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A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $$\mathfrak{p}$$ -local and $$\mathfrak{p}$$ -torsion subcategories of the derived category, for each homogeneous prime ideal $$\mathfrak{p}$$ arising from the action of a commutative ring via Hochschild cohomology.
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The Nakayama functor and its completion for Gorenstein algebras
Duality properties are studied for a Gorenstein algebra that is finite and projective over its center. Using the homotopy category of injective modules, it is proved that there is a local duality theorem for the subcategory of acyclic complexes of such an algebra, akin to the local duality theorems of Grothendieck and Serre in the context of commutative algebra and algebraic geometry. A key ingredient is the Nakayama functor on the bounded derived category of a Gorenstein algebra and its extension to the full homotopy category of injective modules.
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- PAR ID:
- 10411808
- Date Published:
- Journal Name:
- Bulletin de la Société mathématique de France
- Volume:
- 150
- Issue:
- 2
- ISSN:
- 0037-9484
- Page Range / eLocation ID:
- 347-391
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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