Modules that are finite birational algebras
Let \(A\) be a commutative Noetherian ring and let \(B\) be a faithful \(A\) module with a distinguished element \(e \in B\). Assume that \(B\) is a finite birational \(A\) module: that is, that there is \(d \in A\), \(d\) a non-zerodivisor on \(B\) such that \(dB \subseteq Ae \subseteq B\). The authors' first main result is that in the above notation the \(A\) linear double dual \(B^{**}\) is an \(A\) algebra with identity \(e\) if and only if for all \(b \in B\) with \(A\) annihilator \(0\) there is an inclusion \(\text{ann}(B/Ab) \subseteq \text{ann}(B/Ae)\), and that in the second condition it is sufficient for the inclusions to hold for a set of elements \(b_i \in B\) that generate \(B\) as an \(A\) module. If \(A\) is a homomorphic image of a Noetherian ring \(R\), and \(M: R^s \to R^t\) is a presentation matrix for \(B\) whose first row corresponds to \(e \in B\), let \(J\) denote the ideal of \((t-1) \times (t-1)\) minors of \(M\) and let \(I\) denote the ideal of \((t-1) \times (t-1)\) minors of \(M_1\), where the latter matrix is the submatrix of \(M\) obtained by dropping the first row. Then the authors' second main result asserts that if \(B^{**}\) is an \(A\) algebra with identity \(e\) then the radical of \(I\) contains \(J\); and if \(I\) contains \(J\) and \(I\) is either a radical ideal or has grade at least \(s-t+2\) in \(R\), then \(B^{**}\) is an \(A\) algebra with identity \(e\).
scientific article; zbMATH DE number 1028998