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