A note on groups definable in difference fields. (English) Zbl 0983.03033
The paper deals with groups definable in existentially closed difference fields (i.e. models of ACFA) and differentially closed fields. In particular it is shown that any group \(G\) definable in a model \(K\) of ACFA is virtually definably embeddable in an algebraic group, in the following sense: there are a group \(H\) definable in \(K\) as a pure field (hence an algebraic group), a definable subgroup \(G'\) of \(G\) of finite index and a definable finite normal subgroup \(N'\) of \(G'\) such that \(G'/N'\) is definably embeddable into \(H\). This answers positively a question of Zoe Chatzidakis. The proof uses a generalization of the group configuration theorem to the \(\star\)-definable setting.
A new improved proof of the analogous result for groups definable in differentially closed fields \(K\) is also provided. In detail, it is shown that any connected group definable in \(K\) has a definable (in \(K\)) embedding in some suitable connected group definable in the pure field \(K\). The second author gave another proof some years ago. Here the approach avoids any reference to \(\star\)-definability.
Finally the paper extends to the difference field framework some results on the unipotence of definable groups on affine spaces.
A new improved proof of the analogous result for groups definable in differentially closed fields \(K\) is also provided. In detail, it is shown that any connected group definable in \(K\) has a definable (in \(K\)) embedding in some suitable connected group definable in the pure field \(K\). The second author gave another proof some years ago. Here the approach avoids any reference to \(\star\)-definability.
Finally the paper extends to the difference field framework some results on the unipotence of definable groups on affine spaces.
Reviewer: C.Toffalori (Camerino)
MSC:
| 03C60 | Model-theoretic algebra |
| 12L12 | Model theory of fields |
| 12H99 | Differential and difference algebra |
| 20A15 | Applications of logic to group theory |
References:
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