Test ideals in quotients of $F$-finite regular local rings

Abstract:

Let $S$ be an $F$-finite regular local ring and $I$ an ideal contained in $S$. Let $R=S/I$. Fedder proved that $R$ is $F$-pure if and only if $(I^{[p]}:I) \nsubseteq \mathfrak {m}^{[p]}$. We have noted a new proof for his criterion, along with showing that $(I^{[q]}:I) \subseteq (\tau ^{[q]}:\tau )$, where $\tau$ is the pullback of the test ideal for $R$. Combining the the $F$-purity criterion and the above result we see that if $R=S/I$ is $F$-pure then $R/\tau$ is also $F$-pure. In fact, we can form a filtration of $R$, $I \subseteq \tau = \tau _{0} \subseteq \tau _{1} \subseteq \ldots \subseteq \tau _{i} \subseteq \ldots$ that stabilizes such that each $R/\tau _{i}$ is $F$-pure and its test ideal is $\tau _{i+1}$. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let $R=T/I$, where $T$ is either a polynomial or a power series ring and $I= P_{1} \cap \ldots \cap P_{n}$ is generated by monomials and the $R/P_{i}$ are regular. Set $J = \Sigma (P_{1} \cap \ldots \cap \hat {P_{i}} \cap \ldots \cap P_{n})$. Then $J=\tau =\tau _{par}$.