Pragmatic 2023
Cohomology and Frobenius
Catania, Italy
 
June 11th - July 1st, 2023
Logo Indam             Dmi logo
iplab
Research school in
Algebraic Geometry and Commutative Algebra
Sponsored by UNICT, PRIN 2020 "Squarefree Gr÷bner degenerations, special varieties and related topics", GNSAGA, DMI UNICT
Frobenius everywhere
Lecturer:
Luis Nu˝ez-Betancourt (CIMAT, Mexico)
Collaborator:
Eamon Quinlan-Gallego (University of Utah, USA)
Cohomology of line bundles on flag varieties
Lecturer:

Claudiu Raicu (Notre Dame University, USA)
Collaborator:
Alessio Sammartano (Politecnico di Milano, Italy)
Keller VandeBogert
(Notre Dame University, USA)

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P.R.A.G.MAT.I.C. (Promotion of Research in Algebraic Geometry for MAThematicians in Isolated Centres) is a project for stimulating researches in Algebraic Geometry and Commutative Algebra among young people.
Pragmatic aims to give the opportunity of new collaborations and scientific horizons to young talented mathematicians, trying to create a center for training young mathematicians and to help young scientists to find their own area of research.
In pursuit of these goals Pragmatic will organize a period of lectures and seminars on very concrete problems and on techniques to solve them related to fundamental topics in Algebraic Geometry and in Commutative Algebra.


Frobenius everywhere

Abstract:

The Frobenius morphism has proven to be a powerful tool in commutative algebra. Peskine and Szpiro proved Auslander's zero divisor conjecture and answered Bass' question regarding Cohen-Macaulay rings using this map. They also provided a series of important results regarding the structure of local cohomology. Hochster and Roberts proved that rings of invariants are Cohen-Macaulay. Later, Hochster and Huneke developed tight closure theory and used it to solve several homological conjectures in prime characteristic. Since then, the use of the Frobenius map has grown to touch other areas such as differential algebra, combinatorics, representation theory and birational geometry. The first lectures will focus on the basics on the Frobenius maps, its splittings, and local cohomology. Then, we focus on the use of Frobenius in singularity theory and combinatorics. We will also discuss connections with D-modules. There will also be an introduction to Macaulay2 packages in prime characteristic, and related topics. The list of open problems to discuss will try to reflect the diversity of uses of the Frobenius map.


Cohomology of line bundles

Abstract:

The goal of these lectures is to discuss the problem of computing cohomology of line bundles on flag varieties, and to explore various applications to the study of homological invariants in commutative algebra and algebraic geometry. Over fields of characteristic zero, the cohomology calculation is well-understood, and is the subject of the celebrated Borel-Weil-Bott theorem. In positive characteristic however, the description of cohomology is largely unknown, even when it comes to deciding its vanishing and non-vanishing behavior. The question of computing cohomology turns out to be quite versatile – any reasonable restrictions, either on the line bundles considered, or on the flag varieties themselves, lead to special cases that are of interest on their own, and the participants will have the chance to examine a number of particular scenarios. The applications we envision include, but are not limited to: Castelnuovo-Mumford regularity for powers and symbolic powers of determinantal varieties, Hilbert functions of Koszul modules, or the homology of certain arithmetic Koszul-type complexes.