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# 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.

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