On the commutation of the test ideal with localization and completion

Lyubeznik, Gennady; Smith, Karen

doi:10.1090/S0002-9947-01-02643-5

On the commutation of the test ideal with localization and completion
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by Gennady Lyubeznik and Karen E. Smith PDF

Trans. Amer. Math. Soc. 353 (2001), 3149-3180 Request permission

Abstract:

Let $R$ be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of $R$ commutes with localization and, if $R$ is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull $E$ of the residue field of every local ring of $R$ is equal to the finitistic tight closure of zero in $E$. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each $R$-module is introduced and studied. This theory gives rise to an ideal of $R$ which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of $R$ in general, and shown to equal the test ideal under the hypothesis that $0_E^*=0_E^{fg*}$ in every local ring of $R$.

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