Witt vectors and separably closed fields with higher derivations. (English) Zbl 07720260

Hoffmann, Daniel Max

Notre Dame J. Formal Logic 64, No. 2, 173-184 (2023).

The theory \(\mathrm{SCF}_{p,e}\) (separably closed fields of positive characteristic \(p\) and finite inseparability degree \(e\)) was used by E. Hrushovski [J. Am. Math. Soc. 9, No. 3, 667–690 (1996; Zbl 0864.03026)] in his proof of the positive characteristic relative Mordell-Lang conjecture. Hrushovski’s proof of the characteristic version used the theory DCF\(_0\) (differentially closed fields of characteristic 0) instead. It was desirable to have a uniform differential proof valid in all characteristics, therefore people wanted to find the corresponding proper positive characteristic differential set-up.
The theory \(\mathrm{DCF}_p\) is not good here, since the underlying fields of its models have infinite imperfection degree. M. Messmer and C. Wood proposed [J. Symb. Log. 60, No. 3, 898–910 (1995; Zbl 0841.03019)] a theory of fields with one Hasse-Schmidt derivation satisfying certain iterativity condition. M. Ziegler noticed in [J. Symb. Log. 68, No. 1, 311–318 (2003; Zbl 1039.03031)] an error in the approach from [Messmer and Wood, loc. cit.] and suggested the theory of \(e\) commuting Hasse-Schmidt derivations satisfying the standard iterativity rule: \[ D_i\circ D_j=\binom{i+j}{i}D_{i+j}. \] In the paper under review, the author suggests yet another approach which seems to be the closest one to the original idea from [M. Messmer and C. Wood, J. Symb. Log. 60, No. 3, 898–910 (1995; Zbl 0841.03019)]. Namely, a certain iterativity rule is suggested for one Hasse-Schmidt derivation which yields a theory bi-interpretable with SCF\(_{p,e}\).
From the geometric point of view we have the following.

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The approach from [Ziegler, loc. cit.] corresponds to actions of (the formalization of) \(\mathbb{G}_{\mathrm{a}}^e\), where \(\mathbb{G}_{\mathrm{a}}\) is the additive group.

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The approach suggested by the author corresponds to actions of (the formalization of) the algebraic group of Witt vectors of length \(e\).

Reviewer: Piotr Kowalski (Wrocław)

MSC:

03C60	Model-theoretic algebra
13N15	Derivations and commutative rings
20G15	Linear algebraic groups over arbitrary fields
12L12	Model theory of fields

Keywords:

algebraic groups; Hasse-Schmidt derivations; separably closed fields

Citations:

Zbl 0864.03026; Zbl 0841.03019; Zbl 1039.03031

References:

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