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Annihilators of Local Cohomology Modules
Abstract
In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.
Adviser: Thomas Marley
... Related to the question of vanishing is the study of annihilators of local cohomology. In this paper, we study the following question: This question is inspired by a conjecture of Lynch [10,11], which posits that if c is the cohomological dimension of the ideal I of a local ring (R, m, K), and J is the annihilator of the local cohomology module H c I (R), then R/J has the same Krull dimension as R. A number of positive results on Lynch's conjecture, that is cases where Question 1.1 has an affirmative answer, have been established, including that it holds for rings of dimension at most three. We refer the reader to [2] for a summary of some of these results. ...
We study the conditions under which the highest nonvanishing local cohomology module of a domain R with support in an ideal I is faithful over R, i.e., which guarantee that is faithful, where c is the cohomological dimension of I. In particular, we prove that this is true for the case of positive prime characteristic when c is the number of generators of I.
... (i) is well-known. See, e.g.,[Lyn11, Cor. A.11] for a proof. ...
We prove that F-injectivity localizes, descends under faithfully flat homomorphisms, and ascends under flat homomorphisms with Cohen-Macaulay and geometrically F-injective fibers, all for arbitrary Noetherian rings of prime characteristic. As consequences of these results, we show that the F-injective locus is open on rings essentially of finite type over excellent local rings. As a geometric application, we prove that if X is a smooth projective variety of dimension over an algebraically closed field of characteristic embedded via a d-uple embedding for , then every generic projection of X is F-pure, and hence F-injective. This geometric result is the positive characteristic analogue of a theorem of Doherty.
The author would like to correct the errors in the publication of the original article. The corrected details are given below for your reading.
Let B be an affine Cohen-Macaulay algebra over a field of characteristic p. For every prime ideal p ⊂ B, let Hp denote H dim Bp pBp Bp. Each such Hp is an Artinian module endowed with a natural Frobenius map Θ and if Nil(Hp) denotes the set of all elements in Hp killed by some power of Θ then a theorem by Hartshorne-Speiser and Lyubeznik shows that there exists an e ≥ 0 such that Θ e Nil(Hp) = 0. The smallest such e is the HSL-number of Hp which we denote HSL(Hp). The main theorem in this paper shows that for all e > 0, the sets {p ∈ Spec B | HSL(Hp) < e} are Zariski open, hence HSL is upper semi-continuous. An application of this result gives a global test exponent for the calculation of Frobenius closures of parameter ideals in Cohen-Macaulay rings.
One classical topic in the study of local cohomology is whether the
non-vanishing of a specific local cohomology module is equivalent to the
vanishing of its annihilator; this has been studied by several authors,
including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by
Lynch and Zhang, the goal of this paper is to provide some new results about
this topic, which provide some partial positive answers to these questions. The
main technical tool we exploit is the structure of local cohomology as module
over rings of differential operators.
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