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On index theorems for linear ordinary differential operators. (English) Zbl 0901.34012
Let \(D\) be a linear ordinary differential operator of arbitrary order with analytic coefficients in the neighbourhood of the origin in \(\mathbb{C}\). B. Malgrange proved [Enseignement Math., II. Ser. 20, 147-176 (1974; Zbl 0299.34011)] that \(D\) has an index as a linear operator both in the vector space \(\mathbb{C}[[x]]\) of formal power series at the origin, and in the subspace \(\mathbb{C}\{x\}\) of the convergent ones. The authors investigate the sheaf of Deligne \(\widetilde F\) with an application to index theorems for \(D\) acting on a space of sections of this sheaf on various sets. Their method relies on homological algebra. Such theorems as the following play a central role in the paper.
Theorem: The linear maps \(D: H^i(U,\widetilde F)\to H^i(U,\widetilde F)\) for \(i\geq 1\) are isomorphisms when \(U\) is a disk, a sector, a multisector or an annulus.

MSC:
34A30 Linear ordinary differential equations and systems, general
55N30 Sheaf cohomology in algebraic topology
34M99 Ordinary differential equations in the complex domain
34E05 Asymptotic expansions of solutions to ordinary differential equations
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