Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noncommutative Motives II: K-Theory and Noncommutative Motives   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.

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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int