Bounds on the minimal number of generators of the dual module
Let \((A,\mathfrak{m})\) be a Cohen-Macaulay local ring, \(M\) a finitely generated \(A\)-module and \(M^*=\Hom_A(M,A)\). In the paper under review, the authors study bounds on the minimal number of generators of \(M^*\). They prove that if \(M^*\) is a maximal Cohen-Macaulay \(A\)-module, then \(\mu_A(M^*)\leqslant\mu_A(M)e(A)\), where \(\mu_A(M)\) is the cardinality of a minimal generating set of \(M\) as an \(A\)-module and \(e(A)\) is the multiplicity of the local ring \(A\). Furthermore, if the natural map \(M\rightarrow M^{**}\) is an isomorphism, they show that \(\mu_A(M)/e(A)\) is a lower bound for \(\mu_A(M^*)\). As an application, they study the bounds on the minimal number of generators of specific modules over two-dimensional normal local rings.
scientific article; zbMATH DE number 7916059
scientific article