On the behavior of test ideals under finite morphisms

We derive precise transformation rules for test ideals under an arbitrary finite surjective morphism $\pi \colon Y \to X$ of normal varieties in prime characteristic $p > 0$. Specifically, given a ℚ

-divisor

$\Delta _{X}$ on $X$ and any $\mathcal {O}_{X}$-linear map $\mathfrak {T} \colon K(Y) \to K(X)$, we associate a ℚ

-divisor

$\Delta _{Y}$ on $Y$ such that $\mathfrak {T} ( \pi _{*}\tau (Y;\Delta _{Y})) = \tau (X;\Delta _{X})$. When $\pi$ is separable and $\mathfrak {T} = \operatorname {Tr}_{Y/X}$ is the field trace, we have $\Delta _{Y} = \pi ^{*} \Delta _{X} - \operatorname {Ram}_{\pi }$, where $\operatorname {Ram}_{\pi }$ is the ramification divisor. If, in addition, $\operatorname {Tr}_{Y/X}(\pi _{*}\mathcal {O}_{Y}) = \mathcal {O}_{X}$, we conclude that $\pi _{*}\tau (Y;\Delta _{Y}) \cap K(X) = \tau (X;\Delta _{X})$ and thereby recover the analogous transformation rule to multiplier ideals in characteristic zero. Our main technique is a careful study of when an $\mathcal {O}_{X}$-linear map $F_{*} \mathcal {O}_{X} \to \mathcal {O}_{X}$ lifts to an $\mathcal {O}_{Y}$-linear map $F_{*} \mathcal {O}_{Y} \to \mathcal {O}_{Y}$, and the results obtained about these liftings are of independent interest as they relate to the theory of Frobenius splittings. In particular, again assuming $\operatorname {Tr}_{Y/X}(\pi _{*}\mathcal {O}_{Y}) = \mathcal {O}_{X}$, we obtain transformation results for $F$-pure singularities under finite maps which mirror those for log canonical singularities in characteristic zero. Finally, we explore new conditions on the singularities of the ramification locus, which imply that, for a finite extension of normal domains $R \subseteq S$ in characteristic $p > 0$, the trace map $\mathfrak {T} : \operatorname {Frac} S \to \operatorname {Frac} R$ sends $S$ onto $R$.