The case of the submaximal minors: A finishing touch
Let \(F_ 0\) and \(G_ 0\) be the free modules over the commutative ring \(R_ 0\) with finite bases \(\{f_ 1,...,f_ m\}\), \(\{g_ 1,...,g_ n\}\) respectively, where \(m\geq n\). Let R denote the symmetric algebra \(S(F_ 0\otimes G_ 0)\) over \(R_ 0\). Then R is a polynomial ring \(R_ 0[X_{ij}]\) where the indeterminate \(X_{ij}\) corresponds to \(f_ i\otimes g_ i\) for each i, j. This paper is concerned with the free resolutions of the ideals \(I_ p\) generated by the minors of order p of the generic matrix \((X_{ij})\). A construction of a minimal resolution of the ideal of the submaximal minors was given by \textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman} [Adv. Math. 39, 1-30 (1981; Zbl 0474.14035)]. However this construction gave no description of the boundary map of the complex resolving \(I_{n-1}\). Knowledge of this boundary map would be helpful in tackling the resolutions of \(I_{n-2}\), \(I_{n-3}\), etc. The author provides an explicit description of the boundary map and then illustrates this in the specific case when \(m=n+1\).
scientific article; zbMATH DE number 4047847