nLab algebraic K-theory spectrum

The term algebraic K-theory spectrum refers to one of the following two notions:

This entry is about the latter notion; for the former one, see algebraic K-theory.


Suppose that SS is a regular scheme. Then there exists a motivic spectrum KGL SKGL_S with the property that, for every XSm/SX\in Sm/S,

KGL p,q(X)=K 2qp(X), KGL^{p,q}(X) = K_{2q-p}(X),

where K *(X)K_*(X) are the algebraic K-theory groups of XX defined by Quillen. In particular, KGLKGL-cohomology is (2,1)(2,1)-periodic: this is Bott periodicity for algebraic K-theory.

The multiplicative structure of algebraic K-theory makes KGLKGL into a ring spectrum (up to homotopy), which comes from a unique structure of E E_\infty-algebra (see Naumann-Spitzweck-Ostvaer).

Over non-regular schemes, the motivic spectrum KGLKGL is also defined and it represents Weibel’s homotopy invariant version of algebraic K-theory (see Cisinski13).


Last revised on September 12, 2015 at 13:59:28. See the history of this page for a list of all contributions to it.