From algebraic K-theory to motivic cohomology and back

Although homology and cohomology had been a well established theory before the middle of the twentieth century, it proved surprisingly difficult to construct a purely algebraic version. Rather than cohomology, it was K-theory around 1970 that became the first analog of a good topological theory. The corresponding cohomology was conjectured to exist by Beilinson, coming out of his work on regulators and values of L-functions in the early 80’s. We will trace the further development of this theory and its relation to algebraic K-theory and describe how, after more than 30 years since Beilinson first used algebraic K-theory as an approximation to cohomology, algebraic K-theory is again giving direction to new types of algebraic cohomology theories.