Exercise 2, § 5

Let A be a regular local ring, I an ideal of A.

a) If $\mathrm{ht}(I) = 1$, prove that the following conditions are equivalent : (i) the ideal I is principal ; (ii) $A/I$ is a Gorenstein ring ; (iii) $A/I$ is a Macaulay ring. b) Suppose $\mathrm{ht}(I) = 2$; in order that $A/I$ be a Gorenstein ring, it is necessary and sufficient that the ideal I be a complete intersection (if $(L,p)$ is a minimal projective resolution of I, observe that $L_1$ is of rank 1 and consequently $L_0$ of rank 2). c) Let $k$ be a commutative field ; take $A = k[[X,Y,Z]]$ and $I = (XY, YZ, ZX, X^2 - Y^2, Y^2 - Z^2)$. Prove that I is an ideal of height 3 which is not a complete intersection, and that $A/I$ is a Gorenstein ring.