Lyubeznik numbers of local rings and linear strands of graded ideals

Abstract:

The aim of these talks is to review the basics on Lyubeznik numbers and to introduce a new set of invariants associated to the linear strands of a minimal free resolution of a graded ideal in the polynomial ring. It turns out that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. For the case of squarefree monomial ideals we have a close relation between both invariants that allows to interpret Lyubeznik numbers of Stanley-Reisner rings as the obstruction to the acyclicity of the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of Stanley-Reisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field. The non-expository parts of these talks are based on joint works with K.Yanagawa and A.Vahidi respectively.