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Behavior of Test Ideals under Smooth and Étale Homomorphisms

✍ Scribed by A. Bravo; K.E. Smith

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
140 KB
Volume
247
Category
Article
ISSN
0021-8693

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✦ Synopsis

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 τ A D = τ D , 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 τ A D = τ D . 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.

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