Cohomology of Severi--Brauer varieties and the norm residue homomorphism

A. S. Merkur'ev ; A. A. Suslin

Izvestiya. Mathematics , Tome 21 (1983) no. 2, pp. 307-340

Résumé

The basic purpose of this paper is to prove bijectivity of the norm residue homomorphism $R_{F,n}\colon K_2(F)/nK_2(F)\to H^2(F,\mu_n^{\otimes 2})$ for any field $F$ of characteristic prime to $n$. In particular, if $\mu_n\subset F$, then any central simple algebra of exponent $n$ is similar to a tensor product of cyclic algebras. In the course of the proof we obtain partial degeneracy of the Gersten spectral sequence, and we compute some $K$-cohomology groups of Severi–Brauer groups corresponding to cyclic algebras of prime degree. The fundamental theorem also gives us several corollaries. Bibliography: 27 titles.