The behaviour of the differential Galois group on the generic and special fibres: A Tannakian approach
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João Pedro P. dos Santos
Abstract
Let 𝔬 be a complete DVR of fraction field K and algebraically closed residue field k. Let A be an 𝔬-adic domain which is smooth and topologically of finite type. Let 𝒟 be the ring of 𝔬-linear differential operators over A and let ℳ be a 𝒟-module which is finitely generated as A-module. Given an 𝔬-point of Spf(A) we construct using a Tannakian theory of Bruguières-Nori, a faithfully flat 𝔬-group-scheme Π which is analogous—in the sense that its category of dualizable representations is equivalent to a category of 𝒟-modules—to the Tannakian group-scheme (the differential Galois or monodromy group) associated to a 𝒟-module over a field. We show that the differential Galois group G of the reduced 𝒟-module ℳ ⊗ k is a closed subgroup of Π ⊗ k, which coincides with (Π ⊗ k)red when Π is finite, and gives back, in any case, the differential Galois group of ℳ ⊗ K upon tensorisation with K.
© Walter de Gruyter Berlin · New York 2009
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- A short proof of the λg-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves
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Articles in the same Issue
- Trees and mapping class groups
- The Hartogs extension theorem on (n – 1)-complete complex spaces
- On Hartogs' extension theorem on (n – 1)-complete complex spaces
- Cohomological finiteness conditions for elementary amenable groups
- The behaviour of the differential Galois group on the generic and special fibres: A Tannakian approach
- Discrete holomorphic geometry I. Darboux transformations and spectral curves
- Multilinear morphisms between 1-motives
- A short proof of the λg-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves
- Unobstructedness of deformations of holomorphic maps onto Fano manifolds of Picard number 1
- Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials
- Siegel's trace problem and character values of finite groups
