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Quatre descriptions des groupes de Galois différentiels. (Four descriptions of differential Galois groups). (French) Zbl 0651.12015
Sémin. d’algèbre P. Dubreil et M.-P. Malliavin, Proc., Paris 1986, Lect. Notes Math. 1296, 28-41 (1987).
[For the entire collection see Zbl 0624.00008.]
Differential Galois groups are typically defined as the group of automorphisms of the field of solutions of a (linear, homogeneous) differential equation which commute with derivation. The author proposes an intrinsic definition.
He considers the category of finite-dimensional vector spaces with integrable connections. He fixes one such M and considers the smallest subcategory containing M and being closed under direct sums, tensor products, and symmetric and exterior powers. He defines the differential Galois group \(G_{gal}(M)\) to be the stabilizer in \(GL_ M\) of the objects in the subcategory under all polynomial representations of \(GL_ M\). This is shown to be an extension of the usual definition, after scalars are extended to the completion of the field of coefficient functions. It also has direct links with a conjecture of N. Katz on groups associated to p-curvative operators, to monodromy groups and the motivic Galois theory.
This paper is a nearly faithful reproduction, without proofs, of lectures. The style is informal and reportorial. Proofs will appear in a paper entitled “Quelques points de theorie de Galois différentielle”.
Reviewer: A.R.Magid

12H20 Abstract differential equations
14L99 Algebraic groups
12H05 Differential algebra
14A20 Generalizations (algebraic spaces, stacks)