Le groupe de Witt d'une surface réelle. (The Witt group of a real surface)
Let A be an affine \({\mathbb{R}}\)-algebra of dimension 2 and let \(\bar A\) be the normal closure of A. The main result of this paper is that if Coker(Pic A \(\to Pic \bar A)\) is finitely generated, then the Witt group W(A) of A is finitely generated. In particular W(A) is finitely generated if A is normal. The result is in general the best possible: the author constructs examples of non-normal \({\mathbb{R}}\)-algebras A of dimension 2 whose Witt groups are not finitely generated. The author treats successively the cases A regular, A normal and the case Coker(Pic A \(\to Pic \bar A)\) finitely generated. For the regular case, the proof relies on the Elman-Lam theorem on quadratic forms over fields which are extensions of \({\mathbb{R}}\) of transcendence degree two [\textit{R. Elman} and \textit{T. Y. Lam}, ibid. 49, 373-381 (1974; Zbl 0297.15024)] and the fact that the canonical map W(A) \(\to W(K)\), K the field of fraction of A, is injective [\textit{M. Ojanguren}, ibid. 57, 145-157 (1982; Zbl 0487.13005), \textit{W. Pardon}, Lect. Notes Math. 967, 300-315 (1982; Zbl 0506.10020) and 1046, 261-328 (1984; Zbl 0531.10024)].
scientific article; zbMATH DE number 4002175