Acyclicity Versus Total Acyclicity for Complexes over Noetherian Rings

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck's duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of acyclic complexes of projectives by its subcategory of totally acyclic complexes and the corresponding category consisting of injective modules are equivalent. A new characterization is provided for complexes in Auslander categories and in Bass categories of such rings.

2000 Mathematics Subject Classification: Primary: 16E05; Secondary: 13D05, 16E10, 18E30

Keywords and Phrases: Totally acyclic complex, dualizing complex, Gorenstein dimension, Auslander category, Bass category

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