## Essential dimension in mixed characteristic. (arXiv:1801.02245v2 [math.AG] UPDATED)

Suppose \$G\$ is a finite group and \$p\$ is either a prime number or \$0\$. For \$p\$ positive, we say that \$G\$ is weakly tame at \$p\$ if \$G\$ has no non-trivial normal \$p\$-subgroups. By convention we say that every finite group is weakly tame at \$0\$. Now suppose that \$G\$ is a finite group which is weakly tame at the residue characteristic of a discrete valuation ring \$R\$. Our main result shows that the essential dimension of \$G\$ over the fraction field \$K\$ of \$R\$ is at least as large as the essential dimension of \$G\$ over the residue field \$k\$. We also prove a more general statement of this type for a class of \'etale gerbes over \$R\$. As a corollary, we show that, if \$G\$ is weakly tame at \$p\$ and \$k\$ is any field of characteristic \$p >0\$ containing the algebraic closure of \$\mathbb{F}_p\$, then the essential dimension of \$G\$ over \$k\$ is less than or equal to the essential dimension of \$G\$ over any characteristic \$0\$ field. A conjecture of A. Ledet asserts that the essential dimension, \$\math 查看全文>>

﻿