Семинар лаборатории алгебраической геометрии: Александр Павлов

На семинаре выступит Александр Павлов (ВШЭ) с докладом Koszul duality between Betti numbers and homological numbers in Calabi-Yau case

Let R be a commutative Gorenstein ring and M is a finitely generated module over R. Betti numbers of M are classical homological invariants of M, these numbers immediately tell us other invariants e.g. dimension or multiplicity, but in general Betti numbers are hard to compute explicitly. We show that if the Gorenstein parameter of R vanishes (that is Proj(R) is a Calabi-Yau variety) there is a relation between Betti numbers of modules and homological numbers of sheaves on Proj(X). This relation follows from Orlov's equivalence between derived category of Proj(R) and the singularity category of R. In the simplest case when \Proj(R) is an elliptic curve this allows us to find all Betti numbers of maximal Cohen-Macaulay modules over R.