Let [Formula: see text] be a ring (not necessarily commutative) and [Formula: see text] a bi-complete duality pair. We investigate the notions of (flat-typed) [Formula: see text]-Gorenstein rings, which unify Iwanaga–Gorenstein rings, Ding–Chen rings, AC-Gorenstein rings and Gorenstein [Formula: see text]-coherent rings. Using an abelian model category approach, we show that [Formula: see text] and [Formula: see text], the homotopy categories of all exact complexes of projective and injective [Formula: see text]-modules, are triangulated equivalent whenever [Formula: see text] is a flat-typed [Formula: see text]-Gorenstein ring.
In this paper complexes with N -nilpotent differentials are considered. We proceed by generalizing a defining property of injective and projective resolutions to define dg-injective and dg-projective N -complexes, and construct dg-injective and dgprojective resolutions for arbitrary N -complexes. As applications of these results, we prove that the category D N (R) is compactly generated, the category K N (I ) of injectives is compactly generated whenever R is left noetherian, and the category K N (P) of projectives is compactly generated whenever R is a right coherent ring for which every flat left R-module has finite projective dimension. We also establish a recollement of the category K N (R) relative to K ex N (R) and D N (R).
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