Proposition 13

For E to be F-flat (no. 2, Definition 1), it is necessary and sufficient that the following condition be satisfied:

(R) If $(e_i)_{i \in I}$ and $(f_i)_{i \in I}$ are two finite families of elements of E and F respectively such that $\sum_{i \in I} e_i \otimes f_i = 0$ in $E \otimes_A F$, there exists a finite set J, a family $(x_j)_{j \in J}$ of elements of E and a family $(a_{ji})$ ($j \in J, i \in I$) of elements of A, with the following properties:

(1) $\sum_{i \in I} a_{ji} f_i = 0$ for all $j \in J$; (2) $e_i = \sum_{j \in J} x_j a_{ji}$ for all $i \in I$.

Suppose that E is F-flat. Let $(e_i)$ and $(f_i)$ be finite families of elements such that $\sum_{i \in I} e_i \otimes f_i = 0$ in $E \otimes_A F$, and let $F'$ be the submodule of F generated by the $f_i$. Since the canonical mapping $E \otimes_A F' \to E \otimes_A F$ is injective, we have also $\sum_{i \in I} e_i \otimes f_i = 0$ in $E \otimes_A F'$ and Lemma 10 can then be applied to E and F'; thus families $(x_j)$ and $(a_{ji})$ are obtained satisfying the conditions of (R).

Conversely, suppose that condition (R) holds. Let $F'$ be a submodule of F and let $y = \sum_{i \in I} e_i \otimes f_i$ be an element of the kernel of the canonical mapping $E \otimes_A F' \to E \otimes_A F$. Since (R) holds, there exist families $(x_j)$ and $(a_{ji})$ satisfying conditions (1) and (2). We conclude that, in $E \otimes_A F'$,

Hence $E \otimes_A F' \to E \otimes_A F$ is injective.