Noncommutative Motives II: K-Theory and Noncommutative Motives

Abstract

We continue the work initiated in arXiv:1206.3645, where we introduced a new stable symmetric monoidal $(\infty,1)$-category $SH_{nc}$ encoding a motivic stable homotopy theory for the noncommutative spaces of Kontsevich and obtained a canonical monoidal colimit-preserving functor $SH\to SH_{nc}$ relating this new theory to the $(\infty,1)$-category encoding the stable motivic $\mathbb{A}^1$ theory of Morel-Voevodsky. For a scheme $X$ this map recovers the dg-derived category of perfect complexes $L_{pe}(X)$. In this sequel we address the study of the different flavours of algebraic $K$-theory of dg-categories. As in the commutative case, these can be understood as spectral valued $\infty$-presheaves over the category of noncommutative smooth spaces and therefore provide objects in $SH_{nc}$ once properly localized. Our first main result is the description of non-connective $K$-theory of dg-categories introduced by Schlichting as the noncommutative Nisnevich sheafification of connective $K$-theory. In particular it follows that its further $\mathbb{A}^1$-localization is an object in $SH_{nc}$. As a corollary of the recent result in A. Blanc Phd thesis, we prove that this object is a unit for the monoidal structure. Using this, we obtain a precise proof for a conjecture of Kontsevich claiming that $K$-theory gives the correct mapping spaces in noncommutative motives. As a second corollary we obtain a factorization of our comparison map $SH\to SH_{nc}$ through $Mod_{KH}(SH)$ - the $(\infty,1)$-category of modules over the commutative algebra object $KH$ representing homotopy invariant algebraic $K$-theory of schemes in $SH$. If $k$ is a field admitting resolutions of singularities, this factorization is fully faithful, so that, at the motivic level, no information (below $K$-theory) is lost by passing to the noncommutative world.