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.
Categories: math.AG math.NT