On Adically Complete D-Modules in Characteristic Zero

TL;DR

This paper revisits Ogus's theorem on adically complete D-modules in characteristic zero, providing detailed proofs, extending results to non-noetherian settings, and clarifying related issues in existing literature.

Contribution

It offers a detailed proof of Ogus's theorem, extends the theorem to non-noetherian settings, and clarifies a related error in Bjork's book.

Findings

Proof of Ogus's theorem on adically complete D-modules

Extension of the theorem to non-noetherian settings

Correction of an error in Bjork's book

Abstract

Let (X, O_X) be an algebraic manifold in characteristic 0, or an analytic manifold over \C. A standard theorem says that a left D_X-module M, which is coherent as an O_X-module, is locally free. This theorem has a generalization to the adically complete algebraic setting, in a paper by Ogus from 1973. In the present paper we take a new look at the work of Ogus. We provide a detailed proof of the theorem on D-modules, and extend it to the non-noetherian setting. We also give another proof of an interesting result of Ogus about adically complete modules (slightly extended). In the Appendix we discuss a related error in a book by Bjork.