ALGANT Student seminar – Commutative Algebra
Date and time: Wednesday, May 25th, at 4pm
Room: Aula Dottorato
Lecture 1: Federico Binda
Title: the Intersection formula (PDF)
The goal of this lecture is to present the minimum background material that is necessary to understand Serre’s definition of intersection multiplicity, at least in the case of two irreducible subvarieties of an n-dimensional affine space. Following [2], we will quickly review Samuel’s definition of multiplicity of an ideal. Then we will see how the homology modules of the Koszul complex are related to the Tor functor and in which way they can be used to give a “new” definition of multiplicity, as in [1]. We will conclude the talk proving that Serre’s definition is the “right one”, at least in the case of plane curves, and, if time permits, we will also give an example that should convince that the higher Tor are actually useful.
References:
[1] J. P. Serre (2000), Local Agebra, revised edition. Springer Monographs in Mathematics.
[2] P. Roberts (1998), Multiplicities and Chern Classes in Local Algebra. Cambridge Univ. Press. Cambridge.
Lecture 2: Ziyang Gao
Title: Grothendieck-Serre Duality (PDF)
Abstract: This lecture can be viewed as a preliminary part for the next lecture I will give about global Grothendieck duality. In this seminar, we will talk about Koszul complexes of both local and global versions. At the end of the talk, a non-trivial theorem about the canonical bundle of a regular closed immersion will be presented.
Anyone interested is invited.