K 2-Cohomology and the second Chow group

Arithmetic rigidity

“…Now by ([CTR1], Theorem 2.1) the torsion of H 3 (X, Z (2)) is countable, and this completes the proof of the theorem.…”

“…If i = 2j, this last group is of finite index in H i (X, Z(j)), as follows easily from a specialization argument and the Weil conjectures as proved by Deligne (see [CTR1], Theorem 1 for this argument). Thus …”

R-equivalence on three-dimensional tori and zero-cycles

“…The referee has pointed out that using results from [Colliot-Thélène and Raskind 1985] one can deduce that CH 1 (X ) ⊗ F × H 1 (X, K 2 ) for a smooth projective rational variety X over an algebraically closed field F of characteristic zero.…”

On K2 of varieties over number fields

“…where CH 2 .X/ is the Chow group of codimension 2 cycles on X modulo rational equivalence (see [4,Theorem 3.6]). When k is a number field and X is projective and geometrically integral, a theorem of P.Salberger [5,Proposition 4.2,p.235] asserts that H 1 k;K 2 k .X/=H 0 X;K 2 is a group of finite exponent, and this fact, together with the exactness of the sequence displayed above, has been used…”

“…As in [11], our proof is based on a "Hasse principle" theorem of U.Jannsen [8,Theorem 3(d), p.337], together with a local computation of Colliot-Thélène and Raskind [4,Prop. 3.8].…”