Exercise 15, § 4

Let $A$ be the polynomial ring $\mathbf{Z}[X, Y, T, U, V, W]$, $L$ the free $A$-module $A^4, (e_i)_{1 \leq i \leq 4}$ its canonical basis and $M$ the submodule of $L$ generated by the four vectors $Xe_1, Ye_2, Te_3 + Ue_4, Ve_3 + We_2$. Show that

(compare with Exercise 14 (b)).