Reichstein, Zinovy; Youssin, Boris Splitting fields of \(G\)-varieties. (English) Zbl 1054.14063 Pac. J. Math. 200, No. 1, 207-249 (2001). Let \(G\) be an algebraic group defined over an algebraically closed field \(k.\) Let \(X\) be a generically free \(G\)-variety, and let \(K/k\) be a field extension of finite transcendence degree. Let \(u\in H^{1}\left( K,G\right) \) correspond to the class of \(X\). An extension \(L/K\) is a splitting field for \( u\;\)if the natural map \(H^{1}\left( K,G\right) \rightarrow H^{1}\left( L,G\right) \) carries \(u\) to \(1\) (the distinguished element in \(H^{1}\left( L,G\right) \)). Furthermore if \(L/K\) is a finite Galois extension then Gal\((L/K)\) is called the splitting group for \(X\). This paper is a study of the splitting fields of these varieties.For \(X\) a primitive (generically free) \(G\)-variety we can let \(K\) be the set of points in \(k\left( X\right) \) fixed by \(G\). Let \(L/K\) be a splitting field for \(X\). The first main result of this paper is that if \(X\) has a smooth point fixed by a finite abelian \(p\)-subgroup \(H\) of \(G\) then \(\left[ L:K\right] \) is divisible by \([H:H_T]\) for some toral subgroup \(H_T\) of \(H\). This can be viewed as giving a “lower bound” on a splitting field. Several examples of this result are given, including the exceptional groups \(G_2\), \(F_4\), \(3E_6\), \(E_7\) and \(E_8\). Much work is offered as well in the case where \(G=\)P\(\text{GL}_n\). The second main result is the following. Again suppose \(X\) is primitive, and let \(A\) be a splitting group for \(X\). Again suppose \(X\) has a smooth point fixed by a finite abelian \(p\)-subgroup \(H\) of \(G.\) Then \(A\) contains a copy of \(H/H_{T}\) for some toral subgroup \(H_{T}\) of \(H.\) Thus this puts a lower bound on the splitting group for \(X\). The \(E_{7}\) and \(E_{8}\) examples are revisited here.The authors then obtain an application to the second result. Let \(Z\left( p^{r}\right) \) be the center of the universal division algebra UD\(\left( p^{r}\right) .\) Let \(K\) be a field extension of \(Z\left( p^{r}\right) \) and let \(D=\)UD\(\left( p^{r}\right) \otimes _{Z\left( p^{r}\right) }K.\) If \(A\) is a splitting group for \(D\) then \(p^{2r-2e-2}\) divides \(\left| A\right| ,\) where \(p^{e}\) is the highest power of \(p\) dividing \([K:Z(p^r)]\). From this it follows that if \(r\geq 2e+3\) then \(D\) is a noncrossed product.Finally, the paper addresses the problem of constructing noncrossed product examples whose centers are function fields \(K\) with “small” transcendence degree. The result is as follows. For \(r\geq 2\) and \(p\) a prime there exists a division algebra \(D\) of degree \(p^r\) with center \(K\) such that trdeg\(_k(K)=6\) and no prime-to-\(p\) extension of \(D\) is a crossed product. Reviewer: Alan Koch (Decatur) Cited in 1 ReviewCited in 11 Documents MSC: 14L30 Group actions on varieties or schemes (quotients) 16K20 Finite-dimensional division rings 16S35 Twisted and skew group rings, crossed products Keywords:splitting groups; noncrossed product examples; division algebra PDFBibTeX XMLCite \textit{Z. Reichstein} and \textit{B. Youssin}, Pac. J. Math. 200, No. 1, 207--249 (2001; Zbl 1054.14063) Full Text: DOI arXiv