Forcing algebras/Induced torsors/Section

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As is a -torsor, and as every -torsor is represented by a unique cohomology class, there should be a natural cohomology class coming from the forcing data. To see this, let be a noetherian ring and be an ideal. Then on we have the short exact sequence

An element defines an element and hence a cohomology class . Hence defines in fact a -torsor over . We will see that this torsor is induced by the forcing algebra given by and .

Let denote a noetherian ring, let denote an ideal and let be another element. Let be the corresponding cohomology class and let

denote the forcing algebra for these data. Then the scheme together with the natural action of the syzygy bundle on it is isomorphic to the torsor given by .

We compute the cohomology class and the cohomology class given by the forcing algebra. For the first computation we look at the short exact sequence

On , the element is the image of (the non-zero entry is at the th place). The cohomology class is therefore represented by the family of differences

On the other hand, there are isomorphisms

The composition of two such isomorphisms on is the identity plus the same section as before.

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.