On the K-theory Spectrum of a Ring of Algebraic Integers

Suppose that F is a number field (i.e. a finite algebraic extension of the field Q of rational numbers) and that OF is the ring of algebraic integers in F . One of the most fascinating and apparently difficult problems in algebraic K-theory is to compute the groups KiOF . These groups were shown to be finitely generated by Quillen [36] and their ranks were calculated by Borel [7]. Lichtenbaum and Quillen have formulated a very explicit conjecture about what the groups should be [37]. Nevertheless, there is not a single number field F for which KiOF is known for any i ≥ 5. For most fields F , these groups are unknown for i ≥ 3. The groups KiOF are the homotopy groups of a spectrum KOF (“spectrum” in the sense of algebraic topology [1]). In this paper, rather than concentrating on the individual groups KiOF , we study the entire spectrum KOF from the viewpoint of stable homotopy theory. For technical reasons, it is actually more convenient to single out a rational prime number ` and study instead the spectrum KRF , where RF = OF [1/`] is the ring of algebraic `-integers in F . “At `”, that is, after localizing or completing at `, the spectra KOF and KRF are almost the same (cf. [34] and [35, p. 113]).