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