\(K_ 3\) of a field and the Bloch group. (English. Russian original) Zbl 0741.19005

Suslin, A. A.

Proc. Steklov Inst. Math. 183, 217-239 (1991); translation from Tr. Mat. Inst. Steklova 183, 180-199 (1990).

The author gives an exposition of his results, which were got mostly around 1982. The main problem is the computation of \(K_ 3(F)_{nd}=\text{Coker}(K^ M_ 3(F)\to K_ 3(F))\), where \(F\) is an infinite field, \(K^ M(F)\) is the Milnor \(K\)-group and \(K(F)\) is the Quillen \(K\)-group. The result is given in terms of the Bloch group \(B(F)\) describing the non-trivial relations between tensor elements \(x\otimes(1- x)\) in \(F^*\otimes F^*\). It is proved that there is an exact sequence \[ 0\to\widetilde{Tor}(F^*,F^*)\to K_ 3(F)_{nd}\to B(F)\to 0, \] where \(\widetilde{Tor}(F^*,F^*)\) is the unique nontrivial extension of \(Tor(F^*,F^*)\) by \(\mathbb{Z}/2\).

Reviewer: A.N.Parshin (Moskva)

MSC:

19D45	Higher symbols, Milnor \(K\)-theory
11R70	\(K\)-theory of global fields
19D55	\(K\)-theory and homology; cyclic homology and cohomology

Keywords:

Milnor \(K\)-group; Quillen \(K\)-group; Bloch group