Noetherian dimension and co-localization of Artinian modules over local rings
The goal of this short paper is to establish conditions for the Noetherian local ring \(R\) when the Noetherian dimension for any local cohomology module \(H^i_{\mathfrak{m}}(M)\) supported at the maximal ideal is precisely equal to the Krull dimension of \(R\) modulo the annihilator of \(H^i_{\mathfrak{m}}(M)\). To do this, the authors use a co-localization functor \(F_{\mathfrak{p}}\) which is linear on Artinian \(R\)-modules, send Artinian \(R\)-modules to Artinian \(R_{\mathfrak{p}}\)-modules and \(F_{\mathfrak{p}}(A)\) is nonzero for all primes \(\mathfrak{p}\) containing the annihilator of \(A\). Their main result shows that \(N\)-\(\text{dim}_R(H^{i}_{\mathfrak{m}}(M))=\text{dim}_R(R/\text{Ann}_R(H^{i}_{\mathfrak{m}}(M))\) if and only if \(R\) is universally catenary and all its formal fibers are Cohen-Macaulay. The co-localization functor developed by \textit{A. S. Richardson} [Rocky Mt. J. Math. 36, No. 5, 1679--1703 (2006; Zbl 1134.13008)] satisfies the properties mentioned above for all local cohomology modules in the case that \(R\) is a complete ring but the authors are hoping that such a co-localization functor exists in the non complete case.
scientific article; zbMATH DE number 6366788
scientific article