Hostname: page-component-6766d58669-nqrmd Total loading time: 0 Render date: 2026-05-24T17:09:13.896Z Has data issue: false hasContentIssue false
  • English
  • Français

A fiber dimension theorem for essential and canonical dimension

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

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: Roland.Loetscher@mathematik.uni-muenchen.de)
Get access Rights & Permissions [Opens in a new window]

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)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 148 - 174
Copyright
Copyright © The Author(s) 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[BF03]Berhuy, G. and Favi, G., Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330.CrossRefGoogle Scholar
[BR05]Berhuy, G. and Reichstein, Z., On the notion of canonical dimension for algebraic groups, Adv. Math. 198 (2005), 128–171.CrossRefGoogle Scholar
[BR97]Buhler, J. and Reichstein, Z., On the essential dimension of a finite group, Compositio Math. 106 (1997), 159–179.CrossRefGoogle Scholar
[BRV07]Brosnan, P., Reichstein, Z. and Vistoli, A., Essential dimension and algebraic stacks I, http://www.math.uni-bielefeld.de/lag/man/275.html (31 October 2007).Google Scholar
[BRV10]Brosnan, P., Reichstein, Z. and Vistoli, A., Essential dimension, spinor groups and quadratic forms, Ann. of Math. (2) 171 (2010), 533–544.CrossRefGoogle Scholar
[BRV11]Brosnan, P., Reichstein, Z. and Vistoli, A., Essential dimension of moduli of curves and other algebraic stacks, J. Eur. Math. Soc. (JEMS) 13 (2011), 1079–1112 (with an appendix by Najmuddin Fakhruddin).Google Scholar
[CM12]Chernousov, V. and Merkurjev, A., Essential dimension of quadratic forms with trivial discriminant and Clifford invariant, www.mathematik.uni-bielefeld.de/LAG/man/455.pdf(2 January 2012).Google Scholar
[CS06]Chernousov, V. and Serre, J.-P., Lower bounds for essential dimensions via orthogonal representations, J. Algebra 305 (2006), 1055–1070.CrossRefGoogle Scholar
[Gar09]Garibaldi, S., Cohomological invariants: exceptional groups and spin groups, Mem. Amer. Math. Soc. 200 (2009), 67 (with an appendix by Detlev W. Hoffmann).Google Scholar
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, NY, 1977).CrossRefGoogle Scholar
[Kar00]Karpenko, N., On anisotropy of orthogonal involutions, J. Ramanujan Math. Soc. 15 (2000), 1–22.Google Scholar
[Kar05]Karpenko, N., Canonical dimension of orthogonal groups, Transform. Groups 10 (2005), 211–215.CrossRefGoogle Scholar
[Kar10]Karpenko, N., Canonical dimension, in Proceedings of the International Congress of Mathematicians, 2010 (ICM2010) (Hindustan Book Agency, New Delhi, 2010).Google Scholar
[Kar12]Karpenko, N., Incompressibility of quadratic Weil transfer of generalized Severi–Brauer varieties, J. Inst. Math. Jussieu. 11 (2012), 119–131.CrossRefGoogle Scholar
[KM06]Karpenko, N. and Merkurjev, A., Canonical p-dimension of algebraic groups, Adv. Math. 205 (2006), 410–433.CrossRefGoogle Scholar
[KM08]Karpenko, N. and Merkurjev, A., Essential dimension of finite p-groups, Invent. Math. 172 (2008), 491–508.CrossRefGoogle Scholar
[KMRT98]Knus, M.-A., Merkurjev, A., Rost, M. and Tignol, J.-P., The book of involutions (American Mathematical Society, Providence, RI, 1998), (with a preface in French by J. Tits).CrossRefGoogle Scholar
[LM00]Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, vol. 39 (Springer, Berlin, 2000).CrossRefGoogle Scholar
[Lot11]Lötscher, R., Essential dimension of involutions and subalgebras, Israel J. Math., to appear.Google Scholar
[LMMR11]Lötscher, R., MacDonald, M., Meyer, A. and Reichstein, Z., Essential dimension of algebraic tori, J. Reine Angew. Math., to appear.Google Scholar
[LMMR12]Lötscher, R., MacDonald, M., Meyer, A. and Reichstein, Z., Essential p-dimension of algebraic groups whose connected component is a torus, Algebra Number Theory, to appear, www.mathematik.uni-bielefeld.de/LAG/man/461.pdf.Google Scholar
[Mer08]Merkurjev, A., Essential p-dimension of finite groups, Preprint (2008), http://www.math.ucla.edu/∼merkurev/publicat.htm.Google Scholar
[Mer09]Merkurjev, A., Essential dimension, in quadratic forms – algebra, arithmetic, and geometry, Contemporary Mathematics, vol. 493 eds Baeza, R., Chan, W. K., Hoffmann, D. W. and Schulze-Pillot, R. (American Mathematical Society, Providence, RI, 2009), 299–326.CrossRefGoogle Scholar
[MPW96]Merkurjev, A., Panin, I. and Wadsworth, A., Index reduction formulas for twisted flag varieties. I, K-Theory 10 (1996), 517–596.CrossRefGoogle Scholar
[Ngu10]Nguyen, D.-T., On the essential dimension of unipotent algebraic groups, J. Pure Appl. Algebra, to appear, math.NT/1012.2984.Google Scholar
[Oes84]Oesterle, J., Nombre de Tamagawa et groupes unipotents en caractristique p, Invent. Math. 78 (1984), 13–88.CrossRefGoogle Scholar
[PV94]Popov, V. and Vinberg, E., Invariant theory, in Algebraic geometry. IV, Encyclopaedia of Mathematical Sciences, vol. 55 (Springer, Berlin, 1994).Google Scholar
[Rei10]Reichstein, Z., Essential dimension, in Proceedings of the International Congress of Mathematicians, 2010 (ICM2010) (Hindustan Book Agency, New Delhi, 2010).Google Scholar
[Ros99]Rost, M., On the Galois cohomology of spin(14), Preprint (1999), http://www.mathematik.uni-bielefeld.de/∼rost/spin-14.html.Google Scholar
[Ser02]Serre, J.-P., Galois cohomology, Springer Monographs in Mathematics (Springer, Berlin, 2002).Google Scholar
[Stacks] The Stacks Project Authors, Stacks Project, http://math.columbia.edu/algebraic_geometry/stacks-git .Google Scholar
[TV10]Tossici, D. and Vistoli, A., On the essential dimension of infinitesimal group schemes, Amer. J. Math., to appear, math.AG/1001.3988.Google Scholar
[Vis05]Vishik, A., On the Chow groups of quadratic Grassmannians, Doc. Math. 10 (2005), 111–130.CrossRefGoogle Scholar
[Wan11]Wang, J., The moduli stack of G-bundles, Preprint (2011), math.AG/1104.4828.Google Scholar
[Zai07]Zainoulline, K., Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants, Bull. Lond. Math. Soc. 39 (2007), 301–304.CrossRefGoogle Scholar