Proposition 9

Let ρ : A → B be a homomorphism of noetherian rings, M a finitely generated A-module and N a finitely generated B-module, flat over A. Denote by $^a\rho : \mathrm{Spec}(B) \longrightarrow \mathrm{Spec}(A)$ the mapping associated with ρ. The following conditions are equivalent:

(i) the B-module $M \otimes_A N$ is Macaulay; (ii) the $(\kappa(p) \otimes_A B)$-module $\kappa(p) \otimes_A N$ is Macaulay for every $p \in \mathrm{Supp}_A(M)$, and the $A_p$-module $M_p$ is Macaulay for every $p \in ^a\rho(\mathrm{Supp}_B(N))$; (iii) for every maximal ideal of $\mathrm{Supp}_B(N)$ whose inverse image $p$ in A belongs to $\mathrm{Supp}_A(M)$, the $A_p$-module $M_p$ and the $(\kappa(p) \otimes_A B)$-module $\kappa(p) \otimes_A N$ are Macaulay.

If moreover the B-module N is faithfully flat, these conditions imply that the A-module M is Macaulay.

Let q be a prime ideal of B belonging to the support of $M \otimes_A N$. Put $p = \rho^{-1}(q)$. Since the module $(M \otimes_A N)_q$ is identified with $M_p \otimes_{A_p} N_q$, the modules $M_p$ and $N_q$ are non-zero, and the same is true of $\kappa(p) \otimes_A N_q$ (No. 6, Remark). The $A_p$-module $N_q$, being isomorphic to a module of fractions of $N_p$, is flat and we have the equalities

(§ 1, No. 6, prop. 11, b) and remark), in which each term is an integer $\geq 0$. Taking into account the fact that the $B_q$-module $\kappa(p) \otimes_A N_q$ is a Macaulay module if and only if it is so as a $(\kappa(p) \otimes_A B_q)$-module (No. 1, example 4), we deduce the equivalence of the following two conditions:

(α) the $B_q$-module $(M \otimes_A N)_q$ is a Macaulay module; (β) the $A_p$-module $M_p$ and the $(\kappa(p) \otimes_A B_q)$-module $\kappa(p) \otimes_A N_q$ are Macaulay modules.

Let us now prove that (iii) implies (i). Let n be a maximal ideal of B belonging to the support of $M \otimes_A N$, and let $p = \rho^{-1}(n)$. From the foregoing we have $p \in \mathrm{Supp}_A(M) \cap ^a\rho(\mathrm{Supp}_B(N))$; condition (iii) and the remark of No. 6 imply that condition (β) above is satisfied with $q = n$. It follows that the $B_n$-module $(M \otimes_A N)_n$ is a Macaulay module, whence (i).

The implication (ii) ⇒ (iii) is clear; let us prove that (i) implies (ii). Suppose the B-module M ⊗_A N to be a Macaulay module. Let p be an element of Supp_A(M). We may suppose that the (κ(p) ⊗_A B)-module κ(p) ⊗_A N is nonzero, that is to say, that there exists a prime ideal q of Supp_B(N) lying over p (No. 6, remark). The B_q-module (M ⊗_A N)_q is a Macaulay module (No. 1, example 3); it follows from implication (α) ⇒ (β) proved above and from the remark of No. 6 that the A_p-module M_p and the (κ(p) ⊗_A B)-module (κ(p) ⊗_A N) are Macaulay modules, whence (ii).

If moreover N is faithfully flat over A, we have κ(p) ⊗_A N ≠ 0 for every p ∈ Spec(A), whence a_p(Supp_B(N)) = Spec(A) (No. 6, remark), so that (ii) implies that M is a Macaulay module.