Embedding stable groups into algebraic groups
Charlotte Bartnick
TL;DR
The paper addresses which infinite groups definable in stable theories of fields arise from algebraic groups, extending the BD02 embedding result to more general stable field theories. The main method is to reduce to generically rational groups and apply Weil's pre-group theorem, under a definable-closure Property 1 that ensures $\mathrm{dcl}_{\mathcal{L}}(A,B) \subseteq A\cdot B$ when $A \perp^{\mathcal{L}}_N B$, yielding a definable embedding into an algebraic group defined over the base. The key contribution is a general theorem: every type-definable connected group definable over a small model $N$ embeds, up to definable isomorphism, into the $M$-points of an algebraic group defined over $N$, with the result applying to $SCF_{p,e}$ and $DCF_p$ among other theories. This framework also provides a pathway to uniform descriptions of definable closure in these theories, clarifying how algebraic and model-theoretic structures interact in fields with extra structure.
Abstract
Adapting a proof of Bouscaren and Delon, we show that every type-definable connected group in a given stable theory of fields embeds into an algebraic group, under a condition on the definable closure. We also present general hypotheses which yield a uniform description of the definable closure in such theories of fields. The setting includes in particular the theories of separably closed fields of arbitrary degree of imperfection and differentially closed fields of arbitrary characteristic.