Unramified cohomology of quadrics. II. (Q1847802)

Let \(F\) be a field of characteristic \(\neq 2\), \(X\) a projective quadric over \(F\) defined by a non-degenerate quadratic form \(q\), and \(F(X)\) its function field. Let \(H^i(F)\) be the \(i\)th Galois cohomology group with mod \(2\) coefficients. The unramified cohomology group \(H^i_{nr}(F(X)/F)\) is the kernel of the map \(H^i(F(X))\to\bigoplus_{x\in X^{(1)}}H^{i-1}(F_x)\) where \(F_x\) denotes the residue field at a codimension \(1\) point \(x\in X^{(1)}\). Similarly, on defines \(H^i_{nr}(F(X)/F, i-1)\) by replacing \(H^k(\cdot)\) (\(k=i,i-1\)) by \(H^k(\cdot ,k-1)\) in the above map. In the present paper, the authors continue their study of the kernel and the cokernel of the maps \(\eta^i_2\,:\,H^i(F(X))\to H^i_{nr}(F(X)/F)\) and \(\eta^i\,:\,H^i(F(X),i-1)\to H^i_{nr}(F(X)/F,i-1)\) induced by the usual restriction map which they (with M. Rost) began in [Am. J. Math. 120, No. 4, 841--891 (1998; Zbl 0913.11018)] where they showed various results for \(i\leq 4\). In the present paper, they focus on the case \(i=4\) and they treat various cases not covered in their previous paper. They show first that the maps \(\text{Coker}(\eta^i_2)\to \text{Coker}(\eta^i)\) are surjective for all \(i\) provided \(F\) contains all \(2\)-primary roots of unity and they ask whether surjevtivity holds in general. It is proved that if \(\dim(X)\leq 2\), then \(\text{Ker}(\eta^4_2)=H^1F\cdot \text{Ker}(\eta^3_2)\) and that it is generated by symbols. For \(2\leq\dim(X)\leq 3\), it is shown that \(\text{Coker}(\eta^4)=0\) and they give various results on \(\text{Coker}(\eta^4)\), \(\text{Coker}(\eta^4_2)\) in the case \(\dim (X)=4\) where \(X\) comes from a Pfister neighbor or a so-called (virtual) Albert form. Some of the proofs rely on the existence of certain spectral sequences obtained by the first author in [W. Raskind (ed.) et al., Algebraic \(K\)-theory, Seattle, WA, USA, 1997, Proc. Symp. Pure Math. 67, 149--174 (1999; Zbl 0954.19004)] which in turn depend on results of Voevodsky which require resolution of singularities and are therefore a priori only valid in characteristic \(0\) as is pointed out by the authors. A certain spectral sequence for the étale motivic cohomology of an affine quadric needed in one of the cases is constructed in an appendix.