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Published online by Cambridge University Press: 29 May 2025
We give an exposition and generalization of Orlov’s theorem on graded Goren-stein rings. We show the theorem holds for nonnegatively graded rings that are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary noncommutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this level of generality is given in order to prove the theorem. As an application we give an equivalence of the derived category of a commutative complete intersection with the homotopy category of graded matrix factorizations over a related ring.
Let A be a graded Gorenstein ring. Orlov [2009] related the bounded derived category of coherent sheaves on Proj A and the singularity category of graded A-modules via fully faithful functors; the exact relation depends on the a-invariant of A. This is a striking theorem that has found applications in physics, algebraic geometry and representation theory. To give an idea of the scope of the theorem: in the limiting case that A has finite global dimension (so the singularity category is trivial), it recovers (and generalizes to noncommutative rings) Be˘ılinson’s result [1978] that the derived category of Proj A is generated by a finite sequence of twists of the structure sheaf.
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