Z. Chatzidakis: Notions of difference closures of difference fields
Thursday, 23.11.2017 11:00 im Raum SR 1D
It is well known that the theory of differentially closed fields of
characteristic 0 has prime models (over differential subfields) and
that they are unique up to isomorphism.
One can ask the same question for the theory ACFA of existentially
closed difference fields (recall that a difference field is a field
with an automorphism).
In this talk, I will first give the trivial reasons of why this question cannot have a positive answer. It could however be the case that over certain difference fields, prime models (of the theory ACFA)
exist and are unique. Such a prime model would be called a difference closure of the difference field K. I will show by an example that the obvious conditions on K do not suffice.
I will then consider the class of aleph-epsilon saturated models of
ACFA, or of kappa-saturated models of ACFA. There are natural notions of aleph-epsilon prime model and kappa-prime model. It turns out that for these stronger notions, if K is an algebraically closed difference field of characteristic 0, with fixed subfield F aleph-epsilon saturated, then there is an aleph-epsilon prime model over K, and it is unique up to K-isomorphism. A similar result holds for kappa-prime when kappa is a regular cardinal.
None of this extends to positive characteristic.
Angelegt am 08.11.2017 von Martina Pfeifer
Geändert am 08.11.2017 von Martina Pfeifer
[Edit | Vorlage]