Abstract
The present paper deals with algebraic tori and essential dimension in three unrelated contexts. After some preliminaries on essential dimension, versal torsors and tori, we explicitly construct a versal torsor for PGL n , n ⩾ 5 odd, defined over a field of transcendence degree 1 2 ( n − 1 ) ( n − 2 ) over the base field. This recovers a result of Lorenz, Reichstein, Rowen and Saltman. We also discuss the so-called “tori method” which gives a geometric proof of a result of Ledet on the essential dimension of a cyclic p-group. In the last section we compute the essential dimension of the functor K ↦ H 1 ( K , GL n ( Z ) ) , the latter set being in bijection with the isomorphism classes of n-dimensional K-tori.
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