[Paper Review] Acyclicity versus total acyclicity for complexes over noetherian rings

Srikanth B. Iyengar, Henning Krause|ArXiv.org|Jun 14, 2005

Algebraic structures and combinatorial modelsMathematics19 references99 citations

TL;DR

This paper establishes an equivalence between the homotopy categories of projective and injective modules over a commutative noetherian ring with a dualizing complex, using the functor $ D igotimes_R - $, and proves that the quotients of acyclic complexes modulo totally acyclic complexes in both categories are equivalent and compactly generated. The key result identifies these quotients with the thick subcategory quotient $ \operatorname{Thick}(R,D)/\operatorname{Thick}(R) $, providing a new characterization of complexes in Auslander and Bass categories.

ABSTRACT

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.

Motivation & Objective

• To establish a triangulated equivalence between the homotopy categories of projective and injective modules over a commutative noetherian ring with a dualizing complex.
• To characterize the quotient categories of acyclic complexes modulo totally acyclic complexes in both the projective and injective settings.
• To provide a new characterization of complexes in the Auslander and Bass categories using this equivalence.
• To extend Grothendieck duality to the level of homotopy categories via derived functors and compact objects.

Proposed method

• Use of the functor $ D \otimes_R - $, where $ D $ is a dualizing complex, to construct a triangulated equivalence between $ \mathbf{K}(\operatorname{Prj} R) $ and $ \mathbf{K}(\operatorname{Inj} R) $.
• Leveraging the fact that $ D $ is a bounded complex of injectives and that direct sums of injectives are injective, to ensure the functor preserves triangulated structures.
• Establishing a quasi-inverse via the functor $ \mathsf{q} \circ \operatorname{Hom}_R(D, -) $, where $ \mathsf{q} $ is the quotient functor from flat to injective complexes.
• Using the compactness of objects in $ \mathbf{K}(\operatorname{Prj} R) $ and $ \mathbf{K}(\operatorname{Inj} R) $ to relate the equivalence to Grothendieck duality in $ \mathbf{D}^f(R) $.
• Analyzing the quotient categories $ \mathbf{K}_{\mathrm{ac}}(\operatorname{Prj} R)/\mathbf{K}_{\mathrm{tac}}(\operatorname{Prj} R) $ and $ \mathbf{K}_{\mathrm{ac}}(\operatorname{Inj} R)/\mathbf{K}_{\mathrm{tac}}(\operatorname{Inj} R) $ via thick subcategory quotients.
• Applying the theory of compact generation and localization to show that the quotient categories are compactly generated and equivalent to $ \operatorname{Thick}(R,D)/\operatorname{Thick}(R) $.

Experimental results

Research questions

• RQ1 Is there a triangulated equivalence between the homotopy categories of projective and injective modules over a noetherian ring with a dualizing complex?
• RQ2 How do the quotient categories of acyclic complexes modulo totally acyclic complexes in the projective and injective settings relate?
• RQ3 Can the Auslander and Bass categories be characterized via these quotient categories?
• RQ4 To what extent does the derived equivalence $ \mathbf{R}\operatorname{Hom}_R(-,D) $ extend to the homotopy category level?
• RQ5 What conditions ensure that a complex of injectives has finite G-injective dimension?

Key findings

• The functor $ D \otimes_R - $ induces a triangulated equivalence between $ \mathbf{K}(\operatorname{Prj} R) $ and $ \mathbf{K}(\operatorname{Inj} R) $, with quasi-inverse $ \mathsf{q} \circ \operatorname{Hom}_R(D, -) $.
• The quotient categories $ \mathbf{K}_{\mathrm{ac}}(\operatorname{Prj} R)/\mathbf{K}_{\mathrm{tac}}(\operatorname{Prj} R) $ and $ \mathbf{K}_{\mathrm{ac}}(\operatorname{Inj} R)/\mathbf{K}_{\mathrm{tac}}(\operatorname{Inj} R) $ are compactly generated and equivalent to $ \operatorname{Thick}(R,D)/\operatorname{Thick}(R) $.
• Complexes of finite G-injective dimension over $ R $ are precisely those in the Bass category $ \mathcal{B}(R) $, characterized by the existence of an exact triangle $ V \to Y \to T \to \Sigma V $ with $ V \in \operatorname{Loc}(D) $ and $ T $ totally acyclic.
• A complex $ Y $ of injectives has finite G-injective dimension if and only if $ \mathbf{R}\operatorname{Hom}_R(D,Y) $ is homologically bounded on the right, or equivalently, if $ \mathsf{S}(V) $ is in $ \operatorname{Loc}(R) $ for $ V \cong \mathsf{v}(Y) $.
• The equivalence $ \mathbf{K}(\operatorname{Prj} R) \simeq \mathbf{K}(\operatorname{Inj} R) $ recovers Grothendieck duality as a consequence of the compact object correspondence.
• The results extend to non-commutative settings via dualizing complexes over pairs of rings $ \langle S,R \rangle $, identifying G-projective and G-injective dimensions with objects in the Auslander and Bass categories.