On ideals of minors fixing a submatrix
Ideals of minors fixing a submatrix arise from the following situation. Let R be a commutative ring and \(f:\quad F\to G\) be a map of free modules, where rank F\(=m\geq n=rank G\). It is well-known that one may identify the image of the induced map \(\bigwedge^ nf:\quad \bigwedge^ nF\to \bigwedge^ nG\) with the ideal of the \(n\times n\) minors of a matrix of f. Let there be given a decomposition \(F=F'\oplus F''\), where F' and F'' are free modules and rank F'\(=r\leq n-1\). Then the image of the restriction of \(\bigwedge^ nf\) to \(\bigwedge^ rF'\otimes \bigwedge^{n-r}F''\subset \bigwedge^ nF\) is correspondingly identified with the ideal \(J\subset R\) generated by the \(n\times n\) minors fixing r columns of a matrix of f. The authors give, in the generic case, the primary decomposition and the homological dimension of J. In particular, if \(r=n-1\) or if \(m=n+1\), they succeed in constructing an explicit free resolution of J and in proving the following statements: \((i)\quad R[Jt]\) is a normal Cohen-Macaulay domain. \((ii)\quad gr_ J(R)\) is a reduced Cohen-Macaulay ring. \((iii)\quad gr_ J(R)\) is torsionfree over R/J and \(J^{(n)}=J^ n\) for all \(n\geq 1\). Recently, the second author and the reviewer have shown that these statements hold for arbitrary r by using the theory of Hodge algebras (to appear).
scientific article; zbMATH DE number 3955116