The canonical trace of determinantal rings (Q6633586)

In this work, the authors study the canonical trace of generic determinantal rings. In particular, they compute the canonical trace of Cohen-Macaulay rings of codimension two, which are generically Gorenstein. Let \(K\) be a field, and let \((R,\mathfrak{m})\) be a Cohen-Macaulay local ring or a finitely generated Cohen-Macaulay \(K\)-algebra with graded maximal ideal \(\mathfrak{m}\), admitting a canonical module \(\omega_R\). The \textit{canonical trace} of \(R\) is the ideal of \(R\) defined by\N\[\Ntr(\omega_R) = \sum_{\varphi\in \Hom_R(\omega_R,R)}{\varphi(\omega_R)}.\N\]\NAssume \(R\) is a determinantal ring, \textit{i.e.}, \(R = K[X]/I_{r+1}(X)\), where \(X\) is an \(m\times n\) matrix of indeterminates with \(m<n\), and \(I_{r+1}(X)\) is the ideal of \(K[X]\) generated by the \(r+1\)-minors of \(X\) with \(1<r+1\leq m\). Under these conditions, the authors prove that\N\[\Ntr(\omega_R) = I_{r}(X)^{n-m}R.\N\]\NAs a corollary, the singular locus and the non-Gorenstein locus of \(R\) coincide. Furthermore, the authors determine the \textit{Teter number} of \(R\), which is the smallest number of maps such that \(tr(\omega_R) = \sum_{i=1}^{t}{\varphi_i(\omega_R)}\). If \(R\) is a determinantal ring, the Teter number is given by\N\[\N\mathrm{Teter}(R)=\det\left[ \binom{2n-m-j}{n-i} \right]_{1\leq i,j\leq r}.\N\]\NAs an application of the previous results, consider a Cohen-Macaulay local ring \((R,\mathfrak{m})\) or a finitely generated Cohen-Macaulay \(K\)-algebra with graded maximal ideal \(\mathfrak{m}\), admitting a canonical module \(\omega_R\). If \(R\) is generically Gorenstein, then the canonical trace specializes, \textit{i.e.}, \(tr(\omega_R)\overline R = tr(\omega_{\overline R})\) where \(\overline R = R/ \mathtt{x}R \) is generically Gorenstein and \(\mathtt{x}\) is an \(R\)-sequence. Finally, the authors prove that if \(I\) is a graded perfect ideal of height 2 in the polynomial ring \(S = K[x_1, \ldots , x_s]\), minimally generated by \(n\) homogeneous elements, and \(R = S/I\) is generically Gorenstein, then\N\[\Ntr(\omega_R) = I_{n-2}(A)R,\N\]\Nwhere \(A\) is the Hilbert-Burch matrix of \(I\).