Algebraický seminář

Leonid Positselski (IM CAS): MGM equivalence of derived categories over a Noetherian formal scheme

Abstract: The now-classical topic of MGM equivalence is often viewed essentially through the abstract lens of an adjoint pair of an idempotent comonad and an idempotent monad on the derived category of modules, arising in the context of a commutative ring with a finitely generated ideal. The dempotent comonad is the derived torsion functor, and its adjoint idempotent monad is the derived completion functor. The adjunction is called the MGM duality, and the induced equivalence between the full subcategories of derived torsion and derived complete complexes in the derived category of modules is called the MGM equivalence.
Some 10-11 years ago, I published a paper suggesting an interpretation of the MGM equivalence as an equivalence between the derived categories of two abelian categories: the category of torsion modules and the category of contramodules. In this talk I will explain how to globalize my version of the MGM equivalence to Noetherian formal schemes, obtaining an equivalence between the derived category of the abelian category of quasi-coherent torsion sheaves and the derived category of the exact category of contraherent cosheaves of contramodules.