## The K-theory of fields in characteristic p (1996)

Citations: | 39 - 3 self |

### BibTeX

@TECHREPORT{Geisser96thek-theory,

author = {Thomas Geisser and Marc Levine},

title = {The K-theory of fields in characteristic p},

institution = {},

year = {1996}

}

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### Abstract

Abstract. The purpose of this paper is to study the p-part of motivic cohomology and algebraic K-theory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, Hi (k, Z(n)) is uniquely p-divisible for i ̸ = n. This implies that the natural map KM n (k) − → Kn(k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that Kn(k) is p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example Kn(X, Z/pr) = 0 for n> dimX. Another consequence is Gersten’s conjecture with mod p-coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except the vanishing conjecture. 1.