Proposition 5

Let $\rho : A \to B$ be a homomorphism of noetherian rings and $N$ a $B$-module. Suppose that a) the ring $A$ is regular, b) $N$ is a finitely generated $A$-module, c) the $B$-module $N$ is Cohen-Macaulay, d) every minimal prime ideal of $\mathrm{Supp}_B(N)$ lies over a minimal prime ideal of $A$. Then $N$ is a projective $A$-module (of finite type).

The point is to prove that, for every maximal ideal $m$ of $A$, the $A_m$-module $N_m$ is free (II, § 5, No. 2, th. 1). The $A$-module $B/\mathrm{Ann}_B(N)$ is a submodule of the finitely generated $A$-module $\mathrm{End}_A(N)$, and is therefore finitely generated. If one replaces $B$ by $B/\mathrm{Ann}_B(N)$, the hypotheses of the proposition are still satisfied ($§ 2$, No. 1, example 5); one may therefore suppose that $B$ is a finitely generated $A$-module and that $\mathrm{Supp}_B(N) = \mathrm{Spec}(B)$.

Let $m$ be a maximal ideal of $\mathrm{Supp}_A(N)$; put $n = \dim(A_m)$. By Cor. 2 of Prop. 3 of No. 1, it is enough to prove that $N_m$ is a Macaulay $A_m$-module of dimension $n$. Every maximal ideal of $B_m$ is of the form $nB_m$, where $n$ is a prime ideal of $B$ lying over $m$ (V, § 2, No. 1, Lemma 1 and Prop. 1). Let $p$ be a minimal prime ideal of $\mathrm{Supp}_B(N)$, contained in $n$. The closed subset $V(pB_n)$ of $\mathrm{Supp}_{B_n}(N_n)$ is then of codimension zero; since the $B_n$-module $N_n$ is Macaulay, the Cor. of Prop. 2, § 2, No. 2 yields the equality $\dim_{B_n}(N_n) = \dim(B_n/pB_n)$. But $p$ lies over a minimal prime ideal of $A$, contained in $m$, so that