Katrin Tent : Defining $R$ and $G(R)$

Posted by Filippo Calderoni , part of the Logic Seminar.

At

March 2, 2021, 11 a.m.

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Abstract

In joint work with Segal we use the fact that for Chevalley groups $G(R)$ of rank at least $2$ over a ring $R$ the root subgroups are (nearly always) the double centralizer of a corresponding root element to show under mild restrictions on the ring $R$ that $R$ and $G(R)$ are bi-interpretable. (This holds in particular for any field $k$.) For such groups it then follows that the group $G(R)$ is finitely axiomatizable in the appropriate class of groups provided $R$ is finitely axiomatizable in the corresponding class of rings.