Isotropic motivic category

Isotropic motivic category - Alexander Vishik

Alexander Vishik
The University of Nottingham
September 12, 2018
Abstract: It was observed for a while (at least, since the times of E.Witt) that the notion
of anisotropy of an algebraic variety (that is, the absence of points of degree prime to a given p on it) plays an important role (most notably, in the theory of quadratic forms).
Taken to the limit, this leads to the idea of “Isotropic motivic category”. This category is obtained from the motivic category of Voevodsky by, roughly speaking, killing the motives of all varieties anisotropic over the ground field. In practical terms, this is achieved by applying the projector on DM(k) coming from Cech simplicial schemes.
The latter are “forms” of a point and measure how far a given variety is from having a zero cycle of degree 1. These simplicial schemes were heavily used by Vladimir Voevodsky in the proofs of Milnor and Bloch-Kato conjectures. The resulting category, introduced originally for the study of the Picard group of Voevodsky’s category, appears to be related to: numerical equivalence with finite coefficients, Milnor’s operations, and other interesting topics. It can serve as just another illustration of what can be done using the tools provided to us by Vladimir.