Existentially closed fields with<i>G</i>-derivations

We give algebraic conditions about a finite commutative algebra B over a field of positive characteristic, which are equivalent to the companionability of the theory of fields with "B-operators" (i.e. the operators coming from homomorphisms into tensor products with B). We show that, in the most interesting case of a local B, these model companions admit quantifier elimination in the "smallest possible" language and they are strictly stable. We also describe the forking relation there. ♦ SDG.

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“…the L λ0 -extensions) are exactly the field extensions such that M p ∩ K = K p (see e.g. [7,Lemma 4.1]). Proof.…”

Section: Amalgamation Property and Quantifier Eliminationmentioning

confidence: 99%

Model theory of fields with free operators in positive characteristic

Beyarslan

Hoffmann

Kamensky

et al. 2019

Trans. Amer. Math. Soc.

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107

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“…There is an interesting parallel between the paragraph above and the axiomatization of existentially closed fields with iterative Hasse–Schmidt derivations (in the case of positive characteristic). For the necessary background, the reader is advised to consult for example, . Let

G_{a}

be the additive group over a field of positive characteristic

p

{double-struckG}_{normala} false[n false]

be the kernel (considered as a finite group scheme) of the algebraic group morphism

{Fr}_{{double-struckG}_{normala}}^{n}

and

{\hat{double-struckG}}_{a}

be the formal group scheme which is the formalization of

G_{a}

.…”

Section: Going Furthermentioning

confidence: 99%

“…Then we have:

{\overset{G}{true}}_{normala} ≅ {\underset{\to}{trueprefixlim}}_{n} {double-struckG}_{normala} false[n false],

the algebraic actions of

{\hat{double-struckG}}_{a}

correspond to iterative Hasse–Schmidt derivations and the algebraic actions of

{double-struckG}_{normala} false[n false]

correspond to

n

‐truncated iterative Hasse–Schmidt derivations. Then indeed, the theory of existentially closed fields with iterative Hasse–Schmidt derivations can be considered as the limit of the theories of existentially closed fields with

n

‐truncated iterative Hasse–Schmidt derivations (see and, in a more general context, ).…”

Section: Going Furthermentioning

confidence: 99%

Model theory of fields with virtually free group actions

Beyarslan

Kowalski

2018

Proc. London Math. Soc.

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For a group G, we define the notion of a G‐kernel and show that the properties of G‐kernels are closely related with the existence of a model companion of the theory of Galois actions of G. Using Bass–Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP2.

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“…Piotr Kowalski and I in [2] are treating iterative HS-derivations in much more abstract way. Many proofs from [2] would be obvious if canonical bases exist for the HS-derivations considered there (a similar sentence was noted at the end of Section 2. in [3]). Moreover, Section 6. in [2] suggests possible generalisations for the notion of canonical basis.…”

Section: Introductionmentioning

confidence: 99%