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This paper concerns our earlier conjecture about the equivalence of a derivedcompletion construction applied to the representation spectrum of the absoluteGalois group of a geometric field is equivalent to the algebraic K-theory ofthe field. We prove that the p-adic version of the derived completionconstruction can be interpreted as the K-theory of a certain pro-scheme over analgebraically closed field contained within the field. In order to make thisconstruction, we use a special property of absolute Galois groups of geometricfield, namely that of total torsion freeness, and the paper also contains somepoperties of the representation theory of such groups.

K-theoretic descent and a motivic Atiyah-Segal theorem