Galois Module Structure of Unramified Covers

Let G be a finite group. Suppose that Y is a projective algebraic variety over Z (i.e an integral scheme which is projective and flat over Spec (Z)) of relative dimension d. In this paper, we consider finite Galois covers π : X → Y with group G which are everywhere unramified, i.e “G-torsors”. Let F be a G-equivariant coherent sheaf on X. Consider the value of the right derived global section functor at F . This is a complex RΓ(X,F) in the derived category of complexes of modules over the group ring Z[G] whose cohomology groups are the finitely generated G-modules Hi(X,F). The central question in the theory of “additive” Galois module structure is: Existence of a normal integral basis: Is the complex RΓ(X,F) isomorphic to a bounded complex of finitely generated free Z[G]-modules? By an observation of Chinburg [C] (which essentially goes back to Noether), the complex RΓ(X,F) is always perfect, i.e is isomorphic to a bounded complex of finitely generated projective Z[G]-modules. It follows that the obstruction to a positive answer to our question is given by the “stable projective Euler characteristic” of RΓ(X,F). By definition, this is an element χ̄P (F) in the class group Cl(Z[G]) = K0(Z[G])/±{free classes} of finitely generated projective Z[G]-modules. Let N/K be a Galois extension of number fields with group G which is unramified at all finite places. Then we can take X = Spec (ON ), Y = Spec (OK), with ON , OK the corresponding rings of integers. Our question for F = OX amounts to asking if the ring of integers ON is a stably free Z[G]-module; this type of problem has a long history going back to Hilbert and Noether. By “Fröhlich’s conjecture” (shown by M. Taylor; see [Ta], [F]) the answer is positive if in addition either G is abelian or N/K is also unramified at the infinite places. Without these assumptions it is known that ON ⊕ON is always a free Z[G]-module; this is equivalent to the statement 2 · χ̄P (OX) = 0. In this paper we establish a connection between the problem of the vanishing of the obstructions χ̄P (OX) for higher dimensional varieties, and the theory of cyclotomic ideal class groups. As we will explain below, our results indicate a close connection between a positive answer to the above question for all unramified Galois covers X → Y of prime order p > dim(Y ) and the truth of Vandiver’s conjecture for p. Vandiver’s conjecture for the prime number p is the statement that p does not divide the class number hp = #Cl(Q(ζp + ζ −1 p )).