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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 $Q \in \Spec R$, does not generalize to the case of pairs $(R, \ba^t)$. The author's definition of sharp $F$-purity for pairs $(R, \ba^t)$ 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.