Exercise 1, § 5

Let k be a field, A the quotient ring of the polynomial ring $k[X, Y, Z]$ by the ideal generated by $Z^2, ZX, Z(Y-1)$, x and y the classes of X and Y in A. Now prove that the sequence $(x, y)$ is completely secant for A but is not A-regular, though the sequence $(y, x)$ is A-regular.