| Abstract: | Let  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  commutes with localization and, if  is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull  of the residue field of every local ring of  is equal to the finitistic tight closure of zero in  . 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  -module is introduced and studied. This theory gives rise to an ideal of  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  in general, and shown to equal the test ideal under the hypothesis that  in every local ring of  . |
