A Note on Local Cohomology

Extended plus closure in complete local rings

Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the ascending chain condition.

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.

Local cohomology of module of differentials of integral extensions

Applications of dualizing complexes to Buchsbaum rings

Local cohomology based on a nonclosed support defined by a pair of ideals

Cohomological dimension, cofiniteness and Abelian categories of cofinite modules

The content in Chapter 1–3 is a fairly standard one-semester course on local rings with the goal to reach the fact that a regular local ring is a unique factorization domain. The homological machinery is also supported by Cohen–Macaulay rings and depth. In Chapters 4–6 the methods of injective modules, Matlis duality and local cohomology are discussed. Chapters 7–9 are not so standard and introduce the reader to the generalizations of modules to complexes of modules. Some of Professor Iversen's results are given in Chapter 9. Chapter 10 is about Serre's intersection conjecture. The graded case is fully exposed. The last chapter introduces the reader to Fitting ideals and McRae invariants.

Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H i I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q sat is simplicial, and a finitely generated module M over k[Q] that is graded by Q gP , the Bass numbers of H i I(M) are finite for any Q-graded ideal I of k[Q]. Conversely, if Q sat is not simplicial, we find a graded ideal I and graded k[Q]-module M such that the local cohomology module H i I(M) has infinite-dimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup.

On the annihilators of local cohomology modules

Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra  and an embedding i: A →  are constructed such that  is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure  is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field of recursive real numbers.In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt. Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.

Small Cofinite Irreducibles

Let $(R,m)$ be a complete Noetherian local ring, $I$ a proper ideal of R and $M$, $N$ two finitely generated R-modules such that Supp $(N)\subseteq V(I)$. Let $t\geq 0$ be an integer such that for each $0\leq i\leq t$, the $R$-module $H^i_I(M)$ is in dimension $<n$. Then we show that each element $L$ of the set $\mathfrak{J}$, which is defined as: $$ \{{Ext}^j_R(N,H^i_I(M)):j\ge0 \ and \ 0\le i \le t\} \\ \cup\{{Hom}_R(N,H^{t+1}_I(M)),{Ext}^1_R(N,H^{t+1}_I(M))\} $$ is in dimension $<n-2$ and so as a consequence, it follows that the set $$ {Ass}_R(L)\cap\{{\frak{p}} \in Spec(R): :\,\dim (R/{\frak{p}})\ge n − 2\} $$ is finite. In particular, the set $$ {Ass}_R(\oplus_{i=0}^{t+1}H^i_I(R))\cap \{{\frak{p}}\in \,{Spec}(R)\,:\,\dim(R/{\frak{p}})\geq n-2\} $$ is finite. Also, as an immediately consequence of this result it follows that the $R$-modules ${Ext}^j_R(N,H^i_I(M))$ are in dimension $<n-1$, for all integers $i,j\geq 0$, whenever $\dim(M/IM)=n$. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in thecase of complete local rings.

Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties

Some properties of graded local cohomology modules