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Kontsevich’s noncommutative numerical motives. (English) Zbl 1269.18004

M. Kontsevich constructed the rigid symmetric category \(NC_{\mathrm{num}}(k)_{F}\) of noncommutative numerical motives over ground field \(k\) and with coefficients in a field \(F\). This construction relies on the existence of a bilinear form, with good properties, on the Grothendieck group of each smooth and proper \(dg\) category. In their recent work (cf. [the authors, “Noncommutative motives, numerical equivalence, and semi-simplicity”, arxiv:1105.2950]) the authors constructed an alternative rigid symmetric monoidal category \(N\mathrm{Num}(k)_{F}\) of noncommutative motives. Their approach uses Hochschild homology. The main result of the paper is the equivalence of both rigid symmetric monoidal categories. As a consequence they obtain that the category \(NC_{\mathrm{num}}(k)_{F}\) is abelian semi-simple for \(k\) and \(F\) of the same characteristic. This result was conjectured by Kontsevich in the special case \(F={\mathbb Q}\) and \(k\) of characteristic zero.

MSC:

18D20 Enriched categories (over closed or monoidal categories)
18F30 Grothendieck groups (category-theoretic aspects)
18G55 Nonabelian homotopical algebra (MSC2010)
19A49 \(K_0\) of other rings
19D55 \(K\)-theory and homology; cyclic homology and cohomology
14A22 Noncommutative algebraic geometry
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References:

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