On the commutation of the test ideal with localization and completion

Gennady Lyubeznik, Karen E. Smith

Research output: Contribution to journalArticlepeer-review

65 Scopus citations


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 = 0Efg* in every local ring of R.

Original languageEnglish (US)
Pages (from-to)3149-3180
Number of pages32
JournalTransactions of the American Mathematical Society
Issue number8
StatePublished - 2001


  • Frobenius action
  • Localisation
  • Test ideal
  • Tight closure


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