Homological and homotopical aspects of Gorenstein flat modules and complexes relative to duality pairs

TL;DR

This paper explores the homological and homotopical properties of Gorenstein flat modules relative to duality pairs, establishing new closure properties, duality relations, and model structures in this context.

Contribution

It introduces a framework for Gorenstein flat modules relative to duality pairs, proving closure under extensions and establishing duality relations and model structures.

Findings

Relative Gorenstein flat modules are closed under extensions.

A Pontryagin duality relates these modules to Gorenstein injective modules.

Multiple recollements and derived adjunctions are established between homotopy categories.

Abstract

We study homological and homotopical aspects of Gorenstein flat modules over a ring with respect to a duality pair $(\mathcal{L,A})$. These modules are defined as cycles of exact chain complexes with components in $\mathcal{L}$ which remain exact after tensoring by objects in $\mathcal{A} \cap {}^\perp\mathcal{A} = \mathcal{A} \cap \Big( \bigcap_{i \in \mathbb{Z}_{> 0}} {\rm Ker}({\rm Ext}^i_{R^{\rm o}}(-,\mathcal{A})) \Big)$. In the case where $(\mathcal{L,A})$ is product closed and bicomplete (meaning in addition that $\mathcal{L}$ is closed under extensions, (co)products, $R \in \mathcal{L}$, $(\mathcal{A,L})$ is also a duality pair, and $\mathcal{A}$ is the right half of a hereditary complete cotorsion pair) we prove that these relative Gorenstein flat modules are closed under extensions, and that the corresponding Gorenstein flat dimension is well behaved in the sense that it recovers many of the properties and characterizations of its (absolute) Gorenstein flat counterpart (for instance, it can be described in terms of torsion functors). The latter in turn is a consequence of a Pontryagin duality relation that we show between these relative Gorenstein flat modules and certain Gorenstein injective modules relative to $\mathcal{A}$. We also find several hereditary and cofibrantly generated abelian model structures from these Gorenstein flat modules and complexes relative to $(\mathcal{L,A})$. At the level of chain complexes, we find three recollements between the homotopy categories of these model structures, along with several derived adjunctions connecting these recollements.