Lemma 5

Let $(M_i)_{i \in I}$ be a family of left $A$-modules, $M = \bigoplus_{i \in I} M_i$, their direct sum and $E$ a right $A$-module. If, for all $i \in I$, $E$ is flat for $M_i$, then $E$ is flat for $M$.

(a) Suppose first that $I = \{1, 2\}$, and let $M'$ be a submodule of $M = M_1 \oplus M_2$, $M$, and $M$, being canonically identified with submodules of $M$. Denote by $M'_1$ the intersection $M' \cap M_1$, and by $M'_2$ the image of $M'$ in $M_2$ under the canonical projection $p$ of $M$ onto $M_2$. We have a diagram

where $v_1, v, v_2, i, i'$ are the canonical injections and $p'$ is the mapping with the same graph as the restriction of $p$ to $M'$, which is surjective. It is immediately verified that this diagram is commutative and that its rows are exact. With $T(Q)$ and $T(v)$ used in the same sense as in the proof of Lemma 4, we have a commutative diagram

By Lemma 1 of no. 1 the two rows of this diagram are exact; as E is flat for $M_1$ and $M_n$, $T(v_1)$ and $T(v_2)$ are injective; moreover, by Lemma 2 of no. 1, $T(i)$ is injective. Corollary 2 to Proposition 2 of § 1, no. 4 then shows that $T(v)$ is injective and consequently E is M-flat.

(b) If I is a finite set with n elements, the lemma follows by induction on n using (a).

(c) In the general case let $M'$ be a finitely generated submodule of M. Then there exists a finite subset J of the indexing set I such that $M'$ is contained in the direct sum $M_J = \bigoplus_{i \in J} M_i$. By (b) E is flat for $M_J$; the canonical homomorphism $E \otimes_A M' \to E \otimes_A M$, is therefore injective. On the other hand, as $M_J$ is a direct factor of M, the canonical homomorphism $E \otimes_A M_J \to E \otimes_A M$ is injective (no. 1, Lemma 2). Taking the composition, it follows that $E \otimes_A M, \to E \otimes_A M$ is injective and E is flat for M by Lemma 3.