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Grayson, Daniel R.

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

J. \(K\)-Theory 11, No. 3, 527-547 (2013).
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.
Reviewer: Claudio Pedrini (Genova)

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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\textit{D. R. Grayson}, J. \(K\)-Theory 11, No. 3, 527--547 (2013; Zbl 1291.19001)
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