Bounding crystalline torsion from \'etale torsion

TL;DR

This paper establishes bounds on crystalline torsion in terms of étale torsion, cohomological degree, and ramification for smooth proper families over p-adic rings, using Breuil--Kisin prismatic cohomology techniques.

Contribution

It introduces a boundedness result for annihilator ideals of u^{}-torsion in Breuil--Kisin prismatic cohomology, linking crystalline and étale torsion.

Findings

Boundedness of annihilator ideals in prismatic cohomology

Control of crystalline torsion via étale torsion and ramification

Technical core result on torsion in prismatic cohomology

Abstract

In this note, we prove that given a smooth proper family over a $p$-adic ring of integers, one gets a control of its crystalline torsion in terms of its \'{e}tale torsion, the cohomological degree, and the ramification. Our technical core result is a boundedness result concerning annihilator ideals of $u^{\infty}$-torsion in Breuil--Kisin prismatic cohomology, which might be of independent interest.