On certain length functions associated to a system of parameters in local rings (Q1970065)

Let \((A, \mathfrak m)\) be a Noetherian local ring, \(M\) a finitely generated \(A\)-module of dimension \(d\) and \(\mathbf x = x_1\), \dots, \(x_d\) a system of parameters for \(M\). In the present paper the authors consider a submodule of \(M\): \[ Q_M(\mathbf x) = \bigcup_t (x_1^{t+1}, \dots, x_d^{t+1})M: (x_1 \cdots x_d)^t. \] \textit{M. Hochster} [Nagoya Math. J. 51, 25-43 (1973; Zbl 0268.13004)] posed a conjecture: \(A \neq Q_A(\mathbf x)\) for any system of parameters for \(A\). It is called the monomial conjecture and Hochster showed that \(A \neq Q_A(x_1^n, \dots, x_d^n)\) for any system of parameters \(x_1\), \dots, \(x_d\) and sufficiently large \(n\). The authors show the uniformity of \(n\). That is, there is an integer \(N\) such that \(M \neq Q_M(x_1^n, \dots, x_d^n)\) for any system of parameters \(x_1\), \dots, \(x_d\) and integer \(n > N\).