Motives via free abelian categories

Mike Prest (Manchester)

Frank Adams 1,

I will talk about some joint work with Luca Barbieri-Viale.

(Co)homology theories provide algebraic invariants of algebraic varieties.  Roughly, the category of motives should be a universal such theory - an abelian category through which all such theories factor.  Existence (of the simplest version of this) has been shown assuming the (deep and open) Standard Conjectures of Grothendieck.

Some time ago Nori showed how to construct a version of this from a particular cohomology theory. Recently Barbieri-Viale, Caramello and Lafforgue [1], [3] gave a simpler (and more general) construction of Nori's category using categorical logic.  This can also be done [2] with a more algebraic approach, using Freyd's free abelian category construction or the model theory of modules.  I'll describe this.

[1]  L. Barbieri-Viale, O. Caramello and L. Lafforgue: Syntactic categories for Nori motives, arXiv:1506.06113v1, 2015.

[2]  L. Barbieri-Viale and M. Prest, Definable categories and T-motives, arXiv:1604.00153, 2016.

[3]  L. Barbieri-Viale, T-motives, J. Pure Appl. Algebra, to appear, 2017,  http://dx.doi.org/10.1016/j.jpaa.2016.12.017.

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