Behavior of Test Ideals under Smooth and Étale Homomorphisms

We investigate the behavior of the test ideal of an excellent reduced ring of prime characteristic under base change. It is shown that if h: A → D is a smooth homomorphism, then τAD = τD, assuming that all residue fields of A at maximal ideals are perfect and that formation of the test ideal commutes with localization. It is also shown that if h: (A, m) → D is a finite flat homomorphism of Gorenstein normal rings, étale in codimension 1, then τAD = τD. More generally, this last result holds under the assumption that the closed fiber of h: (A, m) → D is Gorenstein, provided one knows that the tight closure of zero and the finitistic tight closure of zero in the injective hulls of the residue fields of A and S are equal.