Breuil $\mathcal {O}$-windows and $\pi$-divisible $\mathcal {O}$-modules

Abstract:

Let $p>2$ be a prime number. Let $\mathcal {O}$ be the ring of integers of a finite extension of $\mathbb {Q}_p$ and let $\pi$ be a uniformizer of $\mathcal {O}$. We prove that, for any complete Noetherian regular local $\mathcal {O}$-algebra $R$ with perfect residue field of characteristic $p$, the category of Breuil $\mathcal {O}$-windows over $R$ is equivalent to the category of $\pi$-divisible $\mathcal {O}$-modules over $R$. We also prove that the category of Breuil $\mathcal {O}$-modules over $R$ is equivalent to the category of commutative finite flat $\mathcal {O}$-group schemes over $R$ which are kernels of isogenies of $\pi$-divisible $\mathcal {O}$-modules. As an application of these equivalences, we then prove a boundedness result on Barsotti-Tate groups and deduce some corollaries. The results generalize some earlier results of Zink, Vasiu-Zink, and Lau.