publication . Preprint . Article . 2020

Tensor structure for Nori motives

Luca Barbieri-Viale; Annette Huber; Mike Prest;
Open Access English
  • Published: 14 Jun 2020
Comment: Revised & updated version, 23 pages
Persistent Identifiers
arXiv: Mathematics::Category Theory
free text keywords: Mathematics - Algebraic Geometry, Mathematics - Representation Theory, General Mathematics, Tensor, Mathematics, Algebra
21 references, page 1 of 2

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.

[Ay] Joseph Ayoub, Note sur les opérations de Grothendieck et la réalisation de Betti. J. Inst. Math. Jussieu 9 (2010), no. 2, 225-263.

[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

[F] Peter Freyd, Representations in abelian categories. 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 95-120 Springer, New York

[GR] Peter Gabriel and Christine Riedtmann, Group representations without groups, Commentarii mathematici Helvetici 54(1979), 240-287.

[Ha] Daniel Harrer, Comparison of the Categories of Motives defined by Voevodsky and Nori, Thesis Freiburg 2016, arXiv:1609.05516

[Hu1] Annette Huber, Realization of Voevodsky's motives. J. Algebraic Geom. 9 (2000), no. 4, 755-799.

[Hu2] Annette Huber, Corrigendum to: "Realization of Voevodsky's motives” [J. Algebraic Geom. 9 (2000), no. 4, 755âĂŞ799]. J. Algebraic Geom. 13 (2004), no. 1, 195-207.

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