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Licensed Unlicensed Requires Authentication Published by De Gruyter November 23, 2009

The behaviour of the differential Galois group on the generic and special fibres: A Tannakian approach

  • 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.

Received: 2007-06-14
Revised: 2008-03-05
Published Online: 2009-11-23
Published in Print: 2009-December

© Walter de Gruyter Berlin · New York 2009

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