Galois Module Structure of Milnor K - Theory

Let E be a cyclic extension of pth-power degree of a field F of characteristic p. For all m, s ∈ N, we determine KmE/p sKmE as a (Z/pZ)[Gal(E/F )]-module. We also provide examples of extensions for which all of the possible nonzero summands in the decomposition are indeed nonzero. Let F be a field of characteristic p. Let KmF denote the mth Milnor K-group of F and kmF = KmF/pKmF . (See, for instance, [M] and [FV, IX.1].) If E/F is a Galois extension of fields, let G = Gal(E/F ) denote the associated Galois group. In [BLMS] the structure of kmE as an FpG-module was determined when G is cyclic of pth-power order. In this paper we determine the Galois module structure ofKmE modulo p s for s ∈ N and these same G. We also provide examples of extensions for which the possible free summands in the decomposition are all nonzero. These examples together with the results in [BLMS] show that the dimensions over Fp of indecomposable Fp[Gal(E/F )]-modules occuring as direct summands of kmE are all powers of p and that all dimensions p, i = 0, 1, . . . , n, indeed occur in suitable examples. Recall the theorem of Bloch-Kato and Gabber (see [BK]): the sequence 0 → kmF → ΩF P → ΩF /d Ω F is exact, where ΩF is themth graded component of the exterior algebra on Kähler differentials and P is the Artin-Schreier operator. In [I, §6], Izhboldin succeeded in providing an analogue of this important interpretation of kmF , as follows: for s ∈ N, the sequence 0 → KmF/pKmF δ → Q(F, s) P → Q(F, s) Date: February 12, 2006.