Noncommutative Motives II: K-Theory and Noncommutative Motives

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We continue the work initiated in arXiv:1206.3645, where we introduced a new stable symmetric monoidal (∞,1)-category SHnc encoding a motivic stable homotopy theory for the noncommutative spaces of Kontsevich and obtained a canonical monoidal colimit-preserving functor SH→ SHnc relating this new theory to the (∞,1)-category encoding the stable motivic A¹ theory of Morel-Voevodsky. For a scheme X this map recovers the dg-derived category of perfect complexes Lpe(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 ∞-presheaves over the category of noncommutative smooth spaces and therefore provide objects in SHnc 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 A¹-localization is an object in SHnc. 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→ SHnc through ModKH(SH) - the (∞,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.