Perfect ideals of grade three defined by skew-symmetrizable matrices (Q2902031)

A square matrix \(M\) is said to be skew-symmetrizable or generalized alternating if there exist nonzero diagonal matrices \(C,D\) such that \(CMD\) is an alternating matrix. The notion of a skew-symmetrizable matrix was introduced by authors in order to describe a structure theorem for complete intersections of grade 4 [\textit{O.-J. Kang} and \textit{H. Ko}, Algebra Colloq.\ 12, No. 2, 181--197 (2005; Zbl 1100.13012)].NEWLINENEWLINEThe purposes of the paper are to construct a skew-symmetrizable matrix \(G_3\) of degree \(r+4\) (Definition 3.6) and an ideal \(I=\overline{Pf_{r+3}(G_3)}\) associated with \(G_3\) (Definition 3.12), to show Theorem 3.17 (see below), and to compute the Hilbert function of \(R/I\) in the case where \(I\) is a homogeneous perfect ideal of grade 3 in a polynomial ring \(R\) over a field (Proposition 4.2). Theorem 3.17: Let \((R,\mathfrak m)\) be a Noetherian local ring and let \(r\) be an odd integer \(>1\). Assume that the entries of matrices defining \(G_3\) are all contained in \(\mathfrak m\). If \(I\) has grade 3, then it is a perfect ideal of type 2.