Operators coming from ring schemes

Abstract

We introduce the notion of a coordinate k-algebra scheme and the corresponding notion of a B-operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it also generalizes the operators related to D-rings from [15]. We classify the (coordinate) kalgebra schemes for a perfect field k and we also discuss the model-theoretic properties of fields with B-operators.

Model theory of derivations of the Frobenius map revisited

“…The proof is completely analogous to the proofs of Theorem 2.4 and Theorem 2.7. Theorem 2.8 is new and strengthens Theorem 4.14 from [3], which states that B − DCF eliminates quantifiers in the language L ∂ λ . This theorem is also a vast generalization of Theorem 2.4.…”

“…This normal form has the property that there is some k-algebra B such that B-operators "twisted" by an appropriate sequence of Frobenius maps are B-operators. In other words, Theorem 2.19 in [3] says that general B-operators are related to B-operators in the same way as derivations of the Frobenius map are related to derivations. Now, in order to speak about B-operators as tuples of maps ∂ 1 , .…”

“…Then the theory B−DCF has quantifier eliminations in the language L ∂ λ 0 . Even more generally, we can get a quantifier-elimination result for so-called B-operators considered in [3]. We will not recall the precise definition of a B-operator, as it is a bit involved, but the rough idea is to replace the functor −⊗ k B in the context of B-operators by an appropriate functor B and the map π by a natural transformation B −→ id.…”

“…. , ∂ d is "Fr n -linear over the constants" (see the proof of Lemma 4.16 in [3] for details), so they fall under the class of operators considered in Lemma 2.1.…”

“…Actually, in Lemma 2.1 and Lemma 2.2 we provide an easy proof of a surprisingly general fact, which implies quantifier elimination results for a very general class of theories of fields with operators. More precisely (see Subsection 2.1), we give a new and shorter proof of quantifier elimination in the framework of Boperators (introduced in [1]) and a stronger quantifier elimination result in the setting of B-operators (introduced in [3]).…”