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18
A guided tour through the garden of noncommutative motives
, 2011
"... These are the extended notes of a survey talk on noncommutative motives given at the 3 era Escuela de Inverno Luis SantalóCIMPA Research ..."
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Cited by 15 (6 self)
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These are the extended notes of a survey talk on noncommutative motives given at the 3 era Escuela de Inverno Luis SantalóCIMPA Research
Noncommutative motives, numerical equivalence, and semisimplicity
, 2011
"... In this article we further the study of the relationship between pure motives and noncommutative motives, initiated in [25]. Making use of Hochschild homology, we introduce the category NNum(k)F of noncommutative numerical motives (over a base ring k and with coefficients in a field F). We prove t ..."
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In this article we further the study of the relationship between pure motives and noncommutative motives, initiated in [25]. Making use of Hochschild homology, we introduce the category NNum(k)F of noncommutative numerical motives (over a base ring k and with coefficients in a field F). We prove that NNum(k)F isabelian semisimpleand that Grothendieck’s category Num(k)Q of numerical motives embeds in NNum(k)Q after being factored out by the action of the Tate object. As an application we obtain an alternative proof of Jannsen’s semisimplicity result, which uses the noncommutative world instead of a Weil cohomology.
CHOW MOTIVES VERSUS NONCOMMUTATIVE MOTIVES
"... Abstract. In this article we formalize and enhance Kontsevich’s beautiful insight that Chow motives can be embedded into noncommutative ones after factoring out by the action of the Tate object. We illustrate the potential of this result by developing three of its manyfold applications: (1) the not ..."
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Cited by 7 (4 self)
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Abstract. In this article we formalize and enhance Kontsevich’s beautiful insight that Chow motives can be embedded into noncommutative ones after factoring out by the action of the Tate object. We illustrate the potential of this result by developing three of its manyfold applications: (1) the notions of Schur and Kimura finiteness admit an adequate extension to the realm of noncommutative motives; (2) GilletSoulé’s motivic measure admits an extension to the Grothendieck ring of noncommutative motives; (3) certain
THE FUNDAMENTAL THEOREM VIA DERIVED MORITA INVARIANCE, LOCALIZATION, AND A 1HOMOTOPY INVARIANCE
, 2011
"... Abstract. We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A 1homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel’s fundamental theorems in homotopy algebraic ..."
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Abstract. We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A 1homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel’s fundamental theorems in homotopy algebraic Ktheory, and periodic cyclic homology, respectively. 1. Statement of results A differential graded (=dg) category, over a fixed commutative base ring k, is a category enriched over cochain complexes of kmodules (morphisms sets are such complexes) in such a way that composition fulfills the Leibniz rule: d(f ◦ g) = (df) ◦g +(−1) deg(f) f ◦(dg). Dg categories enhance and solve many of the technical problems inherent to triangulated categories; see Keller’s ICM address [11]. In noncommutative algebraic geometry in the sense of Bondal, Drinfeld, Kaledin, Kontsevich, Van den Bergh [1, 5, 6, 9, 13, 14, 15] they are considered as dgenhancements of (bounded) derived categories of quasicoherent sheaves on a hypothetic noncommutative space. Given a dg algebra A, we will denote by A the associated dg category with one object and dg algebra of endomorphisms A. Let E: dgcat → T be a functor, defined on the category of dg categories, and with values in an arbitrary triangulated
NONCOMMUTATIVE NUMERICAL MOTIVES, TANNAKIAN STRUCTURES, AND MOTIVIC GALOIS GROUPS
"... Abstract. In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the changeofcoefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schurfiniteness, we prove that the category NNum(k) ..."
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Abstract. In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the changeofcoefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schurfiniteness, we prove that the category NNum(k)F of noncommutative numerical motives is (neutral) superTannakian. As in the commutative world, NNum(k)F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a welldefined symmetric monoidal functor HP ∗ on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues CNC and DNC of Grothendieck’s standard conjectures C and D. Assuming CNC, we
KONTSEVICH’S NONCOMMUTATIVE NUMERICAL MOTIVES
, 1108
"... Abstract. In this note we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [14]. As a consequence, we conclude that NCnum(k)F is abelian semisimple as conjectured by Kontsevich. ..."
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Abstract. In this note we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [14]. As a consequence, we conclude that NCnum(k)F is abelian semisimple as conjectured by Kontsevich.
Introduction to motives
"... This article is based on the lectures of the same tittle given by the first author during ..."
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This article is based on the lectures of the same tittle given by the first author during
Universidad de Buenos Aires
, 2010
"... Cover photo of the front of Pabellón 1 of Ciudad Universitaria courtesy of Adrián ..."
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Cover photo of the front of Pabellón 1 of Ciudad Universitaria courtesy of Adrián