Essential Dimension in Mixed Characteristic
Documenta mathematica, Tome 23 (2018), pp. 1587-1600
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 edK(G)≥edk(G). Here edF(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 X over R. As a corollary, we show that if G is weakly tame at p, then edLG≥edkG for any field L of characteristic 0 and any field k of characteristic p, provided that k contains Fp. We also show that a conjecture of A. Ledet asserting that edkZ/pnZ)=n for a field k of characteristic p>0 implies that edC(G)≥n for any finite group G which is weakly tame at p and contains an element of order pn. To the best of our knowledge, an unconditional proof of the last inequality is out of the reach of all presently known techniques.
Classification :
13A18, 13A50, 14A20
Mots-clés : gerbe, essential dimension, Ledet's conjecture, genericity theorem, mixed characteristic
Mots-clés : gerbe, essential dimension, Ledet's conjecture, genericity theorem, mixed characteristic
@article{10_4171_dm_653,
author = {Patrick Brosnan and Zinovy Reichstein and Angelo Vistoli},
title = {Essential {Dimension} in {Mixed} {Characteristic}},
journal = {Documenta mathematica},
pages = {1587--1600},
year = {2018},
volume = {23},
doi = {10.4171/dm/653},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/653/}
}
Patrick Brosnan; Zinovy Reichstein; Angelo Vistoli. Essential Dimension in Mixed Characteristic. Documenta mathematica, Tome 23 (2018), pp. 1587-1600. doi: 10.4171/dm/653
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