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Fibrés algébriques sur une surface réelle. (Algebraic fibers on a real surface). (French) Zbl 0647.14011

Let A be a real affine algebra of Krull dimension 2. Every projective A- module is of the form \(P\oplus A^ n,\) where P is of rank of most 2. Hence, the classification of projective A-modules is reduced to that of rank 1 and rank 2 modules. Leaving those of rank 1 aside (they are classified by Pic(A)) the authors study the projective A-modules of rank 2 with trivial determinant. To do this, they compute the Witt group W-(A) of antisymmetric bilinear spaces over A and show that if A is the affine coordinate ring of a real compact smooth orientable algebraic surface S, W-(A) is the free abelian group on the set of connected components of S.
If, for instance, S is the real 2-dimensional sphere, the knowledge of W- (A) (an infinite cyclic group, in this case) suffices to classify the projective A-modules up to isomorphism. It turns out that two projective modules over \({\mathbb{R}}[X,Y,Z]/(X^ 2+Y^ 2+Z^ 2-1)\) are isomorphic if and only if the continuous vector bundles over S associated to them are isomorphic.
Reviewer: F.Broglia

MSC:

14Pxx Real algebraic and real-analytic geometry
11E16 General binary quadratic forms
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