Philippe Gille and Zinovy Reichstein: A lower bound on the essential dimension of a connected linear group,

Submission: 2007, Apr 12

Let G be a connected linear algebraic group defined over an algebraically closed field k and H be a finite abelian subgroup of G whose order is prime to char(k). We show that the essential dimension of G is bounded from below by rank(H) - rank C_G(H)^0, where rank C_G(H)^0 denotes the rank of the maximal torus in the centralizer C_G(H). This inequality, conjectured by J.-P. Serre, generalizes previous results of Reichstein - Youssin (where char(k) is assumed to be 0 and C_G(H) to be finite) and Chernousov - Serre (where H is assumed to be a 2-group).

2000 Mathematics Subject Classification: 11E72, 20G10, 14L30

Keywords and Phrases: Linear algebraic group, essential dimension, non-abelian cohomology, group action

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