Summary: Abstracts
2-Motives
Joseph Ayoub
The goal of this talk is to give a reasonable candidate for a category of mixed
2-motives over a field k. By "reasonable" we mean a category M2(k) that shares
some of the mirific properties that the conjectural category of mixed 2-motives
is expected to enjoy. The plan is as follows. First, we give the definition of
M2(k). Then we explain the ideas behind the verification that M2(k) is an abelian
category.
0.1. Definition. Let k be a perfect field. An object M DMeff (k) is called a
mixed 2-motive, or simply a 2-motive, if it satisfies the following conditions:
(a) Hi(M) = 0 for i {0, -1, -2};
(b) H0(M) is a 0-motivic sheaf;
(c) H-1(M) is a 1-motivic sheaf;
(d) H-2(M) is a 2-motivic sheaf which is 1-connected;
(e) if L is a non-zero 0-motivic sheaf, then L[-1] is not a direct factor of M
and Ext1
(H-2(M), L) = 0.
The category of mixed 2-motives is denoted by M2(k).
Some explanations are needed. Here, DMeff (k) is Voevodsky's category of ef-

Source: Ayoub, Joseph - Institut für Mathematik, Universität Zürich

Collections: Mathematics