Proposition 12

Let E be an A-module and $(a_t)_{t \in T}$ a generating system of E. The following conditions are equivalent: (a) E is a projective A-module. (b) There exists a family $(a_t^*)_{t \in T}$ of linear forms on E such that, for all $x \in E$, the family $(\langle x, a_t^* \rangle)_{t \in T}$ has finite support and

There exists a surjective homomorphism $u : L \to E$, where $L = A_s^{(T)}$, such that if $(e_t)_{t \in T}$ is the canonical basis of L then $u(e_t) = a_t$ (§ 1, no. 11, Proposition 17); for E to be projective, it is necessary and sufficient that there exist a linear mapping $v : E \to L$ such that $u \circ v = 1_E$ (no. 2, Proposition 4 and § 1, no. 9, Proposition 15). If such a mapping exists and we write ${}^tv(e_t^*) = a_t^*$, then $\langle x, a_t^* \rangle = \langle x, {}^tv(e_t^*) \rangle = \langle v(x), e_t^* \rangle$, hence the family $(\langle x, a_t^* \rangle)$ has finite support and $x = u \left( \sum_{t \in T} \langle (x), e_t^* \rangle e_t \right) = \sum_{t \in T} \langle x, a_t^* \rangle a_t$ for all $x \in E$. Conversely, if condition (b) of the statement is fulfilled, the sum $\sum_{t \in T} \langle x, a_t^* \rangle e_t$ is defined for all $x \in E$ and $x \to \sum_{t \in T} \langle x, a_t^* \rangle e_t$ is a linear mapping $v : E \to L$ such that $u \circ v = 1_E$.