Galois Module Structure of Galois Cohomology

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F . We determine the structure of the cohomology group H(U, Fp) as an Fp[GF /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H(U, Fp) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. Let F be a field containing a primitive pth root of unity ξp. Let GF be the absolute Galois group of F , U an open normal subgroup of GF of index p, and G = GF /U . Let E be the fixed field of U in the separable closure Fsep of F . Fix a ∈ F such that E = F ( p √ a), and let σ ∈ G satisfy p √aσ−1 = ξp. In the 1960s Z. I. Borevič and D. K. Faddeev classified the possible G-module structures of the first cohomology groups H(U, Fp) in the case F a local field [B]. Quite recently this result was extended for all fields F as above [MS]. For the study of Galois cohomology it is important to extend these results to all cohomology groups H(U, Fp), n ∈ N, and a solution of this problem was out of reach until now. Recently, based on earlier work of A. S. Merkurjev, M. Rost and A. A. Suslin, V. Voevodsky established the Bloch-Kato Conjecture [V1, V2], and it turns out that some of the main theorems in his proof are sufficient to determine the structure of all G-Galois modules H(U, Fp), using only simple arithmetical invariants attached to the field extension E/F . The theorems we use (quoted as Theorems 3 and 4 in section 1 below) had, in fact, been standard conjectures on Galois cohomology. Date: October 29, 2004. Research supported in part by NSERC grant R3276A01. Research supported in part by NSERC grant R0370A01. Supported by the Mathematical Sciences Research Institute, Berkeley. Research supported in part by National Security Agency grant MDA904-02-10061. 1 2 NICOLE LEMIRE, JÁN MINÁČ, AND JOHN SWALLOW It is interesting to point out, however, that the case n = 2 could have already been settled some 20 years ago, thanks to the work of Merkurjev and Suslin [MeSu]. The main ingredient for our determination of the G-module structure of H(U, Fp) is Milnor K-theory. (See [Mi] and [FV, Chap. IX].) For i ≥ 0, let KiF denote the ith Milnor K-group of the field F , with standard generators denoted by {f1, . . . , fi}, f1, . . . , fi ∈ F \ {0}. For α ∈ KiF , we denote by ᾱ the class of α modulo p, and we use the usual abbreviation knF for KnF/pKnF . We write NE/F for the norm map KnE → KnF , and we use the same notation for the induced map modulo p. We denote by iE the natural homomorphism in the reverse direction. We also apply the same notation NE/F and iE for the corresponding homomorphisms between cohomology groups. The image of an element α ∈ KiF in H (GF , Fp) we also denote by α. Voevodsky’s proof of the Bloch-Kato Conjecture establishes a GF -isomorphism H(U, Fp) ∼= knE. We formulate our results in terms of Galois cohomology for intended applications, but we use Milnor K-theory in our proof. We concentrate upon the case when F is a perfect field, and at the end of the paper we indicate how one may reduce the case of an imperfect field F to the case of characteristic 0. Our decomposition depends on four arithmetic invariants Υ1, Υ2, y, z, which we define as follows. First, for an element ᾱ of kiF , let annkn−1F ᾱ = ann ( kn−1F ᾱ·− −−→ kn−1+iF ) denote the annihilator of the product with ᾱ. When the domain of ᾱ is clear, we omit the subscript on the map and write simply ann ᾱ. Because we will often use the elements {a}, {ξp}, {a, a}, and {a, ξp}, we omit the bars for these elements. We also omit the bar in the element { p √a} ∈ knE. Fix n ∈ N and U an open normal subgroup of GF of index p with fixed field E. Define invariants associated to E/F and n as follows: d := dimFp knF / NE/F knE e := dimFp NE/F knE Υ1 := dimFp ann{a, ξp} / ann{a} Υ2 := dimFp kn−1F / ann{a, ξp} GALOIS MODULE STRUCTURE OF GALOIS COHOMOLOGY 3