Proposition 12

Let A, B be commutative rings, $\varphi$ a homomorphism of A to B, $E_1, E_2, E_3$ three A-modules, $f : E_1 \times E \to E_3$ an A-bilinear mapping and

its extension. Consider a submodule $F_2$ of $E_2$, a submodule $F_3$ of $E_3$, and denote by T the submodule of $E_1$ consisting of those $x_1 \in E_1$ such that $f(x_1, x_2) \in F$, for all $x_2 \in F$. Suppose that B is a flat A-module and that $F_2$ is finitely generated. Then $T_{(B)}$ is the set of those $x'_1 \in E_{1(B)}$ such that $f_B(x'_1, x'_2) \in F_{3(B)}$ for all $x'_2 \in F_{2(B)}$.

Let $p$ be the canonical surjection $E_3 \to E_3/F_3$; with each $x_1 \in E_1$ we associate the A-linear mapping $x_2 \mapsto p(f(x_1, x_2))$ of $F_2$ to $E_3/F_3$, which we denote by $g(x_1)$; then $g$ is an A-homomorphism of $E_1$ to $\mathrm{Hom}_A(F_2, E_3/F_3)$ and the kernel of $g$ is precisely T. Since B is a flat A-module, we have the exact sequence

(no. 3, Proposition 1). By Proposition 11 the canonical homomorphism

is injective. On the other hand, as B is a flat A-module, $(E_3/F_3)_{(B)}$ is canonically identified with $E_{3(B)}/F_{3(B)}$; taking the composition of $\omega$ with $1 \otimes g$, we obtain a homomorphism $u$ for which the sequence!

is exact. It follows immediately from the definitions that $u(x'_1)$, where

is the linear mapping which maps each $x'_2 \in F_{2(B)}$ to the class mod. $F_{3(B)}$ of $f_B(x'_1, x'_2)$; by linearity this is also true for all $x'_1 \in E_{1(B)}$; since the kernel of $u$ is $T_{(B)}$, the proposition is proved.