# 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.

## Idea

Suppose that $S$ is a regular scheme. Then there exists a motivic spectrum $KGL_S$ with the property that, for every $X\in Sm/S$,

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

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

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

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

## References

Last revised on September 12, 2015 at 09:59:28. See the history of this page for a list of all contributions to it.