On the stability of the overconvergence under the direct image by a proper smooth morphism

TL;DR

This paper proves Berthelot's conjecture that overconvergence of arithmetic D-modules is preserved under the direct image by a smooth proper morphism over a perfect field of characteristic p>0.

Contribution

It establishes the stability of overconvergence under direct images in the context of arithmetic D-modules, confirming a key conjecture in the field.

Findings

Overconvergence is preserved under direct image by smooth proper morphisms.

The proof involves translating the problem into the language of arithmetic D-modules.

The result confirms a fundamental conjecture of Berthelot.

Abstract

Up to a translation in the language of arithmetic $\D$-modules, we prove a conjecture of Berthelot on the preservation of the overconvergence under the direct image by a smooth proper morphism of varieties over a perfect field of characteristic $p>0$.