# Seminar Algebraische Geometrie

**On Monday, 24th April**

**Arvid Seqveland (USN, Norway) will give a talk on**

**Noncommutative Geometric Invariant Theory**

** Abstract:**A scheme over $k$ can be defined by its local rings, and is a fine moduli for all its closed points $M$. It is well known that the local rings are pro-representing hulls for the deformation functor $Def_M$ from the category $l$ of local Artinian $k$-algebras to the category of sets. Earlier work gives an algorithm for computing pro-representing hulls for the deformation functor, and so gives a (formal) moduli for its objects. Here we will always work in a category of modules, but the deformation theory works for any additive abelian category.

We generalize the ordinary deformation theory: Let $A$ be an associative $k$-algebra, not necessarily commutative, and let $\mathcal M=\{M_1,...,M_r\}$ be a set of $r$ right $A$-modules. We will define the category $a_r$ of $r$-pointed Artinian $k$-algebras and the noncommutative deformation functor $Def_{\mathcal M}:a_ràSets$. We prove that the noncommutative deformation functor has a prorepresenting hull, acting as a semilocal ring for a noncommutative scheme which can be defined. Along the way, we obtain a structure theorem for finitely generated associative algebras from which we can conclude that the basic objects in noncommutative algebra are the matrix polynomial algebras.

**Based on this noncommutative algebraic geometry, we study geometric invariant theory: The classification of orbits under group-actions. It turns out that we can construct noncommutative schemes that are fine moduli of the orbits, also for families of non-stable points. We give some examples where sufficient invariants and their relations are given.**

**The talk will take place in room 119, Arnimallee 3 at 16:15.**

**Schedule and Abstracts: winter semester 2016 ****Forschungsseminar "Algebraische Geometrie" an der HU**