Graded Witt ring and unramified cohomology
Let \(k\) be a field of characteristic different from 2 and \(X\) a smooth variety over \(k\). Let \[ \partial:W(k(X))\to\bigoplus_{x\in X^ 1}W(k(x)) \] denote the map induced by the second residue homomorphism at codimension one points \(X^ 1\) of \(X\). The kernel of \(\partial\) is the unramified Witt group \(W_{nr}(X)\) of \(X\). The powers of the fundamental ideal \(I(k(X))\) give a filtration on \(W_{nr}(X)\). The aim of this paper is to compare the associated graded unramified Witt ring \(\bigoplus_{n\geq 0}I_ n(X)/I_{n+1}(X)\) with the graded unramified cohomology ring \(\bigoplus_{n\geq 0}H^ 0(X,{\mathcal H}^ n)\), where \({\mathcal H}^ n\) denotes the Zariski sheaf associated to the presheaf \(U\to H^ n_{et}(U,\mu_ 2)\). The authors call a variety \(X\) good if for every point \(x\in X\) of codimension \(\leq 1\), the graded Witt ring of \(k(x)\) is isomorphic to the graded cohomology ring of \(k(x)\). Examples of good varieties are provided by curves over local and global fields and varieties of dimension \(\leq 2\) over \(\mathbb{R}\) or \(\mathbb{C}\) [\textit{J. K. Arason}, \textit{R. Elman} and \textit{B. Jacob} in Math. Ann. 274, 649-657 (1986; Zbl 0576.12025) and in Quadratic and Hermitian Forms, Conf. Hamilton/Ont. 1983, CMS Conf. Proc. 4, 17-50 (1984; Zbl 0554.10011)]. If \(X\) is good, there exist homomorphisms \(e_ n:I_ n(X)\to H^ 0(X,{\mathcal H}^ n)\) with kernel \(I_{n+1}(X)\), which induce a homomorphism \(e\) of the graded unramified Witt ring into the graded unramified cohomology ring of \(X\). The main result of this paper is that the map \(e\) is an isomorphism for smooth affine curves over local or global fields, as well as for projective curves over local or global fields if the canonical line bundle of \(X\) is a square in \(\text{Pic}(X)\). However, even for smooth projective curves over local fields, the map \(e_ 2\) is in general not surjective. Thus Merkujev's theorem does not admit a globalisation to curves.
homomorphism of the graded unramified Witt ring into the graded unramified cohomology ring
scientific article; zbMATH DE number 149764
scientific article