Essential Dimension in Mixed Characteristic

Brosnan, Patrick ; Reichstein, Zinovy ; Vistoli, Angelo

Documenta mathematica, Tome 23 (2018), pp. 1587-1600.

Résumé

Let $G$ be a finite group, and let $R$ be a discrete valuation ring with residue field $k$ and fraction field $K$. We say that $G$ is weakly tame at a prime $p$ if it has no non-trivial normal $p$-subgroups. By convention, every finite group is weakly tame at $0$. We show that if $G$ is weakly tame at char(k), then $\operatorname{ed}_K(G) \geq \operatorname{ed}_k(G)$. Here $\operatorname{ed}_F(G)$ denotes the essential dimension of $G$ over the field $F$. We also prove a more general statement of this type, for a class of étale gerbes $\mathcal{X}$ over $R$. As a corollary, we show that if $G$ is weakly tame at $p$, then $\operatorname{ed}_{L} G \geq \operatorname{ed}_{k} G$ for any field $L$ of characteristic $0$ and any field $k$ of characteristic $p$, provided that $k$ contains $\overline {\mathbb{F}}_{p}$. We also show that a conjecture of A. Ledet asserting that $\operatorname{ed}_k \mathbb{Z}/p^n \mathbb{Z}) = n$ for a field $k$ of characteristic $p>0$ implies that $\operatorname{ed}_{\mathbb{C}}(G) \geq n$ for any finite group $G$ which is weakly tame at $p$ and contains an element of order $p^n$. To the best of our knowledge, an unconditional proof of the last inequality is out of the reach of all presently known techniques.

Détail

Classification : 14A20, 13A18, 13A50
Keywords: essential dimension, Ledet's conjecture, genericity theorem, gerbe, mixed characteristic