On the K-theory spectrum of a ring of algebraic integers, by W. G. Dwyer and S. A. Mitchell

Let l be an odd prime number, R the ring of l-integers in an algebraic number field, and KR the algebraic K-theory spectrum of R. (The word spectrum is used here in the sense of stable homotopy theory.) We give a simple formula for the l-adic topological K-theory of KR, in terms of standard number theoretic invariants of R. This formula leads to many other things. A few of them are:

  1. An explicit recipe for using classical invariants of R to construct a candidate C for the algebraic K-theory spectrum of R. If the Lichtenbaum/Quillen conjecture is true, then C essentially is the algebraic K-theory spectrum. The spectrum C is constructed from the arithmetic of R in a way which extends Quillen's construction of FPsi(q) from the Galois theory of a finite field F(q).
  2. Calculations in some cases of the l-adic topological K-theory of BGL(R).
  3. Conjectural calculations of the mod l cohomology of BGL(R). These calculations are contingent on the Quillen/Lichtenbaum conjecture.
There are similar results if R is replaced by an l-adic local field.

W. G. Dwyer <dwyer.1@nd.edu>
S. A. Mitchell <mitchell@math.washington.edu>