Tori and essential dimension
This paper establishes three interesting, but unrelated results about essential dimensions. The authors use geometric tools, in particular, those of algebraic tori, to prove these results. The study of essential dimension of a linear algebraic group (= a smooth affine group scheme) \(G\) over a field \(k\), denoted by \(ed_k(G)\), was initiated by Reichstein and his coauthors [see \textit{J. Buhler} and \textit{Z. Reichstein}, Compos. Math. 106, No. 2, 159--179 (1997; Zbl 0905.12003); \textit{Z. Reichstein}, Transform. Groups 5, No. 3, 265--304 (2000; Zbl 0981.20033); \textit{G. Berhuy} and \textit{G. Favi}, Doc. Math., J. DMV 8, 279--330 (2003; Zbl 1101.14324)]. Note that it is permissible that \(G\) is a finite group. The definition may be summarized as follows: \(\text{ed}_k(G)= \min \{ \dim_k(Y) \}\) where \(Y\) runs over irreducible \(k\)-varieties satisfying that there is is a \(k\)-variety \(X\) and a \(k\)-morphism \(f: X \rightarrow Y\) so that \(f: X \rightarrow Y\) is a versal (or classifying) \(G\)-torsor over \(Y\) (see the first section of the present paper for the precise definitions of \(G\)-torsor and versal \(G\)-torsors). The definition of essential dimension of a linear algebraic group was generalized by Merkurjev (1999) to that of a functor \(F: C_k \rightarrow Sets\), denoted by \(\text{ed}_k(F)\), where \(C_k\) is the category of field extensions of a field \(k\) and \(Sets\) is the category of sets (see Definition 1.14 of the present paper). In particular, if \(G\) is a linear algebraic group over a field \(k\), then \(\text{ed}_k(G)=\text{ed}_k(F)\) where \(F\) is the functor defined by \(F(K)=H^1(K, G)\) for any field extension \(K\) of the field \(k\). The first two sections of the paper under review provides a clear and convenient preliminary of torsors, essential dimensions and tori. If \(n\) is an odd integer and \(n \geq 5\), a versal torsor for \(\text{PGL}_n\) over any base field \(k\) is constructed explictly. In particular, the dimension of this versal torsor over \(k\) is \((n-1)(n-2)/2\). This gives a geometric proof of a previous result of Lorenz, Reichstein etc. [see \textit{M. Lorenz, Z. Reichstein, L. H. Rowen} and \textit{D. J. Saltman}, J. Lond. Math. Soc., II. Ser. 68, No. 3, 651--670 (2003; Zbl 1071.16012); \textit{M. Lorenz} and \textit{Z. Reichstein}, \url{arXiv:math.RA/0001026} (2000)]. Another result is to show that, for an odd prime number \(p\neq \text{char } k\), for a positive integer \(r\), if \(G\) is the semi-direct product of \(\mathbb{Z}/p^r\mathbb{Z}\) with the Galois group \(\text{Gal}(k(\zeta)/k)\) (\(\zeta\) being a primitive \(p^r\)-th root of unity), then \(\text{ed}_k(G)\leq p^d \phi(q)\) where the order of \(\text{Gal}(k(\zeta)/k)\) is equal to \( p^dq\) with \(q \mid p-1\). Again the authors give a geometric proof of a previous result [\textit{A. Ledet}, Can. Math. Bull. 45, No. 3, 422--427 (2002; Zbl 1024.12004)]. The final result is to study \(\text{ed}_k(\text{GL}_n(\mathbb{Z}))\), the essential dimension of the functor \(F\) defined by \(F(K)=H^1(K, \text{GL}_n(\mathbb{Z}))\). It is shown that \(\text{ed}_k(GL_n(Z))= \max \{\text{ed}_k(G) \}\) where \(G\) runs over all the finite subgroups of \(\text{GL}_n(\mathbb{Z})\). A similar result for \(\text{SL}_n(\mathbb{Z})\) is obtained also.
scientific article; zbMATH DE number 5303137
scientific article