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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 primeideal $Q \in \Spec R$, 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 fromthe same defect. We therefore propose different definitions of sharply $F$-pureand strongly $F$-regular pairs. Our new definitions agree with the olddefinitions in several common contexts, including the case that $R$ is a localring.

A refinement of sharply F-pure and strongly F-regular pairs