Symmetric algebras of modules arising from a fixed submatrix of a generic matrix
Let \(K\) be a field and let \(n\leq m\) be positive integers. In the polynomial algebra \(R=K[X]\), where \(X=(X_{ij})\) is a generic \(n\times m\) matrix, let \(I\) be the ideal generated by the \(n\times n\) minors of \(X\) containing the first \(n-1\) columns. In the first part the authors study the divisorial properties of the symmetric algebra \(S(I)\). This is done by first observing that \(S(I)\approx R[T_n,\ldots,T_m]/I_{n+1}(\tilde X)\), where \(\tilde X\) is the matrix obtained by adding the row \((0,\ldots,0,T_n,\ldots,T_m)\) to \(X\) and \(I_{n+1}(\tilde X)\) is the ideal generated by all \((n+1)\times (n+1)\) minors of \(\tilde X\), and then applying the Hochster-Eagon theory of determinantal ideals [\textit{M. Hochster} and \textit{J. A. Eagon}, Am. J. Math. 93, 1020--1058 (1971; Zbl 0244.13012)]. The following results are proved: (1) \(S(I)\) is a Cohen-Macaulay normal domain; (2) for \(n\geq 2\) the divisor class group of \(S(I)\) is \(\mathbb{Z}\oplus \mathbb{Z}\); (3) the canonical module of \(S(I)\) is \(\mathfrak a^{m-n-1}\), where \(\mathfrak a\) is the ideal of \(S(I)\) generated by the \(n\times n\) minors of the first (resp. last) \(n\) columns of \(\tilde X\); (4) the type of \(S(I)\) is \(\binom{m-1}{m-n-1}\). In the second part similar results are proved for the symmetric algebra \(S(M)\), where \(M\) varies over certain submodules of \(\operatorname{coker}(X^*: (R^n)^*\to (R^m)^*)\).
scientific article; zbMATH DE number 4047844
scientific article