On the K-theory spectrum of a smooth curve over a finite field

Abstract

Suppose that X is a nonsingular curve over a finite field F. According to ideas of Quillen and others, there is an algebraic K-theory spectrum KX associated to the category of locally free sheaves on X . The word “spectrum” here designates an object in the stable homotopy category of spaces. If X = SpecO is affine, then KX is equivalent to the usual K-theory spectrum associated to the category of finitely generated projective modules over O. A certain amount is known about the groups KiX = πiKX. They are abelian, and they vanish for i < 0. They are finitely generated and with a few exceptions are actually finite: if X is complete then K0X has rank 2, while if X is affine then K0X has rank 1 and K1X has rank s−1, where s is the number of points at infinity of X. For i ≤ 2, KiX can be described in fairly classical terms (see §2). In general, though, these groups seem very difficult to pin down. In this paper we detour around the groups KiX in favor of studying the entire spectrum KX, or more accurately the `-completion KX of KX, where ` is an odd rational prime different from the characteristic of F. Let K be the `-completion of the periodic topological complex K-theory spectrum and K∗ the associated cohomology theory. We begin by calculating K∗(KX) = K∗(KX) as a module over the ring of cohomology operations in K∗ (5.1, 5.2). The answer is surprisingly simple and leads to a formula for the `-adic topological K-theory K(Ω0 KX) of the basepoint component of the zero space in the associated Ω-spectrum (7.5). If X = SpecO is affine, then K(Ω0 KX) is the same as K∗(BGL(O)), so in this case we obtain a formula for the `-adic topological K-theory of the classifying space of the infinite general linear group of O. Next we compute the homotopy type of a certain localization L(KX) = L(KX) of KX; this is localization in the sense of Bousfield with respect to the cohomology theory K∗ (4.2). The homotopy groups πiL(KX) do not vanish for i < 0, but there is a form of the Lichtenbaum-Quillen conjecture which suggests that the localization map KX −→ L(KX)