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Koszul duality for non-graded derived categories
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Koszul duality for non-graded derived categories
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We are concerned with relating derived categories of all modules of two dual Koszul algebras defined by a locally bounded quiver. We first generalize the well known Acyclic Assembly Lemma and formalize an old method of extending a functor from an additive category into a complex category to its complex category. Applying this to the Koszul functor associated with a Koszul algebra defined by a gradable quiver, we obtain a Koszul complex functor, that descends to an equivalence of a continuous family of pairs of triangulated subcategories of doubly unbounded complexes of the respective derived categories of all modules of the Koszul algebra and its Koszul dual. Under this special setting, this extends Beilinson, Ginzburg and Soegel's Koszul duality. In case the Koszul algebra is right or left locally bounded and its Koszul dual is left or right locally bounded respectively (for instance, the quiver has no right infinite path or no left infinite path), our Koszul duality restricts to an equivalence of the bounded derived categories of finitely supported modules, and an equivalence of the bounded derived categories of finite dimensional modules.
Forward citations
Cited by 4 Pith papers
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A Non-graded Koszul Duality and Its Applications
Finite-dimensional Koszul algebras admit derived Koszul dualities in bounded derived categories of modules both graded and ungraded, reconstructed via dg orbit categories, with applications to all integral blocks of c...
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A Non-graded Koszul Duality and Its Applications
Finite-dimensional Koszul algebras admit non-graded derived Koszul dualities via triangulated hulls of orbit categories, yielding dualities for all integral blocks of category O including singular ones.
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Koszul Duality for Coherent Sheaves
Claims a bounded Koszul duality for infinite-dimensional Koszul algebras and a BGG-type description of D^b(coh(X)), but key identifications and hypotheses are unproved.
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Koszul Duality for Coherent Sheaves
For a Koszul quotient Λ of a polynomial ring, the derived category of coherent sheaves on Proj(Λ) is claimed to be equivalent to a quotient of derived categories over the Koszul dual Λ!.
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