The first syzygies of determinantal ideals
Let \(A\) be a commutative ring, and \(S=A[X]\) the polynomial ring over \(A\) in the indeterminates \(X_{ij}\) forming the \(m\times n\) matrix \(X\). Denote the ideal of \(S\) generated by the \(t\)-minors of \(X\) by \(I_ t\). A free resolution of \(S/I_ t\) as an \(S\)-module was constructed by Eagon and Northcott in the case \(t=\min(m,n)\) and by Akin, Buchsbaum, and Weyman for \(t=\min(m,n)-1\). Furthermore Lascoux describes resolutions for arbitrary \(t\) in the case in which \(A\) contains the rational numbers. Without any assumption on the characteristic of \(A\), \textit{D. W. Sharpe} [Proc. Lond. Math. Soc., III. Ser. 15, 645-679 (1965; Zbl 0119.036)] showed that for \(t=2\) the first module of syzygies of \(I_ t\) has a system of generators whose coefficients are linear in the \(X_{ij}\). For arbitrary \(t\) this was proved by \textit{K. Kurano} [J. Algebra 124, No. 2, 414-436 (1989; Zbl 0703.13011)] using the characteristic free representation theory of the general linear group. In particular the second Betti number of \(S/I_ t\) is independent of the characteristic of \(A\); this is wrong for the third Betti number [\textit{M. Hashimoto}, Nagoya Math. J. 118, 203-216 (1990; Zbl 0707.13005)]. In the paper under review the author derives Kurano's theorem in an `elementary' manner, using a Gröbner basis approach. The heart of the paper is a direct, algorithmic proof of the fact that the \(t\)-minors of \(X\) form a Gröbner basis of \(I_ t\) with respect to a natural term order. [Other proofs of this fact use counting arguments; see for examples \textit{B. Sturmfels}, Math. Z. 205, No. 1, 137-144 (1990; Zbl 0685.13005)].
scientific article; zbMATH DE number 166183