On modules of linear type (Q2352387)

Let \(R\) be a Noetherian ring. If \(A\) is an \(m \times n\) matrix over \(R\), then the ideal of \(R\) generated by the \(k \times k\) minors of \(A\) is denoted by \(I_{k}(A)\). Let \(M\) be the \(R\)-module that is the cokernel of the \(R\)-linear map from \(R^{M}\) to \(R^{n}\) induced by the transpose of \(A\). A result by \textit{L. L. Avramov} [J. Algebra 73, 248--263 (1981; Zbl 0516.13001)] states that, for any positive integer \(q\), \(M\) has rank \(n-m\) and the symmetric algebra of \(M\) is \(q\)-torsion-free if and only if grade \(I_{K}(A) \geq m-k+q+1\) for all \(k=1,2, \ldots , m\). A special case of this result is proved as main result of this paper: \(M\) has rank \(n-m\) and the symmetric algebra of \(M\) is torsion-free (or equivalently, the natural map from the symmetric algebra of \(N\) to the Rees algebra is an isomorphism) if and only if grade \(I_{k}(A) \geq m-k+2\) for all \(k=1,2, \ldots m.\) This paper provides a simple proof of this result, using induction on \(m\). Several lemmas pave the way for the induction proof. The main theorem is then generalized to prove the result of Avramov stated above. An example of a matrix satisfying the determinantal condition in the main theorem concludes the paper.