Back to Commutative Algebra Seminar: Spring 2019
Commutative Algebra Seminar
Date: Friday, Mar 29, 2019
Time: 2:30PM - 3:20PM
Location: LCB 222
Ian Aberbach
University of Missouri, Columbia
Title |
Local cohomology bounds and test ideals |
Abstract |
Let R be a local ring of positive prime characteristic p. The notion of tight closure, developed by Hochster and Huneke, is a powerful tool in understanding Noetherian rings, and is also of intrinsic interest. In particular, we can classify singularities in terms of the “amount” of tight closure over all ideals. The nicest situation is when all ideals in the ring are equal to their own tight closure. Such a ring is called weakly F-regular. Unfortunately, tight closure does not necessarily localize well. In particular, the weak F-regularity property has not been shown to localize. In order to get around this problem, Hochster and Huneke defined strong F-regularity (for F-finite rings), a property which both localizes well and implies weak F-regularity. The converse has only been proven in a limited number of cases - it is known essentially only when R is Q-Gorenstein on the punctured spectrum (which settles the case of dimension at most 3). We show that if R is an excellent local ring and some symbolic power of an anti-canonical ideal has analytic spread two, then weak F-regularity implies strong F-regularity. There is reason to believe that this result combined with results from the MMP will solve the dimension 4 case. The techniques employed differ significantly from any previous methods. As a corollary of the proof, we obtain the result that, in this situation, the F-signature of R may be expressed not just as an infimum of relative Hilbert-Kunz multiplicities, but is achieved as an actual minimum. This work is joint with Thomas Polstra. |
Commutative Algebra