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We study homological and homotopical aspects of Gorenstein flat modules overa ring with respect to a duality pair $(\mathcal{L,A})$. These modules aredefined 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 halfof a hereditary complete cotorsion pair) we prove that these relativeGorenstein flat modules are closed under extensions, and that the correspondingGorenstein flat dimension is well behaved in the sense that it recovers many ofthe properties and characterizations of its (absolute) Gorenstein flatcounterpart (for instance, it can be described in terms of torsion functors).The latter in turn is a consequence of a Pontryagin duality relation that weshow between these relative Gorenstein flat modules and certain Gorensteininjective modules relative to $\mathcal{A}$. We also find several hereditaryand cofibrantly generated abelian model structures from these Gorenstein flatmodules and complexes relative to $(\mathcal{L,A})$. At the level of chaincomplexes, we find three recollements between the homotopy categories of thesemodel structures, along with several derived adjunctions connecting theserecollements.

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