The classification of $p$-divisible groups over $p$-adic formally smooth rings

Details The classification of $p$-divisible groups over $p$-adic formally smooth rings The classification of $p$-divisible groups over $p$-adic formally smooth rings

2011-12-30

arxiv.org/abs/1201.0121v1

Mathematics

Number Theory

Scientific paper

Let $\mathscr{O}_K$ be a $p$-adic discrete valuation ring with residue field admitting a finite $p$-basis, and let $R$ be a formally smooth adic $\mathscr{O}_K$-algebra which satisfies some reasonable finiteness condition. Assume that $p>2$. We construct using crystalline Dieudonn\'e theory an anti-equivalence of categories between the categories of $p$-divisible groups over $R$ and certain semi-linear algebra objects which generalise $(\varphi,\mathfrak{S})$-modules of height $\leqslant1$ (or Kisin modules). We also show compatibility of various construction of ($\mathbb{Z}_p$-lattice) Galois representations, including the relative version of Faltings' integral comparison theorem for $p$-divisible groups. When $p=2$, we prove the results up to isogeny.

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