Unramified Cohomology of Quadrics, Iii

This paper is the third and last instalment of our project of computation of the low-degree unramified cohomology of quadrics. As in the previous papers [15] and [16], we denote by η X the map H(F,Q/Z(n− 1))→ H nr(F (X)/F,Q/Z(n− 1)) for a smooth, projective quadric X defined over a field F of characteristic 6= 2. Recall that in these papers we proved that Ker η X is generated by symbols for n ≤ 4 and any X (this is conjectured to hold without restriction on n). On the other hand, we computed Coker η X for any X when n = 3 and (in case charF = 0) any X of dimension ≤ 4 for n = 4. We also obtained some information on Coker η X when dimX > 4: this group is canonically a subgroup of 2CH (X), which is at most Z/2 and is 0 for dimX > 10 by results of Karpenko. The contents of the present paper are as follows. First we get more precise information on Coker η X when dimX > 4 in case charF = 0. This represents by no means a complete computation of this group, and the interested reader is encouraged to push our investigation further. Second, we prove (still in characteristic 0) that Coker η X = 0 for all n when X is defined by a Pfister neighbour (theorem 3 in section 3). It follows that, for X defined by a Pfister neighbour, the map