Time: 9:30, December 7, 2022.

Venue: Room 612 A6.

Abstract: The first part presents, without proofs but hopefully with sufficient insight for the students to understand the process, the technique of spectral sequences in the case of the two standard filtrations of a double complex. Immediate corollaries are given, as well as some other constructions or simple applications, like the Eagon-Northcott family of complexes, or local duality in the graded case. Also, Cech and Koszul complexes are presented. Spectral sequences will be used in several places; it is often a very powerful tool to establish non trivial isomorphisms, or vanishing results.

The second part is devoted to presenting Castelnuovo-Mumford regularity. We restrict our framework to Noetherian standard graded algebras and modules of finite type; this is enough for many applications. The multigraded versions are not presented. However, this version is sufficient to study the regularity of powers of graded ideals and relate the asymptotic behavior of these to the fibers and stalks of morphisms. In the classical case of projective schemes over a field, we show how one can use liaison to study Castelnuovo-Mumford regularity.