Behavior of Test Ideals under Smooth and Etale Homomorphisms
نویسنده
چکیده
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 A D = D , assuming that all residue elds 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 nite at homomorphism of Gorenstein normal rings, etale in codimension 1, then A D = D. More generally this last result holds under the assumption that the closed ber of h : (A; m) ?! D is Gorenstein, provided one knows that the tight closure of zero, and the nitistic tight closure of zero in the injective hulls of the residue elds of A and S, are equal. 1. Introduction The theory of tight closure, initiated by M. Hochster and C. Huneke about fteen years ago, has been considerably developed since then. The success of this theory lies in the large number of applications to other problems in commutative algebra and algebraic geometry. For example, it provides a short proof of Briann con-Skoda Theorem and captures the obstruction of a ring to being Cohen-Macaulay 7]. In algebraic geometry, it is connected both with vanishing theorems and with the study of singularities 11], 5], 19], 23].
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