  The containment problem for symbolic and ordinary powers of ideals asks for
what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular
ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede
partially answers this question, but the containments it provides are not
always best possible. In particular, a tighter containment conjectured by
Harbourne has been shown to hold for interesting classes of ideals - although
it does not hold in general. In this paper, we develop a Fedder (respectively,
Glassbrenner) type criterion for $F$-purity (respectively, strong
$F$-regularity) for ideals of finite projective dimension over $F$-finite
Gorenstein rings and use our criteria to extend the prime characteristic
results of Grifo-Huneke to singular ambient rings. For ideals of infinite
projective dimension, we prove that a variation of the containment still holds,
in the spirit of work by Hochster-Huneke and Takagi.
