KK-theory as an Eilenberg-Mac Lane spectrum

  • D. Kaledin


For an additive Waldhausen category linear over a ring kk, the corresponding KK-theory spectrum is a module spectrum over the KK-theory spectrum of kk. Thus if kk is a finite field of characteristic pp, then after localization at pp, we obtain an Eilenberg-MacLane spectrum -- in other words, a chain complex. We propose an elementary and direct construction of this chain complex that behaves well in families and uses only methods of homological algebra (in particular, the notions of a ring spectrum and a module spectrum are not used).