Why is this ring Cohen-Macaulay?

Question

Let $q_1, q_2$ be quadratic homogeneous polynomials in $x_0,x_1,x_2,x_3,x_4$ over $\mathbb C$ and let $X_i:=V(q_i)=\{(a_0,\dots,a_4)\in \mathbb{P}_{\mathbb{C}}^4\mid q_i(a_0,\dots,a_4)=0\}$. If $X_1\cap X_2$ is a surface, then $R=\mathbb{C}[x_0,x_1,x_2,x_3,x_4]/(q_1,q_2)$ is Cohen-Macaulay.

I can't understand about Cohen-Macaulay ring well. Please let me know why it is CM.

Answer 1

Any complete intersection is Cohen-Macaulay. Since a surface in $\mathbb{P}^4$ cut out by two polynomials is a complete intersection (4 - 2 = 2), it is automatically CM.