Rodney Y. Sharp

Bass numbers of local cohomology modules

Let A be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of A itself, but with respect to an arbitrary ideal of A. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that A is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

A Method for the Study of Artinian Modules, With an Application to Asymptotic Behavior

If N is a Noetherian module over the commutative ring R (throughout the paper, R will denote a commutative ring with identity), then the study of N in many contexts can be reduced to the study of a finitely generated module over a commutative Noetherian ring, because N has a natural structure as a module over R/(0 : N) and the latter ring is Noetherian. For a long time it has been a source of irritation to me that I did not know of any method which would reduce the study of an Artinian module A over the commutative ring R to the study of an Artinian module over a commutative Noetherian ring. However, during the MSRI Microprogram on Commutative Algebra, my attention was drawn to a result of W. Heinzer and D. Lantz [2, Proposition 4.3]; this proposition proves that if A is a faithful Artinian module over a quasi-local ring (R, M) which is (Hausdorff) complete in the M-adic topology, then R is Noetherian. It turns out that a generalization of this result provides a missing link to complete a chain of reductions by which one can, for some purposes, reduce the study of an Artinian module over an arbitrary commutative ring R to the study of an Artinian module over a complete (Noetherian) local ring; in the latter situation we have Matlis’s duality available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

On annihilators and associated primes of local cohomology modules

We establish the Local-global Principle for the annihilation of local cohomology modules over an arbitrary commutative Noetherian ring R at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 4. We explore interrelations between the principle and the Annihilator Theorem for local cohomology, and show that, if R is universally catenary and all formal fibres of all localizations of R satisfy Serres condition (Sr), then the Annihilator Theorem for local cohomology holds at level r over R if and only if the Local-global Principle for the annihilation of local cohomology modules holds at level r over R. Moreover, we show that certain local cohomology modules have only finitely many associated primes. This provides motivation for the study of conditions under which the set ⋃m,n∈NAss(M/(xm,yn)M) (where M is a finitely generated R-module and x,y∈R) is finite: an example due to M. Katzman shows that this set is not always finite; we provide some sufficient conditions for its finiteness.

Secondary representations for injective modules over commutative Noetherian rings

There have been several recent accounts of a theory dual to the well-known theory of primary decomposition for modules over a (non-trivial) commutative ring A with identity: see ( 4 ), ( 2 ) and ( 9 ). Here we shall follow Macdonalds terminology from ( 4 ) and refer to this dual theory as “ secondary representation theory ”. A secondary representation for an A -module M is an expression for M as a finite sum of secondary submodules; just as the zero submodule of a Noetherian A -module X has a primary decomposition in X , it turns out, as one would expect, that every Artinian A -module has a secondary representation.

Dualizing complexes for commutative Noetherian rings

The theory of dualizing complexes of Grothendieck and Hartshorne ((5), chapter v) has turned out to be a useful tool even in commutative algebra. For instance, Peskine and Szpiro used dualizing complexes in their (partial) solution of Basss conjecture concerning finitely-generated (f.-g.) modules of finite injective dimension over a Noetherian local ring ((7), chapitre I, §5); and the present author first obtained the results in (9) by using dualizing complexes.

Local cohomology and modules of generalized fractions

The purpose of this paper is to provide additional evidence to support our view that the modules of generalized fractions introduced in [8] are worth further investigation: we show that, for a module M over a (commutative, Noetherian) local ring A (with identity) having maximal ideal m and dimension n , the n -th local cohomology module may be viewed as a module of generalized fractions of M with respect to a certain triangular subset of A n + 1 , and we use this work to formulate Hochsters ‘Monomial Conjecture’ [2, Conjecture 1]; in terms of modules of generalized fractions and to make a quick deduction of one of Hochsters results which supports that conjecture.

On the dimension and multiplicity of local cohomology modules

This paper is concerned with a finitely generated module

On the attached prime ideals of certain Artinian local cohomology modules

M

Associated primes of graded components of local cohomology modules

over a(commutative Noetherian) local ring

Asymptotic behaviour of certain sets of prime ideals

R