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The behaviour of the differential Galois group on the generic and special fibres: a Tannakian approach. (English) Zbl 1242.12005
Summary: Let $$\mathfrak o$$ be a complete DVR of fraction field $$K$$ and algebraically closed residue field $$k$$. Let $$A$$ be an $$\mathfrak o$$-adic domain which is smooth and topologically of finite type. Let $$\mathcal D$$ be the ring of $$\mathfrak o$$-linear differential operators over $$A$$ and let $$\mathcal M$$ be a $$\mathcal D$$-module which is finitely generated as $$A$$-module. Given an $$\mathfrak o$$-point of $$\text{Spf}(A)$$ we construct using a Tannakian theory of Bruguières-Nori, a faithfully flat $$\mathfrak o$$-group-scheme $$\Pi$$ which is analogous-in the sense that its category of dualizable representations is equivalent to a category of $$\mathcal D$$-modules-to the Tannakian group-scheme (the differential Galois or monodromy group) associated to a $$\mathcal D$$-module over a field. We show that the differential Galois group $$G$$ of the reduced $$\mathcal D$$-module $$\mathcal M \otimes k$$ is a closed subgroup of $$\Pi \otimes k$$, which coincides with $$(\Pi \otimes k)_{\text{red}}$$ when $$\Pi$$ is finite, and gives back, in any case, the differential Galois group of $$\mathcal M \otimes K$$ upon tensorisation with $$K$$.

##### MSC:
 12H05 Differential algebra 14L15 Group schemes 13N15 Derivations and commutative rings
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##### References:
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