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Schema/Lokal freie Garbe/Torsor/Beschreibung mit Ext und projektiven Bündeln/en/Bemerkung

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Let denote a locally free sheaf on a scheme . For a cohomology class one can construct a geometric object: Because of , the class defines an extension

This extension is such that under the connecting homomorphism of cohomology, is sent to . The extension yields a projective subbundle

If is the corresponding geometric vector bundle of , one may think of as which consists for every base point of all the lines in the fiber passing through the origin. The projective subbundle has codimension one inside , for every point it is a projective space lying (linearly) inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement

is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when is projective, in an entirely projective setting.