Syntactic categories for Nori motives

• Luca Barbieri-Viale ^[1] ; Olivia Caramello ^[2] ; Laurent Lafforgue ^[3]
  1. [1] University of Milan

     University of Milan

     Milán, Italia

     
  2. [2] Università dell'Insubria

     Università dell'Insubria

     Varese, Italia

     
  3. [3] Institut des Hautes Études Scientifiques

     Institut des Hautes Études Scientifiques

     Arrondissement de Palaiseau, Francia

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 4, 2018, págs. 3619-3648
• Idioma: inglés
• DOI: 10.1007/s00029-018-0425-z
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• Resumen

  • We give a new construction, based on categorical logic, of Nori’s Q-linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over Q. This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a Q-linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, -adic, p-adic or motivic cohomology. We prove that the Q-linear abelian categories of mixed motives associated to different (co)homology functors are equivalent if and only a family (of logical nature) of explicit properties is shared by these different functors. The problem of the existence of a universal cohomology theory and of the equivalence of the information encoded by the different classical cohomology functors thus reduces to that of checking these explicit conditions.