If the socle fits
In studying ideals that are obtained in some fashion from minors of matrices one is frequently in the position of having two ideals \(K\subset J\) for which the generators of \(K\) are known, perfection of \(J\) is known, and there is a strong suspicion of equality. The authors prove that for graded artinian \(k\)-algebras a simple comparison of socle dimensions suffices: if \(\dim_ k\operatorname{socle}(R/K)_ i\leq \dim _ k \operatorname{socle}(R/J)_ i\) for all \(i\), then \(K=J\). The principal new application is to identify a set of generators for the generic residual intersection of a generic grade three Gorenstein ideal; but the power of the method is also illustrated by a new proof (using residual intersection theory) of the perfection of the determinantal ideal generated by the maximal minors of a generic matrix.
scientific article; zbMATH DE number 37121
scientific article