2 problems

Let $X/\mathcal{O}_K$ be the fixed scheme, let $\mathfrak{S}$ be the Breuil–Kisin ring, and let $E$ be its Eisenstein polynomial. For the Breuil–Kisin cohomology module … write…

Assume $K=K_{{\mathrm{cyc}}}\cap K_\infty$. Let $\mathfrak M$ be a crystalline Breuil–Kisin module of rank $d$, let $\lambda:=\frac{\xi}{E([\pi^{\flat}])}$, and let $\widetilde A$…