On the commutation of the test ideal with localization and completion

By Gennady Lyubeznik, Karen E. Smith. 2001

118 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/56daf6c7d325cc3398aeffef5f5bda3390d1aeb4

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Citations in the corpus

The Frobenius structure of local cohomology (2007, methodology): “Let us note that the finitistic tight closure of 0 in H is an an F-stable strongly proper submodule of H d m (R), as shown in [ LySm2 ], Proposition (4.2).”
The intersection homology D–module in finite characteristic (2001, background): “neral, it is a hard question to decide if the 18 MANUEL BLICKLE tight closure equals the ﬁnitistic tight closure, and it is related to aspects of the localization problem in tight closure theory (cf. [LS01]). As our focus lies on modules with Frobenius actions we study the above closure operations i”
Parameter-test-ideals of Cohen–Macaulay rings (2007, methodology): “tation of this module. To ﬁnd the map uin (5) we use the fact that F HdimS mS (S) is the R-algebra with one generator corresponding to the S[T;f]-module structure deﬁned above (cf. Example 3.7 in [LS]). Hence the S-linear maps Extδ R(R/I,R) → F Extδ R (R/I,R) form a rank-one Sfree module and the”
A survey of test ideals (2011, methodology): “e ∈I[pe] for all e≥0. It is called a big test element if z∈0∗ M implies that 0 = c 1/pe ⊗z∈R1/pe ⊗Mfor all R-modules M. It is an open question whether these two deﬁnitions are equivalent. See [HH90], [LS01] and [Hoc07]. Completely stable (ﬁnitistic) test element: An element rin a local ring R, such ”
ON THE BEHAVIOR OF TEST IDEALS UNDER FINITE MORPHISMS (2010, background): “g a more general deﬁnition we need some additional language. Remark 6.1. The big test ideal and the ﬁnitistic test ideal have been shown to agree in many cases, for example the Q-Gorenstein case; see [LS01] and [AM99]. See Remark 6.11 for the precise statement in full generality. Deﬁnition 6.2. A tr”
Globally F-regular and log Fano varieties (2009, background): “stion 7.3. Suppose that Ris a normal reduced F-nite ring. Does there exist an eective Q-divisor on Spec Rsuch that K R+ is Q-Cartier and e˝(R) = ˝e(R;)? For the denition of ~˝we refer to the paper [LS01] for the original boundary-less denition and to [Tak04b] for the general case, also see [HT04]. T”
F-signature of pairs and the asymptotic behavior of Frobenius splittings (2011, background): “ the subalgebra of p−e-linear maps inside CR that they generate, will be denoted by Cϕ. It is easy to see that Cϕ≃ R{ϕ}/hrϕ− ϕrpei . Example 2.6. [Sch09b, Remark 4.4] (cf. [Sch09a, Corollary 3.9] and [LS01]) If Ris a Gorenstein local ring with canonical module ωR ≃ R, then we have that HomR(Fe ∗ R,R”
Test ideals in non-Q-Gorenstein rings (2009, methodology): “In proving our result, we also develop a theory of pairs ( R, T ), where T is a graded subalgebra of the non-commutative algebra (cid:5) e ≥ 0 Hom R ( F e ∗ R, R ) (which is Matlis-dual to the non-commutative algebra F ( E R ( k )) as studied by Lyubeznik and Smith, [LS01]).”
Cartier modules: Finiteness results (2009, methodology): “In this section, we explain the relationship of our theory with the theory of co-finite left R1⁄2F -modules for a local ring ðR;mÞ as studied before by many authors, mainly in connections to questions about local cohomology, see for example [19], [30], [31], [28], [16], [27].”
p −1-Linear Maps in Algebra and Geometry (2012, unknown): “ M= ωR (or is obtained functorially from T: ωY −→ωY from some other variety Y), there probably is no natural choice of ϕ. The obvious solution is to choose all possible ϕ, see [Sch11b, Bli09] and cf. [LS01] for a dual formulation. For any ﬁnitely generated module M, we set Ende(M) to be the set of p”
Observations on the F-signature of local rings of characteristic p (2006, background): “It is shown in [Sm, 7.1.2] that R is strongly F -regular () 0 E = 0, where E := E(k) is the injective hull of the residue eld k = R=m (see also [ LS2 , Proposition 2.9]).”