Exterior powers of Lubin-Tate groups (Q259495)

The main objective of this paper is to prove the existence of ``exterior powers'' of \(p\)-divisible groups of dimension at most one. More specifically, let \(S\) be a scheme and \(G\) a \(p\)-divisible group over \(S\) of height \(h\) and dimension at most one. The following theorem is the main theorem of this paper:NEWLINENEWLINETheorem. There exists a \(p\)-divisible group \(\bigwedge^rG\) over \(S\) of height \(\binom{h}{r}\), and an alternating morphism \(\lambda:G^r\to\bigwedge^rG\) such that for every morphism \(f:S'\to S\) and every \(p\)-divisible group \(H\) over \(S'\), we have the following isomorphism: NEWLINE\[NEWLINE\mathrm{Hom}_{S'}(f^\ast\bigwedge^rG,H)\to\mathrm{Alt}_{S'}^r(f^\ast G,H),\quad \psi\mapsto\psi\circ f^\ast\lambdaNEWLINE\]NEWLINE Moreover, the dimension of \(\bigwedge^rG\) at \(s \in S\) is \(\binom{h-1}{r-1}\) (resp. \(0\)) if the dimension of \(G\) at \(s\) is \(1\) (resp. \(0\)). The above theorem can be generalized to the so-called \(\pi\)-divisible modules over a locally Noetherian \(\mathcal O\)-scheme where \(\mathcal O\) is the ring of integers of a non-Archimedean local field of characteristic zero with uniformizer \(\pi\).NEWLINENEWLINEThere are many applications of exterior powers of \(\pi\)-divisible modules and \(p\)-divisible groups described in this paper and we mention one here. The exteriors powers of \(\pi\)-divisible modules together with their universal property can be used to construct explicitly the Lubin-Tate tower at infinite level in the equal characteristic case and the Rapoport-Zink spaces at infinite level in the mixed characteristic case.