A criterion for integral dependence of modules
The purpose of the paper is to give a multiplicity based criterion for a finitely generated module to be integral over a submodule. Let \((R,\mathfrak m)\) be a noetherian local ring of dimension \(d\) and \(E\) a submodule of a free \(R\)-module \(F = R^e\). The function \(\Sigma(n) = \sum_{i=0}^{n-1} \lambda_R(H_{\mathfrak m}^0(E^iF^{n-i}/E^{i+1}F^{n-i-1}))\) is a polynomial of degree at most \(d+e-1\) for \(n \gg 0\), where \(\lambda_R\) denotes length and the products are taken in the polynomial ring \(\text{Sym}(F) = R[x_1,\ldots,x_e]\). The \(j\)-multiplicity \(j(E)\) of \(E\) is defined as follows: \[ \Sigma(n) = \dfrac{j(E)}{(d+e-1)!}\,n^{d+e-1} + (\text{terms of lower degrees}). \] Then the main result is stated as follows. Theorem 1.1. Let \(R\) be a universally catenary locally equidimensional noetherian ring, let \(U \subset E\) be submodules of a free \(R\)-module \(F=R^e\), and assume that \(U_{\mathfrak p} = F_{\mathfrak p}\) for every minimal prime \(\mathfrak p\) of \(R\). The following are equivalent: (i) \(j(U_{\mathfrak q}) = j(E_{\mathfrak q})\) for every \(\mathfrak q \in \text{Spec}(R)\). (ii) \(j(U_{\mathfrak q}) \leq j(E_{\mathfrak q})\) for every \(\mathfrak q \in \text{Supp}(F/U)\) with \(\dim R_{\mathfrak q} = \ell(U_{\mathfrak q})-e+1\), where \(\ell\) denotes the analytic spread. (iii) \(U\) is a reduction of \(E\), i.e. \(E^{i+1} = UE^i\) for some \(i \geq 0\). In section 2, the \(j^\ast\)-multiplicity \(j^\ast(I,M)\) is defined for an ideal \(I\) generated by linear forms in a standard graded noetherian algebra \(A\) over a noetherian local ring and a finitely generated graded \(A\)-module \(M\), and then the implication (iii) \(\Rightarrow\) (i) is proven in a more general setting. In section 3, the \(j^\sharp\)-multiplicity \(j^\sharp(I,M)\) is defined, and the implication (ii) \(\Rightarrow\) (iii) is shown. In the last section 4, the \(j\)-multiplicity \(j(I,M)\) is defined, and a generalization of theorem 1.1 is given.
scientific article; zbMATH DE number 5302021