Galois Modules, Ideal Class Groups and Cubic Structures

We establish a connection between the theory of cyclotomic ideal class groups and the theory of “geometric” Galois modules and obtain results on the Galois module structure of coherent cohomology groups of Galois covers of varieties over Z. In particular, we show that an invariant that measures the obstruction to the existence of a virtual normal integral basis for the coherent cohomology of such covers is annihilated by Bernoulli numbers and the orders of even K-groups of Z. We also show that the existence of such a normal integral basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Our main tool is a theory of “hypercubic structures” for line bundles over group schemes.