Exercise 3, § 7

Let $k$ be a field, and let $f_1, ..., f_r$ be homogeneous elements of the graded ring $k[X_1, ..., X_n]$ forming a completely transversal sequence. We shall say that the quotient ring

is a graded ring of complete intersection. Let $I$ be a nonzero graded ideal of the ring $H = k[X_1, ..., X_n]/(f_1, ..., f_r)$ endowed with the quotient grading. Now prove that one has $\dim(I) = \dim(H)$. Deduce that one has either $\dim(H/I) < \dim(H)$, or $\dim(H/I) = \dim(H)$ and $c_{M/I} < c_M$ (§ 4, n° 2). (One will consider the ideal $b$ formed by the $s \in H$ such that $s.I = \{0\}$ and one will prove that if one has $\dim(I) < \dim(H)$, then $b$ contains a nonzero divisor element, by examining the position of $b$ with respect to the minimal prime ideals of $H$ and to $H_+ = H_{\geq 1}$.)