On local cohomology of pseudo Cohen-Macaulay modules
Let \((A, \mathfrak m)\) be a Noetherian local ring, \(M\) a finitely generated \(A\)-module of dimension~\(d\) and \(x_1\), \dots, \(x_d\) a system of parameters for~\(M\). We put \[ Q_M(x_1, \dots, x_d) = \bigcup_n (x_1^{n+1}, \dots, x_d^{n+1})M : (x_1 \cdots x_d)^n. \] The \(A\)-module \(M\) is said to be pseudo Cohen-Macaulay if \[ e(x_1, \dots, x_d; M) = \ell(M/Q_M(x_1, \dots, x_d)) \] for some system of parameters \(x_1\), \dots, \(x_d\) for~\(M\). This definition is analogue of one of generalized Cohen-Macaulay modules. We know that \(M\) is generalized Cohen-Macaulay if and only if \(H_{\mathfrak m}^0(M)\), \dots, \(H_{\mathfrak m}^{d-1}(M)\) are of finite length, in other words, they are of Noetherian dimension at most~\(0\). In the present paper, the author gives some properties of local cohomology modules of pseudo Cohen-Macaulay modules, in particular, their Noetherian dimension.
polynomial type of fraction, dimension of Artinian module
scientific article; zbMATH DE number 2144209