Bounding crystalline torsion from étale torsion

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 étale torsion, the cohomological degree, and the ramification. Our technical core result is a boundedness result concerning annihilator ideals of $u^{\mathrm{\infty }}$-torsion in Breuil–Kisin prismatic cohomology, which might be of independent interest.