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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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