Narrow Results

We show some results about local homology modules and local cohomology modules concerning Grothendieck’s conjecture and Huneke’s question. We also show some equivalent properties of $I$-separated modules and of minimax local homology modules. By duality, we get some properties of Grothendieck’s local cohomology modules.

We study the generalized local homology for linearly compact modules which is a generalization of the local homology theory. By duality, we get some properties of the generalized local cohomology and extend well-known properties of the local cohomology theory of Grothendieck.

Let $𝔞$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\mathrm{\text{V}}(𝔞)$. Let $n$ be the supermum of the integers $i$ for which $H_i^{𝔞}(M)\ne 0$. We show that $M$ is $𝔞$-cofinite if and only if the $R$-module ${Tor}_i^R(R/𝔞,M)$ is finitely generated for every $0\le i\le n$. This provides a hands-on and computable finitely-many-steps criterion to examine $𝔞$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.

Let $𝔞$ be an ideal of a commutative noetherian ring $R$ and $M$ an $R$-module with Cosupport in $\mathrm{\text{V}}(𝔞)$. We show that $M$ is $𝔞$-coartinian if and only if ${\mathrm{\text{Ext}}}_R^i(R/𝔞,M)$ is artinian for all $0\le i\le \mathrm{\text{cd}}(𝔞,M)$, which provides finite steps to examine $𝔞$-coartinianess. We also consider the duality of Hartshorne’s questions: for which rings $R$ and ideals $𝔞$ are the modules ${\mathrm{\text{H}}}_i^{𝔞}(M)$$𝔞$-coartinian for every artinian $R$-module $M$ and all $i\ge 0$; whether the category ${𝒞}{(R,𝔞)}_{\mathrm{\text{coa}}}$ of $𝔞$-coartinian modules is an abelian subcategory of the category of $R$-modules, and establish affirmative answers to these questions in the case $\mathrm{\text{cd}}(𝔞,R)\le 1$ and $\mathrm{\text{dim}}R/𝔞\le 1$.

Let (R,𝔪) be a local ring, X an artinian R-module of noetherian dimension d; let x₁,…,x_d ∈ 𝔪 be such that 0:_X (x₁,…,x_d)R has finite length. We show by an example that is not finite as an R-module in general; it is finite if we assume R is complete. This answers a question posed by Tang. As a first application of the latter finiteness result, we give a necessary condition for a finite module to be Cohen–Macaulay; secondly we propose a notion of Cohen–Macaulayfication and prove its uniqueness; finally we show that this new notion of Cohen–Macaulayfication is a direct generalization of a notion of Cohen–Macaulayfication introduced by Goto.

We introduce generalized local homology which is in some sense dual to generalized local cohomology, and study some properties of generalized local homology modules for artinian modules, such as the artinianness, noetherianness and the characterization of Width_I(M) by generalized local homology. By using duality, we get back some properties of generalized local cohomology.

In this paper we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser [16].