Exercise 17, § 4

Let k be a field, A the polynomial ring $k[X, Y]$, L the free A-module $A^2$ and $(e_1, e_2)$ the canonical basis of L. Let M be the submodule of L generated by the two vectors $(X + 1)e_1 + Ye_2, Ye_1 + Xe_2$. Show that there exists no pseudo-isomorphism from L to itself, whose restriction to M is a pseudo-isomorphism from M to the submodule N of the form $au + bv$ where $(u, v)$ is a basis of the A-module L and a, b two ideals of A, or whose restriction to N is a pseudo-isomorphism from N to M. (By considering the determinantal ideals (Exercise 10) and noting that M is reflexive, attention may be confined to the case where N would also be reflexive and then necessarily $a = A, b = AP$, where

is an extremal element of A; show finally that, for every basis $(u, v)$ of L, $M \cap Au$ can neither contain Au nor be contained in $APu$.)