The case of the submaximal minors: A finishing touch

Abstract

Let F,, and G, be any two finitely generated free modules over the commutative ring R,, of ranks m and n, respectively (m 2 n, say). Let R denote the symmetric algebra S,,,(F,,@G,); R is a polynomial ring R,[X,] with 1 6 i < m, 1 <j < n. Each X, corresponds to fi@gj, where {j;} and {g,} denote bases of F, and G,, respectively. Take the modules F = R 0 R,, F, and G = R OR0 G,, finitely generated and free over R, and let {h} and { gj} also denote, by abuse of notation, the obvious R-bases of F and G. A distinguished element in Hom,(FO G, R) is the map cp defined by cp(fi@g,) = X,. Let q also denote the corresponding map G + F* obtained via the canonical isomorphism