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Essential dimension and canonical dimension of gerbes banded by groups of multiplicative type

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Abstract

We study gerbes \(\mathcal X\) over a field banded by groups of multiplicative type \(A\) and prove the formula \({\text {ed}}_{p}(\mathcal X)={\text {cdim}}_{p}(\mathcal X)+{\text {ed}}_{p}(A)\) when \(A\) is the kernel of a homomorphism of algebraic tori \(Q\rightarrow S\) with \(Q\) invertible and \(S\) split. Here \(p\) is either a prime or \(0\). This result is applied to prove new results on the essential dimension of algebraic groups.

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Acknowledgments

I would like to thank Zinovy Reichstein for useful discussions about the topic of this paper. Moreover I am grateful to Philippe Gille for helping me with the proof of Corollary 2.

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Authors and Affiliations

  1. Mathematisches Institut der Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 , Munich, Germany

    Roland Lötscher

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  1. Roland Lötscher
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Correspondence to Roland Lötscher.

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The author acknowledges support from the Deutsche Forschungsgemeinschaft, GI 706/2-1.

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Lötscher, R. Essential dimension and canonical dimension of gerbes banded by groups of multiplicative type. Math. Z. 280, 469–483 (2015). https://doi.org/10.1007/s00209-015-1433-8

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