Essential dimension: A functorial point of view (after A. Merkurjev)

Abstract

In these notes we develop a systematic study of the essential dimension of functors. This approach is due to A. Merkurjev and can be found in his unpublished notes [12]. The notion of essential dimension was earlier introduced for finite groups by J. Buhler and Z. Reichstein in [3] and for an arbitrary algebraic group over an algebraically closed field by Z. Reichstein in [14]. This is a numerical invariant depending on the group $G$ and the field $k$. This number is denoted by ${\mathrm{ed}}_k(G)$. In this paper we insist on the behaviour of the essential dimension under field extension $k^{\prime }/k$ and try to compute ${\mathrm{ed}}_k(G)$ for any $k$. This will be done in particular for the group $\mathbb{Z}/n$ when $n\le 5$ and for the circle group. Along the way we define the essential dimension of functor with versal pairs and prove that all the different notions of essential dimension agree in the case of algebraic groups. Applications to finite groups are given. Finally we give a proof of the so-called homotopy invariance, that is ${\mathrm{ed}}_k(G)={\mathrm{ed}}_{k(t)}(G)$, for an algebraic group $G$ defined over an infinite field $k$.