  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.
