Critères d'induction et de coinduction pour certains anneaux d'opérateurs différentiels. (Induction and coinduction criteria for certain differential operator rings) (Q1093707)

In the context of certain differential operator rings, the author develops criteria for modules to be either induced or coinduced. These results are general enough to imply corresponding results established for Lie algebras by \textit{R. J. Blattner} [Trans. Am. Math. Soc. 144, 457-474 (1969; Zbl 0295.17002)] and \textit{A. Beilinson} and \textit{J. Bernstein} [C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)], using systems of imprimitivity. Here (A,\({\mathfrak M})\) is a commutative Noetherian regular local \({\mathbb{Q}}\)-algebra, \({\mathcal L}\) is a suitable A- module Lie algebra acting by derivations on A, and \(V=V(A,{\mathcal L})\) is the corresponding differential operator ring constructed by \textit{G. S. Rinehart} [Trans. Am. Math. Soc. 108, 195-222 (1963; Zbl 0113.262)]. Moreover, \({\mathcal L}_ 0\) is the Lie algebra \(\{\) \(d\in {\mathcal L}|\) d(\({\mathfrak M})\subseteq {\mathfrak M}\}/{\mathfrak M}{\mathcal L}\), which acts on A/\({\mathfrak M}\), and \(V_ 0=V(A/{\mathfrak M},{\mathcal L}_ 0)\); it is assumed that the rank of \({\mathcal L}\) over A minus the dimension \({\mathcal L}_ 0\) over A/\({\mathfrak M}\) equals the dimension of A. The author proves that the right V-modules induced from right \(V_ 0\)-modules by (-)\(\otimes (V/{\mathfrak M}V)\) are precisely those in which every element is annihilated by a power of \({\mathfrak M}\). Dually, he proves that the left V-modules coinduced from left \(V_ 0\)-modules by Hom(V/\({\mathfrak M}V\),-) are precisely those which are Hausdorff and complete in the \({\mathfrak M}\)-adic topology.