Georgia Institute of Technology College of Sciences

A fundamental conjecture of tight closure theory is every weakly F-regular ring is strongly F -regular. There has been incremental progress on this conjecture since the inception of tight closure. Most notably, the conjecture has been resolved for rings graded over a field by Lyubeznik and Smith. Otherwise, known progress around the conjecture have required assumptions on the ring that are akin to being Gorenstein. We extend known cases by proving the equivalence of F -regularity classes for rings whose anti-canonical algebra is Noetherian on the punctured spectrum. The anti-canonical algebra being Noetherian for a strongly F -regular ring is conjectured to be a vacuous assumption. This talk is based on joint work with Ian Aberbach and Craig Huneke.