Reduced group schemes as iterative differential Galois groups
- PDF / 259,981 Bytes
- 19 Pages / 429.408 x 636.768 pts Page_size
- 54 Downloads / 216 Views
REDUCED GROUP SCHEMES AS ITERATIVE DIFFERENTIAL GALOIS GROUPS
BY
Andreas Maurischat Lehrstuhl A f¨ ur Mathematik, RWTH Aachen University, 52056 Aachen, Germany e-mail: [email protected]
ABSTRACT
This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants. We also introduce the notion of equivalence of iterative derivations on a given field—a condition which implies that the inverse Galois problem over equivalent iterative derivations are equivalent.
1. Introduction In differential Galois theory in positive characteristic, classical derivations are replaced by iterative derivations in order to keep the constants “small”, and to obtain an appropriate Galois theory. For example in characteristic zero, the d rational function field C(t) with the derivation ∂t = dt has as constants C = {f ∈ C(t) | ∂t (f ) = 0}. In positive characteristic p, however, the constants would be the subfield C(tp ). Therefore, one considers the iterative derivation θt instead which is given by a Received June 3, 2019
437
438
A. MAURISCHAT
Isr. J. Math.
(n)
collection of C-linear maps (θt )n≥0 determined by (n)
θt (tk ) =
k k−n t n
(n)
1 n for all k, n ∈ N. Morally, θt equals n! ∂t although the latter expression is not meaningful in characteristic p < n. Then the constants (n)
{f ∈ C(t) | ∀n > 0 : θt (f ) = 0} equal C, as one is used to from characteristic zero. Using these iterative derivations, a Galois theory for linear iterative differential equations (usually called Picard–Vessiot theory) has been developed by Matzat and van der Put [MvdP03b] quite analogous to the Galois theory for linear differential equations in characteristic zero. This theory was improved by the author in [Mau10a] (see also [Mau10b]) to a Galois theory whose Galois correspondence also takes into account intermediate fields over which the solution field is inseparable. It even happens that the solution field itself is inseparable over the base field, in which case the Galois group is a non-reduced affine algebraic group scheme. The inverse Galois problem asks which groups or group schemes, respectively, arise as Galois groups for given iterative differential fields. In this article, we consider the inverse Galois problem over base differential fields which are finitely generated over their field of constants. Our main theorem is the following. Main Theorem (Theorem 4.1): Let (L, θ) be an iterative differential field of positive characteristic p with algebraically closed field of constants C such that L is finitely generated over C, and L = C. Then every reduced affine algebraic group scheme over C is the Galois group of some iterative differential module over (L, θ). Here, we use “algebraic” synonymous to “of finite type”. The proof of this theorem is an adaptation of the proof of the corresponding statement in ch
Data Loading...
