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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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null (Ed.)Abstract It is proved that a map $${\varphi }\colon R\to S$$ of commutative Noetherian rings that is essentially of finite type and flat is locally complete intersection if and only if $$S$$ is proxy small as a bimodule. This means that the thick subcategory generated by $$S$$ as a module over the enveloping algebra $$S\otimes _RS$$ contains a perfect complex supported fully on the diagonal ideal. This is in the spirit of the classical result that $${\varphi }$$ is smooth if and only if $$S$$ is small as a bimodule; that is to say, it is itself equivalent to a perfect complex. The geometric analogue, dealing with maps between schemes, is also established. Applications include simpler proofs of factorization theorems for locally complete intersection maps.more » « less
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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.more » « less
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A duality theorem for the stable module category of representations of a finite group scheme is proved. 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 stable category, for each homogeneous prime ideal $$\mathfrak{p}$$ in the cohomology ring of the group scheme.more » « less
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