Seminar of Algebra

Gregor Böhm - Artin Approximation and Gabrielov's counterexample

This talk will first introduce Artin's approximation theorem about approximating formal power series solutions to systems of algebraic (or respectively convergent) power series. There will be a rough indication of how the theorem is proven.

Then, Popescu's nested approximation theorem for algebraic systems will be stated, before showing Grabielov's counter example for the convergent case from 1971 showing that formal relations between convergent power series can not always be approximated by convergent relations.

Luisa Gietl - On the General Néron Desingularization

This talk aims to offer a concise overview of Popescu's General Néron desingularization theorem which establishes that a homomorphism of noetherian rings $S\to R$ is geometrically regular if and only if $R$ is a filtered colimit of smooth $S$-algebras. After a brief recap of some concepts relevant to Popescu’s theorem, we'll focus on its applications, particularly on projective modules over polynomial rings and on the Artin Approximation theorems.