Induced complete hereditary cotorsion pairs in D(R) with respect to Cartan-Eilenberg exact sequences

Given a complete hereditary cotorsion pair (A,B) in ModR, we construct a complete hereditary cotorsion pair in the derived category D(R) of unbounded complexes with respect to the proper class ξ of cohomologically ghost triangles induced by the Cartan-Eilenberg exact sequences. More specifically, we prove that, each of the classes of projectively coresolved ξ-Gflat complexes PGF(ξ), ξ-Gflat complexes GF(ξ), ξ-Ginjective complexes GI(ξ), ξ-Gprojective complexes GP(ξ) (the last when R is virtually Gorenstein), forms one half of a complete hereditary cotorsion pair in D(R) with respect to ξ. Moreover, various homological dimensions offer additional way to obtain such cotorsion pairs in D(R) with respect to ξ.

💡 Research Summary

This paper presents a systematic study on lifting complete hereditary cotorsion pairs from the module category ModR to the unbounded derived category D(R) of complexes. The central theme is the use of a “proper class ξ of triangles” in D(R), specifically the class of cohomologically ghost triangles induced by Cartan-Eilenberg (CE) exact sequences, which provides a homological framework for D(R) that is richer than its standard triangulated structure but more amenable to module-theoretic techniques.

The introduction contextualizes the work within the historical development of cotorsion pair theory, its link to model category structures via Hovey’s correspondence, and the known fact that important classes like Gorenstein projective, injective, and flat modules form complete hereditary cotorsion pairs in ModR. The paper’s goal is to induce analogous structures in D(R).

The first major result, Theorem A, establishes a general lifting principle. It states that given any complete hereditary cotorsion pair (A, B) in ModR, one can construct a new pair (A_P, B_I) in D(R) with respect to the proper class ξ. Here, A_P consists of complexes in D(R) that admit a semi-projective resolution whose cokernels of differentials lie in A, and B_I consists of complexes that admit a semi-injective resolution whose kernels of differentials lie in B. The proof hinges on establishing equivalences between vanishing conditions for the derived functor Ext^1 in the category of complexes and the ξ-relative extension functor ξxt^1_ξ in D(R), leveraging properties of CE-exact sequences and resolutions.

The second major result, Theorem B (detailed in Corollaries 3.8 and 3.11), applies Theorem A to a host of specific, homologically significant classes. This yields a comprehensive list of new complete hereditary cotorsion pairs in (D(R), ξ). The classes include:

• Complexes of bounded ξ-projective, ξ-flat, and ξ-injective dimension (P(ξ)_n, F(ξ)_n, I(ξ)_n).
• The full classes of ξ-Gorenstein projective (GP(ξ)), ξ-Gorenstein injective (GI(ξ)), ξ-Gorenstein flat (GF(ξ)), and projectively coresolved ξ-Gorenstein flat (PGF(ξ)) complexes.
• Complexes of bounded ξ-Gorenstein dimensions (GP(ξ)_n, GI(ξ)_n, GF(ξ)_n, PGF(ξ)_n).

For each class, the paper explicitly describes its ξ-orthogonal complement, confirming the cotorsion pair structure. A notable condition is that for GP(ξ) to form such a pair, the base ring R must be virtually Gorenstein.

The paper’s methodology is noteworthy for operating within the framework of extriangulated categories—a unification of exact and triangulated categories. The proper class ξ endows D(R) with an extriangulated structure, and the authors develop the theory of cotorsion pairs in this general setting. The implications are significant: each complete hereditary cotorsion pair constructed can, via a theorem of Hu-Zhang-Zhou, be used to generate a model structure on the extriangulated category (D(R), ξ). Thus, the work not only deepens our understanding of the derived category’s homological algebra but also provides a powerful machine for generating a wealth of homotopical structures on D(R), bridging classical module theory, Gorenstein homological algebra, and abstract homotopy theory.