NCMW - Hilbert Functions and Local Cohomology (2023)

Venue: IIT, Bombay

Dates: 27 Nov 2023 to 2 Dec 2023

Hilbert functions were introduced for graded modules by David Hilbert in his study of invariant subrings.  Pierre Samuel introduced the Hilbert-Samuel coefficients of any m-primary ideal of a Noetherian local ring. The Hilbert coefficients have been found to be very useful in the study of singularities of algebraic varieties. Grothendieck and Serre found a deep connection between local cohomology and Hilbert functions. This helped understanding the computation of the postulation number in terms of largest non-vanishing degree of graded local cohomology modules of blowup algebras. A related notion is that of Castelnuovo-Mumford regularity of graded modules which helps in the study of Hilbert functions. Wolmer Vasconcelos introduced the Sally modules of ideals which have been effectively used to find properties of Hilbert polynomials and and blowup algebras. This workshop will introduce the fundamental techniques of local cohomology modules, Sally modules and Castelnuovo-Mumford regularity with applications to Hilbert functions of blowup algebras, bounds on Hilbert coefficients, behaviour of Hilbert functions under perturbations and connections with Sheaf cohomology and local cohomology.