Graded Witt ring and unramified cohomology
type: Item
instance of: scholarly article
title: Graded Witt ring and unramified cohomology
zbMATH Open document ID: 0773.14012
DOI: 10.1007/BF00961333
author: R. Parimala
author: R. Sridharan
published in: \(K\)-Theory
publication date: 1993-04-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 149764
zbMATH Keywords: good variety
zbMATH Keywords: homomorphism of the graded unramified Witt ring into the graded unramified cohomology ring
reviewed by: M.-A. Knus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 149764