Topics in commutative algebra
Lead Research Organisation:
University of Sheffield
Department Name: Mathematics and Statistics
Abstract
BETTI NUMBERS STANLEY REISNER IDEALS
Any graded, finitely generated module affords a minimal graded resolution, and these resolutions provide a wealth of information about the original module. One such piece of information are the Betti numbers of the module; these are the degrees of the free generators in a minimal free resolution.
One of the most exciting recent developments in commutative algebra is the emergence of the Boij-Soderberg Theory which describes the convex cone generated by all the Betti numbers of Cohen-Macaulay modules over a polynomial ring, in particular it describes the extremal rays of this cone and ways to express any given betti diagram as a convex combination of points on these rays.
David Carey's project studies the subcone generated by Betti diagrams of square free monomial ideals in general, and edge ideals in particular. The aim of the project is to exploit the combinatorial structure of these ideals to gain an understanding of the betti cones, e.g., the classification of extremal rays in the Betti cones of edge ideals in terms of the properties of the underlying graphs.
Any graded, finitely generated module affords a minimal graded resolution, and these resolutions provide a wealth of information about the original module. One such piece of information are the Betti numbers of the module; these are the degrees of the free generators in a minimal free resolution.
One of the most exciting recent developments in commutative algebra is the emergence of the Boij-Soderberg Theory which describes the convex cone generated by all the Betti numbers of Cohen-Macaulay modules over a polynomial ring, in particular it describes the extremal rays of this cone and ways to express any given betti diagram as a convex combination of points on these rays.
David Carey's project studies the subcone generated by Betti diagrams of square free monomial ideals in general, and edge ideals in particular. The aim of the project is to exploit the combinatorial structure of these ideals to gain an understanding of the betti cones, e.g., the classification of extremal rays in the Betti cones of edge ideals in terms of the properties of the underlying graphs.
Organisations
People |
ORCID iD |
| Mordechai Katzman (Primary Supervisor) | |
| David Carey (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/R513313/1 | 01/10/2018 | 30/09/2023 | |||
| 2264817 | Studentship | EP/R513313/1 | 01/10/2019 | 31/03/2023 | David Carey |
| Description | My project has been focussed primarily on studying the convex cones generated by Betti diagrams corresponding to graphs and simplicial complexes, because these diagrams contain important information about the graphs and complexes themselves. I have managed to find a formula for the dimensions of these cones, based on the number of vertices in the corresponding graphs and complexes. I have also found two distinct families of highly symmetric simplicial complexes whose corresponding Betti diagrams are pure, which means they have a particularly simple shape. Pure diagrams play a central role in Boij-Söderberg Theory, because the ones corresponding to Cohen-Macaulay modules are extremal rays of the Betti cone. |
| Exploitation Route | These results add to our understanding of the Betti cones corresponding to graphs and simplicial complexes on various numbers of vertices, as well as providing a wide variety of specific examples of Betti diagrams, which other algebraists working in this area may be able to utilise in proving / providing counter-examples for various theorems. In particular, an understanding of the dimensions of these Betti cones is an important step towards being able to characterize the cones more comprehensively, via a description of their defining halfspaces. |
| Sectors | Other |