Tight closure/Two-dimensional/Standard-graded/Syzygy bundle/Introduction/Section

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Let be a two-dimensional standard-graded normal domain over an algebraically closed field . Let be the corresponding smooth projective curve and let

be an -primary homogeneous ideal with generators of degrees . Then we get on the short exact sequence

Here is a vector bundle, called the syzygy bundle, of rank and of degree

Recall that the degree of a vector bundle on a projective curve is defined as the degree of the invertible sheaf , where is the rank of . The degree is additive on short exact sequences.

A homogeneous element of degree defines an element in and thus a cohomology class , so this defines a torsor over the projective curve. We mention an alternative description of the torsor corresponding to a first cohomology class in a locally free sheaf which is better suited for the projective situation.

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[1]

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.

  1. denotes the dual bundle. According to our convention, the geometric vector bundle corresponding to a locally free sheaf is given by and the projective bundle is , where denotes the th symmetric power.