Invariants of finite groups acting non-linearly on rational function fields

Abstract

This paper has its origins in an investigation of the following problem: Let G be a finite group of automorphisms of a rational function field, G c Autk(k(xl, . . . , x,)). Suppose G also stabilizes the “flag of subfields” k(x,, . . . , Xi) 1 sic n. We denote the flag by p,, and write G c Autr,(gn). Under what conditions is the fixed field k(xi,. . . ,x,)~ a pure (i.e., purely transcendental) extension of k? A special case of this question was answered by Miyata [9], who generalized a well-known theorem of Fischer by showing that for G c Autk($“), k(xi,. . . , x,)G/k is pure whenever G acts “linearly,” i.e., via a representation p: G L, Autk(kxi 0 0 kx,). If Gc Autr,(g,,), the flag F,, induces another flag of fixed fields SF = #(xi,. . . , Xi)G}, and since purity of extensions is transitive, k(xl, . . . , x,)~ will be pure whenever each of the intermediate extensions k(xl, . . . , xi)G/k(xl, . . . , Xi-l)G is pure. Since each of these extensions has the form F(x)~/F~ for F = k(x,, . . . , xi-l) and x = xi, we are led to investigate the invariants of groups G c Aut(F(x), F) (the subgroup of Aut(F(x)) which stabilizes F). We find that extensions of the form F(x)~/F” are parameterized in a natural way by the cohomology set H*(G, PGL2(F)). This result is then used to prove that F(x)~/F~ is pure whenever G has odd order or F is a Ci field. We also show that the purity of F(x)~/F~ is completely determined by the 2-Sylow subgroups of G. Finally, using a non-rational variety constructed by Artin and Mumford, we produce an example of G c Autc(g3) where C(x, y, z)~/C is not pure. This example shows that the hypothesis of linear action cannot be dropped from Miyata’s theorem. I am indebted to Hyman Bass and Daniel Grayson for their valuable help on this paper.