Introduction to motivic cohomology and motives

Abstract

The concept of sheaf cohomology in algebraic geometry has been known for a long time. In the 1960s, Grothendieck predicted another cohomology theory, called motivic cohomology. It was later conjectured to be the hypercohomology of complex of sheaves, with properties related to etale cohomology, K-theory, etc. 

In the 1990s, Voevodsky defined the correct form of motivic cohomology satisfying most of the previous conjecture, together with a framework of the triangulated category of motives and the $A^1$-homotopy category. Voevodsky applied it to prove the Milnor conjecture on the K-theory of fields, and he was awarded the 2002 Fields medal for these works. In this talk I will introduce the definition of motivic cohomology, together with the construction of the triangulated category of motives.