Semilinear actions of Galois groups and the algebraic K-theory of fields

We verify a special case of a conjectural description of the completed algebraic K-theory spectrum of a field F that relies solely on the semilinear representation theory of the absolute Galois group GF.;The conjecture applies a completion construction of G. Carlsson, called the derived completion, to K-theory spectra of categories of semilinear representations. Let k be a field equipped with an action by GF. Then we denote by Rep k⟨GF⟩ the category of finite dimensional continuous k-semilinear representations of GF and by &parl0;KRepk&angl0;GF &angr0;&parr0;∧ap the derived completion at a prime p of the K-theory spectrum KRepk⟨ GF⟩. Let -∧ p denote the p-adic completion [BK72].;Conjecture (Carlsson). If k is any algebraically closed subfield of F, then extension of scalars induces a weak equivalence, KRepk GF ∧a p→&parl0;KRepF &angl0;GF&angr0;&parr0;∧ ap≃ KF∧ p.;Let ℓ be a prime different from p and F the colimit of all extensions of degree prime to both ℓ and p of the field of Laurent series Fℓ ((x)). Denote by GtF the tame Galois group and by Ft the maximal tame extension. In Chapter 4, we prove the following theorem.;Theorem. The map E:&parl0;KRepF ℓ&angl0;Gt F&angr0;&parr0;∧ap →&parl0;KRepFt &angl0;GtF&angr0;&parr0;∧ ap≃&parl0;KF&parr0; ∧p, is a weak equivalence of ring spectra.;An essential step in the proof was computing the Grothendieck ring pi 0KRepk⟨ GtF ⟩. This ring can be expressed as a colimit of rings of the form pi 0KE⟨G⟩, where E is a field and G is a finite group which acts on E and whose order is invertible in E. We therefore devote Chapter 2 to the study of skew group rings.;In Chapter 2, which is joint work with K. Ribet, we describe a skew group ring E⟨G⟩ as a product of matrix rings over division rings when the subgroup N of elements of G that act trivially on E is abelian. There is a natural element of H2(Gal( E/EG),N) described by the action of G on E. We show that this element determines the image of E⟨G⟩ in the Brauer group of the center of E⟨G⟩. If W and W' are finitely generated modules over E⟨G⟩, we decompose the E⟨ G⟩-module W⊗EW' as a direct sum of simple modules where the individual summands are described in terms of combinatorial data.