Breuil–kisin modules via crystalline cohomology

For a perfect field k of characteristic p > 0 and a smooth and proper formal scheme X over the ring of integers of a finite and totally ramified extension K of W (k)[1/p], we propose a cohomological construction of the Breuil–Kisin module attached to the p-adic étale cohomology H _ét ⁱ (X _K , Z _p ). We then prove that our proposal works when p > 2, i < p − 1, and the crystalline cohomology of the special fiber of X is torsion-free in degrees i and i + 1.