Links between tight closure and deformation theory

Question

I am looking for links between tight closure and deformation theory. As a sample question:

Question 1. Are there geometric interpretations in terms of deformation theory of Frobenius rationality?

Any references or helps are appreciated.

Searchign the Mathscinet, gave the the following articles, but I am looking for more explicit results:

1) Brenner, Holger Forcing algebras, syzygy bundles, and tight closure. Commutative algebra—Noetherian and non-Noetherian perspectives, 77–99, Springer, New York, 2011.

2) Sharif, Tirdad On a result of Faltings via tight closure. Proc. Amer. Math. Soc. 138 (2010), no. 10, 3495–3499.

3) Brenner, Holger Tight closure and vector bundles. Three lectures on commutative algebra, 1–71, Univ. Lecture Ser., 42, Amer. Math. Soc., Providence, RI, 2008.

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Edit.

As requested, I know ask something maybe more explicit.

Question 2. What is the relation between F-rationality in tight closure and deformation theory in the sense of  Mazur?

Answer 1

It strongly depends on what you mean by "geometric interpretations". For example, normal affine varieties $X = Spec R$ where $R$ is $F$-rational have pseudo-rational singularities, and F-rational singularities deform in the sense of @NeilEpstein excellent comment. This can be done by calculating the parameter test submodule of the canonical module. Does this count for you as a deformation? As geometric?

In some ways, while these test submodules are coming from tight closure, they are also interpretable without mentioning the closure operation itself, and so its a bit unclear if what you are asking about is really tight closure or more broadly Frobenius methods.

It is important, I think, to clarify also what you mean in the sense of deformation. By this paper by Schwede and Zhang, one needs to be careful as Bertini theorems usually used in classic deformation theory for complex varieties aren't available for all classes of F-singularities -- notably F-injective singularities.

So to summarize there is a bit of tension. The usual thing one wants to do is consider a ring as a total space for a deformation and see that the singularities along a special fiber R/fR for f a regular element, extend near by fibers. This happens in two steps, first show the total space has at worst those types of singularities, and then show via Bertini that the near by fibers do too. For F-regular, its possible we can't extend to the total space in the first place and for other classes which do extend, like Cohen-Macaulay F-injective singularities, we won't have a Bertini statement available.

If you can give a more precise version of your question, maybe someone can say more.

Answer 2

There are many papers in the characteristic $p$ literature about which F-singularities `deform'. Let P be a singularity property. We say that P "deforms" if whenever you have a nice enough local ring $R$ and an $R$-regular element $t \in R$, if $R/tR$ satisfies P then $R$ satisfies P. I think geometrically this means that in an affine scheme $X$, if a closed well-behaved hypersurface in $X$ satisfies P, then so does all of $X$; this is good behavior in deformation theory I suppose.

Anyway, normality deforms in this sense (a folk theorem for many years; cf. the first theorem in Heitmann's article "lifting seminormality'' for a proof). One can think of normality as an F-singularity by the way, as a ring is normal if and only if every principal ideal generated by a regular element is tightly closed.

As for the newer F-singularities: F-injectivity deforms under some conditions on local cohomology (c.f. Horiuchi, Miller, and Shimomoto's article "Deformation of F-injectivity and local cohomology"). F-regularity doesn't deform even for nice Cohen-Macaulay rings (cf. Singh's article "F-regularity does not deform"), but it does in the Gorenstein case and somewhat more generally (cf. Aberbach, Katzman, and MacCrimmon's article "Weak F-regularity deforms in \mathbb{Q}-Gorenstein rings". F-purity does not deform in general, but again it does in the Gorenstein case (cf. for both facts Fedder's article "F-purity and rational singularity").

As @lemiller points out, F-rationality does deform, though I don't know a reference.