  For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a
uniform description of an ideal which in characteristic zero yields the
multiplier ideal, and in positive characteristic the test ideal of the pair
$(X,\Delta)$. Our description is in terms of regular alterations over $X$, and
one consequence of it is a common characterization of rational singularities
(in characteristic zero) and F-rational singularities (in characteristic $p$)
by the surjectivity of the trace map $\pi_* \omega_Y \to \omega_X$ for every
such alteration $\pi \: Y \to X$.
  Furthermore, building on work of B. Bhatt, we establish up-to-finite-map
versions of Grauert-Riemenscheneider and Nadel/Kawamata-Viehweg vanishing
theorems in the characteristic $p$ setting without assuming $W2$ lifting, and
show that these are strong enough in some applications to extend sections.
