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\(K\)-theory and the bridge from motives to noncommutative motives. (English) Zbl 1315.14030
The theory of motives since the introduction of the Grothendieck program is one of the most important areas of research in modern mathematics. In the last decade of twentieth century, V. Voevodsky [Ann. Math. Stud. 143, 188–238 (2000; Zbl 1019.14009)] constructed motivic cohomology, and F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)] a general theory of motives. One of the main features of their theory is that it is universal with respect not only to the Weil cohomology, but for all generalized cohomology theories for schemes. The Morel-Voevodsky construction of motivic stable homotopy theory of schemes is based on the homotopy theory of spaces, its stabilization and then inversion of the Tate motive with respect to the monoidal multiplication. One of the main results of the paper is a precise formulation of the universal property for the Morel-Voevodsky construction. This is achieved by a characterization, for a given symmetric monoidal category \(\nu\), of the underlying symmetric monoidal \((\infty, 1)\)-category of the symmetric monoidal model category \(Sp^{\Sigma}(\nu , X)\) (cf. [M. Hovey, J. Pure Appl. Algebra 165, No. 1, 63–127 (2001; Zbl 1008.55006)]). This characterization is done by means of a universal property for symmetric monoidal \((\infty , 1)\)-categories. In the second part of the work the author constructs a stable motivic homotopy theory for the non-commutative spaces considered by Kontsevich. The main feature of this construction is a generalization of Nisnevich topology to the non-commutative setting so that it is compatible with the classical definition. This compatibility along with the mentioned above universal property yield the existence of the (monoidal) lower horizontal arrow in the following diagram \[ \begin{tikzcd} {\text{Classical Schemes}/k}\ar[r]\ar[d] & {\text{NC-Spaces}/k}\ar[d] \\ {\text{Stable Motivic Homotopy}/k} \ar[r, dashed] & {\text{NC-Stable Motivic Homotopy}/k} \end{tikzcd} \] In the third part of the paper the author proves that the non-connective \(K\)-theory of dg-categories introduced by M. Schlichting [Math. Z. 253, No. 1, 97–134 (2006; Zbl 1090.19002)] is the non-commutative Nisnevich sheafification of connective algebraic \(K\)-theory. Its further \({\mathbb A}^{1}\)-localization is a tensor unit for the noncommutative motives.The main consequence of this is a proof of the conjecture of Kontsevich which states that \(K\)-theory provides the right mapping space in noncommutative motives. The author finds also a canonical factorization of what he calls the motivic bridge through the \((\infty, 1)\)-category of modules over the commutative algebra object representing homotopy invariant algebraic \(K\)-theory of schemes. This turns out to be fully faithful over a field \(k\) with resolution of singularities. It is worth noting that different approach to non-commutative motives due to D.-C. Cisinski and G. Tabuada is presented in [J. K-Theory 9, No. 2, 201–268 (2012; Zbl 1256.19002)], [G. Tabuada, Duke Math. J. 145, No. 1, 121–206 (2008; Zbl 1166.18007), J. Noncommut. Geom. 7, No. 3, 767–786 (2013; Zbl 1296.14019)].

MSC:
14F42 Motivic cohomology; motivic homotopy theory
14F20 Étale and other Grothendieck topologies and (co)homologies
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
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