Exterior Powers of Barsotti-Tate Groups

Abstract

Let \CO be the ring of integers of a non-Archimedean local field and $\pi$ a fixed uniformizer of \CO . We establish three main results. The first one states that the exterior powers of a $\pi$-divisible \CO -module scheme of dimension at most 1 over a field exist and commute with algebraic field extensions. The second one states that the exterior powers of a $p$-divisible group of dimension at most 1 over arbitrary base exist and commute with arbitrary base change. The third one states that when \CO has characteristic zero, then the exterior powers of $\pi$-divisible groups with scalar \CO -action and dimension at most 1 over a locally Noetherian base scheme exist and commute with arbitrary base change. We also calculate the height and dimension of the exterior powers in terms of the height of the given $p$-divisible group or $\pi$-divisible \CO -module scheme