  For a perfect field $k$ of characteristic $p>0$ and a smooth and proper
formal scheme $\mathscr{X}$ 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 modules attached to the $p$-adic \'etale cohomology
$H^i_{\mathrm{\'et}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove
that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology
of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
