The Frobenius structure of local cohomology

By Florian Enescu, M. Hochster. 2007

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All given rings in this paper are commutative, associative with identity, and Noetherian. Throughout, p denotes a positive prime integer. For the most part, we shall be studying local rings, i.e., Noetherian rings with a unique maximal ideal. Likewise our main interest is in rings of positive prime characteristic p. If (R, m) is local of characteristic p, there is a natural action of the Frobenius endomorphism of R on each of its local cohomology modules H m(R). We call an R-submodule N of one of these local cohomology modules F-stable if the action of F maps N into itself. One of our objectives is to understand when a local ring, (R, m), especially a reduced Cohen-Macaulay local ring, has the property that only finitely many R-submodules of its local cohomology modules F-stable. When this occurs we say that R is FH-finite. Of course, when the ring is Cohen-Macaulay there is only one non-vanishing local cohomology module, H m(R), where d = dim (R). The problem of studying the F-stable submodules of H m(R) arises naturally in tight closure theory, taking a point of view pioneered by K. Smith [Sm1,2,3]. E.g., if R is complete, reduced, and Gorenstein, the largest proper F-stable submodule of H m(R) corresponds to the tight closure of 0 (in the finitistic sense: see [HH2], §8), and its annihilator is the test ideal of R. Also see Discussion 2.10 here.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/96244f91ce6ffa85cb681261c30d7e164dbf0f35

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Citations in the corpus

Cartier modules: Finiteness results (2009, methodology): “In the same spirit we show the following result, which generalizes [14], Corollary 4.”
p −1-Linear Maps in Algebra and Geometry (2012, result): “ corollary which was obtained in [KM09] and also independently obtained by the second author in [Sch09]. In the case that X= SpecRand Ris local, proofs of this fact were ﬁrst obtained in [Sha07a] and [EH08]. Corollary 8.1.8. Let X be Frobenius split, then OX has only ﬁnitely many ideals with are com”
A survey of test ideals (2011, background): “ has been an important object of study in commutative algebra for several decades. In particular, several questions about their ﬁniteness have been asked, and also answered, see for example [Ene03a], [EH08] and [Sha07]. These questions are closely related to the ﬁniteness of F-pure centers or compat”
Centers of F-purity (2008, background): “The argument of the proof is essentially the same as in [ 6, Theorem 4.1 ], we simply generalize it to the context of triples (R ,� , a•).”