The Nakayama functor and its completion for Gorenstein algebras. (Le foncteur de Nakayama et sa complétion pour les algèbres de Gorenstein.) (English. French summary) Zbl 1505.16020
Summary: 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.
MSC:
| 16G30 | Representations of orders, lattices, algebras over commutative rings |
| 13C60 | Module categories and commutative rings |
| 13D45 | Local cohomology and commutative rings |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 18G65 | Stable module categories |
Keywords:
Gorenstein algebra; Gorenstein-projective module; local duality; Nakayama functor; Serre duality; stable module categoryReferences:
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