Determinantal ideals of linear type (Q1088748)

Let R be a noetherian domain, \(X=(X_{ij})\) an \(n\times n\) generic matrix over R and let I be the ideal \(I_{n-1}(X)\) of \(n-1\) size minors of X in \(R[X_{ij}]\). Then the main theorem 2.4 of this paper says that I is of linear type, i.e. the symmetric algebra \(S(I)\) is isomorphic to the Rees algebra \(\oplus I^ n\) of \(I\). For \(R={\mathbb{Z}}\) this yields the only missing piece in a determination of which ideals of the form \(I_ k(X)\), where X is a generic \(r\times s\) matrix, are of linear type (see corollary 2.6). The proof of theorem 2.4 is essentially based on the fact that for a noetherian reduced (commutative) ring R and an ideal \(I\subset R\) of grade \(I\geq 1\) the following are equivalent: (1) \(S(I)^{red}=\oplus I^ n;\) (2) the analytic spread of \(I_ p\) equals the least number of generators of \(I_ p\) (or \(I_ p=R_ p)\) for all \(P\in \text{Spec}(R)\).