Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

Tabuada, Gonçalo

doi:10.1007/s00029-015-0203-0

Cited by 3 publications

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References 41 publications

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“…Moreover, it is naturally enriched over the derived category D(R); we denote this enrichment by Hom D(R) (−, −). Given dg categories A and B, with A smooth and proper, recall from [39,Prop. 4.4] that we have a natural isomorphism…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Modified mixed realizations, new additive invariants, and periods of dg categories

Tabuada¹

2016

Preprint

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To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, étale, Hodge, etc) as well as its ring of periods. In this note, following an insight of Kontsevich, we prove that, after suitable modifications, these classical constructions can be extended from schemes to the broad setting of dg categories. This leads to new additive invariants of dg categories, which we compute in the case of differential operators, as well as to a theory of periods of dg categories. Among other applications, we prove that the ring of periods of a scheme is invariant under projective homological duality. Along the way, we explicitly describe the modified mixed realizations using the Tannakian formalism.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Modified mixed realizations, new additive invariants, and periods of dg categories

Tabuada¹

2016

Preprint

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Noncommutative rigidity

Tabuada

2017

Math. Z.

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Abstract. In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin's rigidity theorem, as well as of Yagunov-Østvaer's equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the way, we introduce the category of n-adic noncommutative mixed motives. IntroductionLet l/k be a field extension and X an algebraic k-variety. On the one hand, it is (well-)known that when the field extension l/k is primary 1 and the algebraic k-variety X is smooth and proper, base-change induces an isomorphism between the Q-vector spaces of algebraic cycles up to numerical equivalence:On the other hand, when l/k is an extension of algebraically closed fields, a remarkable result of Suslin [12] asserts that, for every integer n ≥ 2 coprime to char(k), base-change induces an isomorphism in mod-n G-theory:Among other applications, the isomorphism (1.2) (with X = Spec(k)) enabled Suslin to describe the torsion of the algebraic K-theory of every algebraically closed field of positive characteristic, thus solving a longstanding conjecture of QuillenLichtenbaum; consult Suslin's ICM address [13] for further applications. The main goal of this article is to establish far-reaching noncommutative generalizations of the above rigidity isomorphisms (1.1)-(1.2); consult §2 for applications. Statement of results.A differential graded (=dg) category A, over a base field k, is a category enriched over complexes of k-vector spaces; see §3.1. Every (dg) kalgebra A gives naturally rise to a dg category with a single object. Another source of examples is proved by algebraic varieties (or more generally by algebraic stacks) since the category of perfect complexes perf(X), resp. the bounded derived category of coherent O X -modules D b (coh(X)), of every algebraic k-variety X admits a canonical dg enhancement perf dg (X), resp. D b dg (coh(X)); see [5, §4.6]. Following

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Modified Mixed Realizations, New Additive Invariants, and Periods of DG Categories

Tabuada

2016

Int Math Res Notices

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Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

Cited by 3 publications

References 41 publications

Modified mixed realizations, new additive invariants, and periods of dg categories

Modified mixed realizations, new additive invariants, and periods of dg categories

Noncommutative rigidity

Modified Mixed Realizations, New Additive Invariants, and Periods of DG Categories

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