Tight closure/Regular ring/Solid closure/Motivation/Section

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An important property of tight closure is that it is trivial for regular rings, i.e.

for every ideal . This rests upon Kunz's theorem saying that the Frobenius homomorphism for regular rings is flat. This property implies the following cohomological property of torsors.

Let denote a regular local ring of dimension and of positive characteristic, let be an -primary ideal and be an element with . Let be the corresponding forcing algebra. Then the extended ideal satisfies

This follows from fact and .

In dimension two this is true in every (even mixed) characteristic.

Let denote a two-dimensional regular local ring, let be an -primary ideal and an element with . Let

be the corresponding forcing algebra. Then for the extended ideal we have

In particular, the open subset is an affine scheme if and only if


The main point for the proof of this result is that for , the natural mapping

is not injective by a Matlis duality argument. Since the local cohomology of a regular ring is explicitly known, this map annihilates some cohomology class of the form where are parameters. But then it annihilates the complete local cohomology module and then is an affine scheme.

For non-regular two-dimensional rings it is a difficult question in general to decide whether a torsor is affine or not. A satisfactory answer is only known in the normal two-dimensional graded case over a field, which we will deal with in the final lectures.

In higher dimension in characteristic zero it is not true that a regular ring is solidly closed (meaning that every ideal equals its solid closure), as was shown by the following example of Paul Roberts.

Let be a field of characteristic and let

Then the ideal has the property that . This means that in , the element belongs to the solid closure of the ideal , and hence the three-dimensional polynomial ring is not solidly closed.

This example was the motivation for the introduction of parasolid closure, which has all the good properties of solid closure but which is also trivial for regular rings.