Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

Let F be a field of characteristic p and let F[V ] denote the symmetric algebra on the dual V of V = Fn. Let H be an unstable Noetherian integral domain and denote by P p H the inseparable extension of H over the Steenrod algebra P of reduced powers. In [Mem. Amer. Math. Soc. 146 (2000), no. 692, x+158 pp.; MR1693799 (2000m:55023)] M. D. Neusel showed that H can be embedded into F[V ] and P p H = F[V ]G, for some finite subgroup G of GL(V ). In the paper under review the author extends these results and those in [C. W. Wilkerson, Jr., in Recent progress in homotopy theory (Baltimore, MD, 2000), 381–396, Contemp. Math., 293, Amer. Math. Soc., Providence, RI, 2002; MR1890745 (2003d:55020)], proving that, if the extension H ,!F[V ]G is of exponent e, then V decomposes as a direct sum of subspaces W0,W1, . . . ,We such that G acts on the flags W0 W1 · · · Wi, for i = 0, . . . , e. Further, the integral closure H of H is equal to the G-invariant subalgebra (F[W0] F[W1]p · · · F[We]pe)G, and the field of fractions H of H is equal to (F(W0) F(W1)p · · · F(We)pe)G. It turns out that G consists of flag matrices whose form depends on the vector space dimensions of the Wi’s. Neusel also proves the following results: H is Cohen-Macaulay if and only if P p H is Cohen-Macaulay; H is polynomial if and only if P p H is polynomial and the generators are pth powers/roots of one another. Combining these results, a twenty-year-old conjecture formulated byWilkerson is settled (see Conjecture 5.1 in [C.W. Wilkerson, Jr., op. cit.]).