Dual of Faltings' Theorems on Finiteness of Local Cohomology

TL;DR

This paper establishes dual versions of Faltings' Theorems for local homology modules of Artinian modules, demonstrating finiteness properties and introducing the Artinianness dimension in the context of semi-local complete rings.

Contribution

It introduces the dual of Faltings' Theorems for local homology and defines the $n$th Artinianness dimension, extending finiteness results for Artinian modules over semi-local complete rings.

Findings

Finiteness of the coassociated primes of certain local homology modules.

Introduction of the $n$th Artinianness dimension for Artinian modules.

Equivalence of the Artinianness dimension with the non-minimax property of local homology modules.

Abstract

Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian module over semi-local complete ring $R$ and $j$ is an integer such that $H_i^{\fa}(A)$ is Artinian for all $i<j$, then the set $\Coass_R(H_j^{\fa}(A))$ is finite. We also introduce the notion of the $n$th Artinianness dimension $g_n^\fa(A)=\inf\{g^{\fa R_{\fp}}(^\fp A): \fp\in\Spec(R) \ \ \text{and} \ \ \dim R/\fp\geq n\}$, for all $n\in\mathbb{N}_{0}$ and prove that $g_1^\fa(A)=\inf\{i\in\mathbb{N}_0: H_i^\fa(A) \ \ \text{is not minimax}\}$, whenever $R$ is a semi-local complete ring. Moreover, in this situation we show that $\Coass_R(H_{g_1^\fa(A)}^\fa(A))$ is a finite set.