Algebra and Topology Seminar

4pm Tuesday 11 November 2003

Daniel Chan
UNSW

Canonical Singularities for Orders

Let G be a finite subgroup of SL₂ which acts naturally on k[[x,y]]. The quotient variety Spec k[[x,y]]^G is a singularity called a Kleinian, du Val or canonical singularity. There have been many generalisations of this to the noncommutative setting, most based on algebraic characterisations of these singularities. We examine here a generalisation based on geometry motivated by Mori theory and its adaptation to orders over surfaces (c.f. Johan de Jong's talk on "Stable Orders over Surfaces").

We will show that these canonical singularities for orders are all invariant rings with respect to a group acting on the matrix algebra over k[[x,y]]. We will also show they satisfy various algebraic properties that their commutative counterparts do e.g. they have finite representation type and are Gorenstein. A key tool is the homological determinant for noncommutative rings.

This is joint work with Colin Ingalls.

Being a talk in the algebra seminar I will concentrate on the algebraic aspects of the project and assume the audience has only a passing acquaintance with algebraic geometry.

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