**Tohru Kohrita**

TIFR, Mumbai

November 23, 2017

**The representability of motivic cohomology**:
Any algebraic variety $X$ over an algebraically closed field $k$ is
associated with its Albanese variety $Alb_X.$ According to Rojtman, for
smooth proper $X,$ the torsion part of the group of rational points
$Alb_X(k)$ is canonically isomorphic to $CH_0(X)_{tor}^0,$ the torsion
part of the degree zero part of the Chow group of zero cycles. For a curve $X,$ this isomorphism agrees with the Abel-Jacobi isomorphism
$CH^1(X)_{alg}longrightarrow Pic_X(k),$ where $CH^1(X)_{alg}$ is the
subgroup of $CH^1(X)$ consisting of algebraically trivial cycles and
$Pic_X$ is the Picard variety.
To extend this picture to other Chow groups, Samuel introduced the concept of regular homomorphisms. For divisors and zero cycles, the map $alb_X$ and the Abel-Jacobi isomorphism are universal with respect to regular homomorphisms. The case of codimension $2$ cycles was also treated by Murre.
In this talk, we explain how to extend this picture to other motivic
invariants. If time permits, we explain the relati