Koszul duality

doi:10.1016/0393-0440(88)90028-9

Journal of Geometry and Physics

Volume 5, Issue 3, 1988, Pages 317-350

Dedicated to I.M. Gelfand on his 75th birthday

https://doi.org/10.1016/0393-0440(88)90028-9 Get rights and content

Abstract

This paper tries to describe a natural framework for the canonical equivalence between derived categories of graded modules over symmetric and exterior algebra, which has been established by J.N. Bernstein, I.M. Gelfand and S.I. Gelfand.

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The Koszul rings in the sense of [BGS1] will be called sometimes classically Koszul later in this paper. The concept of Koszul algebras has been generalized to higher Koszul algebras [Ber,GMMZ,HY,Lü,LHL] and to categories [Man,BGS2,MOS], which are not the topics in this paper. A linear projective resolution of a graded module M can be characterized by the property of (the radical filtration of) syzygies of M [GM1, Proposition 3.1 and Lemma 5.1] (see Proposition 3.2).
We define generalized Koszul modules and rings and develop a generalized Koszul theory for $N$ -graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martinéz-Villa. Let A be a left finite $N$ -graded ring generated in degree 1 with $A_{0}$ noetherian semiperfect, J be its graded Jacobson radical. By the Koszul dual of A we mean the Yoneda Ext ring ${\underline{Ext}}_{A}^{•} (A / J, A / J)$ . If A is a generalized Koszul ring and M is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of A is the associated graded ring $G r_{J} A$ and the Koszul dual of the Koszul dual of M is the associated graded module $G r_{J} M$ . If A is a locally finite algebra, then the following statements are proved to be equivalent: A is generalized Koszul; the Koszul dual ${\underline{Ext}}_{A}^{•} (A / J, A / J)$ of A is (classically) Koszul; $G r_{J} A$ is (classically) Koszul; the opposite ring $A^{o p}$ of A is generalized Koszul. As an application, it is proved that if A is generalized Koszul with finite global dimension then A is generalized AS regular if and only if the Koszul dual of A is self-injective.
Reduction techniques of singular equivalences
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Ladders of recollements of abelian categories
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Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory.
Eventually homological isomorphisms in recollements of derived categories
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Owing to Corollary 1, we restrict our discussions to standard recollement in the following text. The idea of this definition comes from [1,3], where the concept “ladder” was introduced to study mixed categories. Obviously, a 1-recollement is nothing but a recollement, and a 2-recollement is a perfect recollement, that is, the two functors in the second layer preserve compactness (see [1, Proposition 3.2]).
Eventually homological isomorphisms are introduced between derived categories of algebras, and these lead to a special recollement situation, that is, $(D B, D A, D C)$ with the functor $j^{⁎} : D A \to D C$ being an eventually homological isomorphism. This recollement turns out to be a very good context to compare the algebras A and C with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. The results are applied to stratifying ideals, triangular matrix algebras and derived discrete algebras.
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Koszul duality

Abstract

On the derived category of perverse sheaves

Lect. Notes in Math.

How to glue perverse sheaves

Lect. Notes in Math.

Notes on motivic cohomology

Duke Math. J.

Algebraic vector bundless on Pn and problems of linear algebra

Funct. anal. and appl.

On a certain category of g-modules

Funct. anal. and appl.

Representable functors, Serre functors and perestroika

Quantum groups

Algebraic vector bundless on Pⁿ and problems of linear algebra