A relative basis for mixed Tate motives over the projective line minus three points

Ismaël Soudères (1, 2)

1 MPI - Max-Planck-Institut für Mathematik
2 FB6/Institut für Mathematik - Institut für Mathematik [Osnabrück]

In a previous work, the author have built two families of distinguished algebraic cycles in Bloch-Kriz cubical cycle complex over the projective line minus three points. The goal of this paper is to show how these cycles induce well-defined elements in the $\HH^0$ of the bar construction of the cycle complex and thus generated comodules over this $\HH^0$, that is a mixed Tate motives as in Bloch and Kriz construction. In addition, it is shown that out of the two families only ones is needed at the bar construction level. As a consequence, the author obtains that one of the family gives a basis of the tannakian coLie coalgebra of mixed Tate motives over $\ps$ relatively to the tannakian coLie coalgebra of mixed Tate motives over $\Sp(\Q)$. This in turns provides a new formula for Goncharov motivic coproduct, which arise explicitly as the coaction dual to Ihara action by special derivations.

Mots clés: coLie; cobar construction; algebraic cycles; bar construction; mixed Tate motives; polylogarithms

Domaines: Géométrie algébrique [math.AG]; K-théorie et homologie [math.KT]; Combinatoire [math.CO]

Ismaël Soudères. A relative basis for mixed Tate motives over the projective line minus three points. Communications in Number Theory and Physics, 2016, 10 (1), pp.87-131. ⟨hal-00914982v3⟩