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

Abstract

This article is the sequel to (Marcolli and Tabuada in Sel Math 20(1):315–358, 2014). We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub’s weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin–Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub’s weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky’s category of mixed motives.

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Notes

  1. 1.

    Recall that two Quillen model categories are called Quillen equivalent if there exists a zigzag of Quillen equivalences relating them. The 2-step zigzag of Quillen equivalences constructed in [41, Thm. 5.1.6] does not preserves the monoidal structures. An alternative 3-step zigzag of weak monoidal Quillen equivalences was constructed in [42].

  2. 2.

    We are implicitly strictifying the tensor product of bimodules in order to make it strictly associative; see Garkusha–Panin [13, §3] for instance.

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Acknowledgments

The author is very grateful to Joseph Ayoub for kindly teaching him his beautiful weak Tannakian formalism [4, 5] and also for comments and corrections on a previous version of this article. The author also would like to thank Clark Barwick, Guillermo Cortiñas, Haynes Miller, Amnon Neeman, and Niranjan Ramachandran for useful conversations, and to the anonymous referee for his/her comments that greatly allowed the improvement of the article.

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Correspondence to Gonçalo Tabuada.

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The author was supported by the National Science Foundation CAREER Award #1350472 and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project Grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações).

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Tabuada, G. Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras. Sel. Math. New Ser. 22, 735–764 (2016). https://doi.org/10.1007/s00029-015-0203-0

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Keywords

  • Hopf dg algebra
  • Weak Tannakian formalism
  • Hochschild homology
  • Algebraic K-theory
  • Mixed Artin–Tate motives
  • Orbit category
  • Noncommutative algebraic geometry

Mathematics Subject Classification

  • 14A22
  • 14C15
  • 16E40
  • 16T05
  • 19D55