Tensor structure for Nori motives

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Barbieri-Viale, Luca; Huber, Annette; Prest, Mike;
  • Subject: Mathematics - Algebraic Geometry | Mathematics - Representation Theory
    arxiv: Mathematics::Category Theory

We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative construction for the tensor product on Nori motives.
  • References (21)
    21 references, page 1 of 3

    1.11. Proposition. Let C be an additive tensor category, A an abelian tensor category with a right exact tensor product, and M : C → A an additive tensor functor.

    Further assume that M factors through a ♭-subcategory A♭ ⊂ A (see Definition 1.7).

    2.14. Lemma. ZD⊗,sgn and ZD⊗,sgn,+ are well-defined tensor categories.

    2.15. Definition. Let (D, ⊗) be a graded ⊗-quiver. Let A be an additive commutative tensor category. A graded tensor representation of (D, ⊗) is a representation T : D → A of the underlying quiver together with a choice of natural isomorphisms [ASS] Ibrahim Assem, Daniel Simson and Andrzej Skowroński, Elements of the Representation Theory of Associative Algebras. 1: Techniques of Representation Theory, London Math. Soc. Student Texts, Vol. 65, Cambridge University Press, 2006.

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    [Bo] Francis Borceux, Handbook of Categorical Algebra: Vol. 1, Basic category theory, Cambridge Univ. Press, 1994

    [BV] Luca Barbieri-Viale, T-motives, J. Pure Appl. Algebra 221 (2017) pp. 1495-1898.

    [BVP] Luca Barbieri-Viale & Mike Prest, Definable categories and T-motives, to appear in Rend. Sem. Mat. Univ. Padova (2018)

    [BVCL] Luca Barbieri-Viale, Olivia Caramello & Laurent Lafforgue, Syntactic categories for Nori motives, arxiv:1506.06113 (2015)

    [DMT] Deligne, P. & Milne, J.S., Tannakian Categories, in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, 1982, pp. 101Ð228

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