nLab algebraic K-theory spectrum

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

• an object $K(C) \in \Spt$ in the stable (infinity,1)-category of spectra, associated to a stable (infinity,1)-category or symmetric monoidal (infinity,1)-category $C$, whose homotopy groups give the algebraic K-theory of $C$;

• a motivic spectrum $KGL \in SH(S)$ in the stable motivic homotopy category, representing an $\mathbf{A}^1$-homotopy invariant version of algebraic K-theory, where $S$ is a scheme.

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).

Related concepts

• algebraic K-theory
• motivic homotopy theory
• motivic cohomology
• algebraic cobordism

References

• Fabien Morel, Vladimir Voevodsky, $\mathbb{A}^1$-homotopy theory of schemes K-theory, 0305 (web pdf)

• Vladimir Voevodsky, $\mathbf{A}^1$-Homotopy Theory, Doc. Math., Extra Vol. ICM 1998(I), 417-442, web.

• Denis-Charles Cisinski, Descente par éclatements en K-théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), no. 2, pp. 425–448 (pdf)

• Niko Naumann, Markus Spitzweck, Paul Arne Østvær, Existence and uniqueness of E-infinity structures on motivic K-theory spectra, (arXiv)

• Ivan Panin, K. Pimenov, Oliver Röndigs, On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory, Inventiones mathematicae, 2009, 175(2), 435-451, DOI, arXiv.

• David Gepner, Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic K-theory, Documenta Math. 2008 (arXiv:0712.2817).