Exercise 6, § 5

Let $A$ be a local noetherian ring, $I$ an ideal of $A$. a) Prove that the following conditions are equivalent: (i) the ideal $I$ is a complete intersection; (ii) the $A/I$-module $I/I^2$ is free and one has $dp_A(A/I) < +\infty$. (Under assumption (ii), deduce from exerc. 2 of § 3 that there exists a cancellable element x in $I$ such that $x \not\in m_A I$; reason by induction on the dimension of $I/I^2$.) b) Prove that the set of prime ideals $p$ of $A$ where the ideal $I_p$ is a complete intersection is open in $\mathrm{Spec}(A)$. c) Let $B$ be a presentable ring; prove that the set of prime ideals q of $B$ such that $B_q$ is a complete intersection ring is open in $\mathrm{Spec}(B)$.