The higher K-theory of Real Curves, by Claudio Pedrini and Charles A. Weibel

Let X be a smooth curve defined over the real numbers. Classical calculations dating to 1883 clearly show the relation between the Picard group of X (hence K_0X) and the two topological invariants of the underlying space X(R) of real points of X: the number of connected components and the number of loops of X(R); these coincide if X is projective.

In this paper we show that if n>0 the groups K_n(X) are the direct sum of a divisible group and an elementary abelian 2-group. We also compute the torsion subgroup explicitly; it is periodic of period 8. The rank of the elementary 2-group is determined by the topological invariants, and whether or not X is projective. The divisible torsion is a finite sum of copies of (Q/Z), determined by the genus and the number of real and complex points at infinity.

The methods use motivic cohomology and the spectral sequence of Friedlander and Suslin. The results show that (with finite coefficients) the K-groups of X coincide with Atiyah's Real K-theory of the space with involution X(C), including K_0.

This paper has appeared in K-theory 27 (2002), 1-31.

Claudio Pedrini <>
Charles A. Weibel <>