## Quillen’s work in algebraic $$K$$-theory.(English)Zbl 1291.19001

This paper surveys the genesis and development of algebraic $$K$$-theory starting from Quillen’s fundamental results in the early seventies. It contains an interesting account of Quillen’s ideas that opened a field of research with applications to topology, algebraic geometry and arithmetic. The author also reviews the main results in $$K$$-theory such as Bloch’s formula, relating algebraic cycles and $${\mathcal K}$$-cohomology, the $$K$$-theory of Severi-Brauer varieties, the fundamental theorem for a regular ring $$R$$, relating $$K_*R[t,t^{-1}]$$ and $$K_*(R)$$, the finite generation of $$K_*(R)$$ for the ring of integers $$R$$ in a number field.
There also is a short account on the more recent results by Friedlander , Suslin and Voevodsky which leaded to the definition of motivic cohomology and motivic homotopy theory. A very significant application of the developments of algebraic $$K$$-theory is the proof by Voevodsky and Rost of the Bloch-Kato’s conjecture on the isomorphism between the Milnor K-theory and the étale cohomology of a field.

### MSC:

 19-03 History of $$K$$-theory 01A60 History of mathematics in the 20th century 19Dxx Higher algebraic $$K$$-theory

### Keywords:

higher algebraic K-theory; Quillen
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### References:

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