A REFINEMENT OF SHARPLY F-PURE AND STRONGLY F-REGULAR PAIRS

We point out that the usual argument used to prove that R is strongly F-regular if and only if R-Q is strongly F-regular for every prime ideal Q is an element of Spec R, does not generalize to the case of pairs (R, a(t)). The author's definition of sharp F-purity for pairs (R, a(t)) suffers from the same defect. We therefore propose different definitions of sharply F-pure and strongly F-regular pairs. Our new definitions agree with the old definitions in several common contexts, including the case that R is a local ring.