Models for chain homotopy category of relative acyclic complexes

TL;DR

This work studies relative acyclic complexes with respect to a balanced pair $(X,Y)$ in an abelian category, focusing on their chain- and derived-category realizations. It realizes the relative acyclic categories $K_{E-ac}(X)$ and $K_{E-ac}(Y)$ as the homotopy categories of hereditary model structures on the category of chain complexes, via the Hovey correspondence and complete cotorsion pairs. These realizations enable relative recollements of Krause and Neeman-Murfet, generalizing known decompositions in acyclic and derived settings. In the Gorenstein setting (rings with finite Gorenstein weak dimension) one obtains recollements $K_{E-ac}(GP)\to K(GP)\to D_{GP}(R)$ and $K_{E-ac}(GI)\to K(GI)\to D_{GI}(R)$ with $D_{GP}(R)=D_{GI}(R)$, the Gorenstein derived category. Overall, the paper provides a model-category framework for relative acyclicity and Gorenstein homological theory, enabling robust comparisons and constructions in relative derived contexts.

Abstract

Let $(\mathcal{X}, \mathcal{Y})$ be a balanced pair in an abelian category $\mathcal{A}$. Denote by ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X})$ the chain homotopy category of right $\mathcal{X}$-acyclic complexes with all items in $\mathcal{X}$, and dually by ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y})$ the chain homotopy category of left $\mathcal{Y}$-acyclic complexes with all items in $\mathcal{Y}$. We establish realizations of ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X})$ and ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y})$ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further give applications to Gorenstein projective and Gorenstein injective modules.

Theorems & Definitions (23)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Definition 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7