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A fiber dimension theorem for essential and canonical dimension

Part of: Algebraic geometry: Foundations Linear algebraic groups and related topics Forms and linear algebraic groups

Published online by Cambridge University Press:  04 December 2012

Roland Lötscher
Roland Lötscher*
Affiliation:
Mathematisches Institut der Ludwig-Maximilians-Universität München, Theresienstraße 39, D-80333 München, Germany (email: [email protected])
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Abstract

The well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:𝒳→𝒴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

Keywords

essential dimensioncanonical dimensionalgebraic groupfibercategory fibered in groupoidsalgebraic stackalgebraic torus

MSC classification

Secondary: 20G15: Linear algebraic groups over arbitrary fields 11E72: Galois cohomology of linear algebraic groups 14A20: Generalizations (algebraic spaces, stacks)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 148 - 174
Copyright
Copyright © The Author(s) 2012

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