Zero-dimensional non-Artinian local cohomology modules (Q2203984)

Let \(R\) be a commutative Noetherian ring with non-zero identity. For each integer \(i\), Grothendieck defined the \(i\)-th local cohomology of an \(R\)-module \(X\) with respect to an ideal \(\mathfrak {a}\) as follows: \[\text{H}^i_{\mathfrak {a}}(X):={\underset{n}{\varinjlim}\,\text{Ext}^i_R\left(R/\mathfrak {a}^n, X\right)}.\] Now, assume \(R\) is a local ring of dimension \(d\) with maximal ideal \(\mathfrak m\). A sequence \(x_1,\dots,x_d\) of elements of \(R\) is called a system of parameters of \(R\) if \(\mathfrak{m}=\sqrt{Rx_1+\cdots+Rx_d}\). In the paper under review, the authors show that when \(4\leq d\), \(2\leq i\leq d-2\) and \(x_1,\ldots,x_i\) is a part of system of parameters of \(R\), then there exist infinitely many prime ideals \(\mathfrak p\) with \(\text{dim}_R(R/\mathfrak p)=i+1\) such that the top local cohomology module \(\text{H}^i_{(x_1,\ldots,x_i)}(R/\mathfrak p)\) with support in \(\{\mathfrak m\}\) is non-Artinian.