Tight closure/Arithmetic deformation/Brenner and Katzman/Section

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Consider and take the ideal and the element . Consider reductions . Then


In particular, the bundle is semistable in the generic fiber, but not strongly semistable for any reduction . The corresponding torsor is an affine scheme for infinitely many prime reductions and not an affine scheme for infinitely many prime reductions.

In terms of affineness (or local cohomology) of quasiaffine schemes, this example has the following properties: the open subset given by the ideal

has cohomological dimension if and has cohomological dimension (equivalently, is an affine scheme) if .