On the commutation of the test ideal with localization and completion

Abstract

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 f g * E in every local ring of R.

Test ideals and base change problems in tight closure theory

“…The latter equality is obtained for R, a reduced ring that is a homomorphic image of an excellent regular ring in [16]. We show that in fact this equality holds for all excellent reduced local rings (see Theorem 3.6).…”

“…They show that τ (R) (called here the CS test ideal) behaves as expected with respect to localization and completion for rings that are images of excellent regular local rings. Although able to prove the preceding statement, Lyubeznik and Smith expected this to be true for all excellent local rings (see the first paragraph after Theorem 7.1 in [16]). …”

Addendum to “Frobenius and Cartier algebras of Stanley–Reisner rings” [J. Algebra 358 (2012) 162–177]

“…The Cartier algebra of an F -finite complete Gorenstein local ring R is principally generated as a consequence 1 of [5,Example 3.7]. The converse holds true for F -finite normal rings (see [3]).…”

“…the left R-module structure given by r · m := r p e m. One should mention that, using Matlis duality, the Cartier algebra of R corresponds to the Frobenius algebra of the injective hull of the residue field E R (R/m) introduced by G. Lyubeznik and K. E. Smith in [5].…”

The intersection homology D?module in finite characteristic

“…Is the localization/completion of the (parameter) test ideal/module of A the (parameter) test ideal/module of the localization/completion of A? For the test ideal some known cases where it, in fact, commutes with localization and completion are A an isolated singularity, A Q-Gorenstein or A Gorenstein on its punctured spectrum [LS01]. Since the parameter test ideal, the parameter test module, and the test ideal are equal, in the case that A is Gorenstein, all three commute with localization and completion in this case.…”