Localisation and quotient categories / Non-commutative projective lines and elliptic curves

Expository talk: Localisation and quotient categories.

Abstract

In algebraic geometry, much of the geometry can be reduced to algebra. In this talk, we look at the geometric notion of restricting to open subsets from an algebraic perspective. This "localisation" procedure involves inverting elements and gives rise to the notion of a quotient category. With this technology, we obtain a purely algebraic way of studying projective geometry which can be generalised to the noncommutative context. More precisely, quasi-coherent sheaves on projective varieties can now be studied via their Serre modules over the homogeneous coordinate ring. 

Research talk: Non-commutative projective lines and elliptic curves. 

Abstract: Noncommutative algebraic geometry is based partly on Grothendieck's philosophy that to do geometry on an algebraic variety, we need only study the category of quasi-coherent sheaves on it. In this talk, we begin with a brief introduction of noncommutative projective geometry, where we look at Artin-Zhang-Polischchuk's theory of categories which can be considered categories of quasi-coherent sheaves on some noncommutative projective scheme. In particular, there is a notion of a homogeneous coordinate ring. We then give some simple examples of noncommutative analogues of projective lines and elliptic curves and explore how they are related to Ringel's notion of species and commutative elliptic curves. 

This is a report on joint work with Adam Nyman.