A characterization of the overcoherence

TL;DR

This paper characterizes overcoherence of certain arithmetic D-modules on smooth formal schemes, showing it is equivalent to stability under pullback by smooth morphisms, thus clarifying its geometric nature.

Contribution

It provides a criterion for overcoherence in terms of stability under pullback, linking it to coherence after any smooth morphism.

Findings

Overcoherence is equivalent to stability under pullback by smooth morphisms.

The criterion simplifies checking overcoherence in practice.

Supports the understanding of overcoherence as a geometric property.

Abstract

Let $\mathcal{P}$ be a proper smooth formal $\mathcal{V}$-scheme, $X$ a closed subscheme of the special fiber of $\mathcal{P}$, $\mathcal{E} \in F\text{-}D ^\mathrm{b}_\mathrm{coh} (\D ^\dag_{\mathcal{P},\mathbb{Q}})$ with support in $X$. We check that $\mathcal{E}$ is $\D ^\dag _{\mathcal{P},\mathbb{Q}}$-overcoherent if and only if, for any morphism $f : \mathcal{P}' \to \mathcal{P}$ of smooth formal $\mathcal{V}$-schemes, $f ^! (\mathcal{E}) $ is $\D ^\dag_{\mathcal{P}', \mathbb{Q}}$-coherent.