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**Algebraic K-theory of fields.**
*(English)*
Zbl 0675.12005

Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 222-244 (1987).

The author gives a beautiful survey of recent achievements in the algebraic K-theory of fields and its relations to étale cohomology (the so-called Quillen-Lichtenbaum conjectures), K-theory of division algebras, Milnor K-theory, Bloch’s group, higher Chow groups and étale K-theory.

After a one page introduction on the meaning of algebraic K-theory of fields in the general setting of K-theory and cohomology of schemes two proofs of the now famous theorem of Merkurjev and Suslin are given, both relying on the computation of certain K-cohomology groups of Brauer- Severi varieties. A general rigidity principle is discussed which implies e.g. the independence of the algebraically closed field F of the K-groups \(K_ i(F,{\mathbb{Z}}/m)\). Combined with a stability result one obtains the \(K_ i({\mathbb{R}})\) and \(K_ i({\mathbb{C}})\), \(i\quad mod\quad 8,\) modulo uniquely divisible groups. This same stability result gives rise to a homomorphism \(f:\quad K_ i(F)\to K_ i^ M(F),\) where F is an infinite field and \(K_ i^ M(F)\) denotes the Milnor K-groups. Both compositions \[ K_ i(F)\to K_ i^ M(F)\to K_ i(F)\quad and\quad K_ i^ M(F)\to K_ i(F)\to K_ i^ M(F) \] can be explicitly described. Next, Bloch’s group B(F), F any field, is defined and its relation to \(K_ 3(F)\) is derived by means of homological methods and a spectral sequence argument.

As a corollary one obtains the fact that B(F) does not change under purely transcendental extensions. For algebraically closed F, B(F) is uniquely divisible. Beilinson’s conjecture (and Lichtenbaum’s counterpart in the étale case) on the existence of complexes of sheaves \(\Gamma\) (i) with suitable properties to provide absolute (or motivic) cohomology, is repeated and Bloch’s generalized Chow groups \(CH^ i(X,n)\) are discussed. Several questions remain unanswered. In the final paragraph a link between algebraic and étale K-theory is quoted in relation with the Quillen-Lichtenbaum conjectures for number fields. Several footnotes update the text.

After a one page introduction on the meaning of algebraic K-theory of fields in the general setting of K-theory and cohomology of schemes two proofs of the now famous theorem of Merkurjev and Suslin are given, both relying on the computation of certain K-cohomology groups of Brauer- Severi varieties. A general rigidity principle is discussed which implies e.g. the independence of the algebraically closed field F of the K-groups \(K_ i(F,{\mathbb{Z}}/m)\). Combined with a stability result one obtains the \(K_ i({\mathbb{R}})\) and \(K_ i({\mathbb{C}})\), \(i\quad mod\quad 8,\) modulo uniquely divisible groups. This same stability result gives rise to a homomorphism \(f:\quad K_ i(F)\to K_ i^ M(F),\) where F is an infinite field and \(K_ i^ M(F)\) denotes the Milnor K-groups. Both compositions \[ K_ i(F)\to K_ i^ M(F)\to K_ i(F)\quad and\quad K_ i^ M(F)\to K_ i(F)\to K_ i^ M(F) \] can be explicitly described. Next, Bloch’s group B(F), F any field, is defined and its relation to \(K_ 3(F)\) is derived by means of homological methods and a spectral sequence argument.

As a corollary one obtains the fact that B(F) does not change under purely transcendental extensions. For algebraically closed F, B(F) is uniquely divisible. Beilinson’s conjecture (and Lichtenbaum’s counterpart in the étale case) on the existence of complexes of sheaves \(\Gamma\) (i) with suitable properties to provide absolute (or motivic) cohomology, is repeated and Bloch’s generalized Chow groups \(CH^ i(X,n)\) are discussed. Several questions remain unanswered. In the final paragraph a link between algebraic and étale K-theory is quoted in relation with the Quillen-Lichtenbaum conjectures for number fields. Several footnotes update the text.

Reviewer: W.W.J.Hulsbergen

### MSC:

11R70 | \(K\)-theory of global fields |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

13D15 | Grothendieck groups, \(K\)-theory and commutative rings |

14C05 | Parametrization (Chow and Hilbert schemes) |