# Geometric algebras and their properties

Lead Research Organisation: University of Edinburgh
Department Name: Sch of Mathematics

### Abstract

A geometric algebra is a noncommutative ring constructed from geometric data, such as a projective variety $X$ and an automorphism $\sigma$ of $X$. Such algebras are important sources of examples in noncommutative ring theory.
Geometric algebras constructed from a surface are mostly understood. However, those built from higher-dimensional varieties are still mysterious in general. This project will investigate these algebras and determine their properties. There are potential applications to algebraic geometry as well as to noncommutative algebra.

### ORCID iD

Susan Judith Sierra (Primary Supervisor)
Ruth Alice Reynolds (Student)

### Publications

10 25 50
Reynolds R (2019) Idealisers in skew group rings in Journal of Algebra

### Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509644/1 01/10/2016 30/09/2021
1783868 Studentship EP/N509644/1 01/10/2016 31/03/2020 Ruth Alice Reynolds

Description My research is on noncommutative ring theory, in particular a type of noncommutative ring called an idealiser subring. During my research I have proved that the noetherianity (a particularly nice condition for rings) of these rings in rings built from suitable geometric data is completely controlled by interesting geometrical conditions. This relates a very abstract algebraic concept to a geometric one, and finding links between areas of mathematics is useful.This paper has now been submitted to the Journal of Algebra.
This paper has now been published in the Journal of Algebra.
Since finishing this project, I have completed a second project on the noetherianity of idealizer subrings. In particular, I have proved that certain idealizers at singular curves in the second Weyl algebra are noetherian, which generalises a result of McCaffrey. I am currently writing this result up in the form of a paper.
Exploitation Route Whilst I have worked out an example in the two dimensional case to show that genus >0 curves always give noetherian idealisers in my particular construction. It would be interesting to get a classification of how genus 0 curves behave. The rings I have worked with, skew group rings, also fall into a bigger set of rings, Ore extensions. So a natural question would be how to extend this, this is something I am working on now. Another natural question is what are the other properties these rings possess.
My second project has a natural question about generalising to curves with more complicated singularities, this would require different techniques from the ones I have used as simplicity, an important property used in my proof, no longer holds here.
Sectors Other

URL https://arxiv.org/abs/1811.10366