Séminaires : Séminaire d'Algèbre

The (derived) category of coherent sheaves on a scheme encodes rich information about the underlying geometry. P. Gabriel showed that for noetherian schemes X and Y, if Coh X and Coh Y are equivalent as abelian categories, then X and Y are isomorphic. Furthermore, A. Bondal and D. Orlov proved that for smooth projective schemes X and Y with (anti-)ample canonical bundles, if D^b(Coh X) and D^b(Coh Y) are equivalent as triangulated categories, then X and Y are isomorphic. On the other hand, J.-P. Serre showed that the category of coherent sheaves on a projective scheme can be described as the quotient category of finitely generated graded modules over the homogeneous coordinate ring by the subcategory of torsion modules. Motivated by the results of Gabriel and Serre, the quotient category of finitely generated graded modules over a (not necessarily commutative) graded ring by the subcategory of torsion modules is called a noncommutative projective scheme. In this talk, I will present an analogue of Bondal–Orlov’s reconstruction theorem in the setting of noncommutative projective geometry.

This talk will take place in hybrid mode at the Institut Henri Poincaré.