Extensions of motives and higher Chow groups

Higher Tits indices of linear algebraic groups

Let G be a semisimple algebraic group over a field k. We introduce the higher Tits indices of G as the set of all Tits indices of G over all field extensions K/k. In the context of quadratic forms this notion coincides with the notion of the higher Witt indices introduced by M. Knebusch and classified by N. Karpenko and A. Vishik. We classify the higher Tits indices for exceptional algebraic gr...

Mixed Motives and Geometric Representation Theory in Equal Characteristic

Let k be a field of characteristic p. We introduce a formalism of mixed sheaves with coefficients in k and apply it in representation theory. We construct a system of k-linear triangulated category of motives on schemes over Fp, which has a six functor formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of mixed sheaves that one is used to. We attempt to ...

Regulators on Additive Higher Chow Groups

As an attempt to understand motives over k[x]/(xm), we define the cubical additive higher Chow groups with modulus for all dimensions extending the works of S. Bloch, H. Esnault and K. Rülling on 0-dimensional cycles. We give an explicit construction of regulator maps on the groups of 1-cycles with an aid of the residue theory of A. Parshin and V. Lomadze.

Noncommutative Artin Motives

In this article we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k)Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridg...

Motives and Chow Groups of Quadrics with Application to the U-invariant (after Oleg Izhboldin)