules wDI (studied by Iacob) and show that the fibrant objects of the resulting model structure are precisely the right orthogonal of wDI, denoted Z. In the Ding‑Chen setting—coherent rings with finite self‑FP‑injective dimension—the trivial class 𝒲 coincides with both the left orthogonal of Ding‑injectives and the right orthogonal of FP‑injectives, i.e., modules of finite flat (equivalently finite FP‑injective) dimension.

The main results for Ding‑Chen rings are collected in Theorem 1.1:

• Proposition 8.1 proves that Z equals the class of all cycles of acyclic complexes of FP‑injective modules, and any such complex is automatically Hom‑acyclic with respect to FP‑pro‑injective modules.
• The same proposition shows that the intersection FP‑projective ∩ Z coincides with the class GFP of Gorenstein FP‑pro‑injective modules; consequently every acyclic complex of FP‑pro‑injectives is totally acyclic.
• Theorem 8.3 establishes that the triple (FP‑projective, 𝒲, Z) is a cofibrantly generated abelian model structure, whose homotopy category is triangulated‑equivalent to the stable module category Stmod(R). Moreover, Stmod(R) can be realized as the stable category of the Frobenius category of GFP modules.
• Several equivalent characterizations of fibrant objects are given: (i) being a cycle of an FP‑injective acyclic complex, (ii) fitting into a short exact sequence 0→D→A→M→0 with A FP‑injective and D Ding‑injective, (iii) vanishing of Extⁱ_R(W,M) for all FP‑pro‑injective W and i≥1, and (iv) existence of a short exact sequence 0→M→D→A→0 that remains exact after applying Hom_R(M,–) for any FP‑pro‑injective M.
• Theorem 8.4 shows that (GF, Z) and (Z, GF) form complete duality pairs, mirroring the flat‑cotorsion duality in the Ding‑Chen context.

Consequently, the stable module category of a Ding‑Chen ring admits four equivalent descriptions: as the homotopy category of complexes of projective, injective, flat‑cotorsion, or FP‑pro‑injective modules, each yielding totally acyclic complexes. This unifies and extends earlier work on Ding‑projective/​injective modules, Gorenstein flat modules, and the flat‑cotorsion theory.

In a separate direction, the paper proves that any complete cotorsion pair—hereditary or not—can be lifted to an abelian model structure for the derived category D(R). Using the merging technique, the authors extend the construction of model structures on chain complexes (originally requiring hereditary cotorsion pairs) to the non‑hereditary case, thereby providing non‑trivial model structures whose homotopy categories are equivalent to D(R).

Overall, the work contributes (1) a versatile method for constructing new model structures from existing cotorsion data, (2) a comprehensive homological description of Ding‑Chen rings via FP‑injective complexes and Gorenstein FP‑pro‑injective modules, and (3) a resolution of the long‑standing question of whether non‑hereditary cotorsion pairs can yield meaningful derived‑category models. These results deepen the interplay between cotorsion theory, model category theory, and Gorenstein homological algebra.