Plus closure/Two-dimensional/Graded/Torsor/Introduction/Section

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Let be a field and let be a normal two-dimensional standard-graded domain over with corresponding smooth projective curve . A homogeneous -primary ideal with homogeneous ideal generators and another homogeneous element of degree yield a cohomology class

Let be the corresponding torsor. We have seen that the affineness of this torsor over is equivalent to the affineness of the corresponding torsor over (and to the property of not belonging to the tight closure). Now we want to understand what the property means for and for . Instead of the plus closure we will work with the graded plus closure , where holds if and only if there exists a finite graded extension such that . The existence of such an translates into the existence of a finite morphism

such that . Here we may assume that is also smooth. Therefore, we discuss the more general question when a cohomology class , where is a locally free sheaf on , can be annihilated by a finite morphism

of smooth projective curves. The advantage of this more general approach is that we may work with short exact sequences (in particular, the sequences coming from the Harder-Narasimhan filtration) in order to reduce the problem to semistable bundles which do not necessarily come from an ideal situation.

Let denote a smooth projective curve over an algebraically closed field , let be a locally free sheaf on and let

be a cohomology class with corresponding torsor . Then the following conditions are equivalent.
  1. There exists a finite morphism

    from a smooth projective curve such that .

  2. There exists a projective curve .

If (1) holds, then the pull-back is trivial (as a torsor), as it equals the torsor given by . Hence is isomorphic to a vector bundle and contains in particular a copy of . The image of this copy is a projective curve inside .

If (2) holds, then let be the normalization of . Since dominates , the resulting morphism

is finite. Since this morphism factors through and since annihilates the cohomology class by which it is defined, it follows that


We want to show that the cohomological criterion for (non)-affineness of a torsor along the Harder-Narasimhan filtration of the vector bundle also holds for the existence of projective curves inside the torsor, under the condition that the projective curve is defined over a finite field. This implies that tight closure is (graded) plus closure for graded -primary ideals in a two-dimensional graded domain over a finite field.