Overcoherence implies holonomicity

TL;DR

The paper proves that overcoherence after any change of basis implies holonomicity for certain modules over a formal scheme, and characterizes overholonomicity of complexes via their cohomology modules.

Contribution

It establishes a link between overcoherence and holonomicity under change of basis, and characterizes overholonomicity of complexes through their cohomology.

Findings

Overcoherent modules after any change of basis are holonomic.

A bounded complex is overholonomic after any change of basis iff its cohomology modules are.

The result applies to smooth formal schemes over mixed characteristic DVRs.

Abstract

Let $\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field. Let $\X$ be a smooth formal scheme over $\V$. We prove than a $\D ^\dag_{\X,\Q} $-module which is overcoherent after any change of basis is an holonomic $\D ^\dag_{\X,\Q} $-module. Furthermore, we check that this implies than a bounded complex $\E$ of $\D ^\dag_{\X,\,\Q}$-modules is overholonomic after any change of basis if and only if, for any integer $j$, $\mathcal{H} ^{j} (\E) $ is overholonomic after any change of basis.