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Vektorbündel/A^1/Punktierte affine Fläche/Erzwingende Algebra/Variante 1/Beispiel/en

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Let denote a two-dimensional normal local noetherian domain and let and be two parameters in . On we have the short exact sequence

and its corresponding long exact sequence of cohomology,

The connecting homomorphism sends an element to . The torsor given by such a cohomology class can be realized by the forcing algebra

Note that different forcing algebras may give the same torsor, because the torsor depends only on the spectrum of the forcing algebra restricted to the punctured spectrum of . For example, the cohomology class defines one torsor, but the two fractions yield the two forcing algebras and , which are quite different. The fiber over the maximal ideal of the first one is empty, whereas the fiber over the maximal ideal of the second one is a plane.

If is regular, say (or the localization of this at or the corresponding power series ring) then the first cohomology classes are -linear combinations of , .

They are realized by the forcing algebras
Since the fiber over the maximal ideal is empty, the spectrum of the forcing algebra equals the torsor. Or, the other way round, the torsor is itself an affine scheme.