Theorem 1 (D. Lazard)

For every A-module E, the following conditions are equivalent: (i) E is flat. (ii) For every A-module P of finite presentation, the canonical homomorphism

is surjective. (iii) For every A-module P of finite presentation and every homomorphism $u : P \to E$, there exists a free A-module L of finite type and homomorphisms $v : P \to L$ and $w : L \to E$ such that $u = w \circ v$. (iv) There exists a filtered ordered set J, an inductive system of free modules of finite type $(L_j)_{j \in J}$ and an isomorphism of E onto $\lim \overrightarrow{L_j}$. (iv) $\Rightarrow$ (i) : this follows from prop. 4 (ii) of X, p. 8. (i) $\Rightarrow$ (ii) : this follows from prop. 8b) of X, p. 12. (ii) $\Rightarrow$ (iii) : let P be a A-module of finite presentation and $u : P \to E$ a homomorphism; by (ii), there exist $v_1, \ldots, v_n \in \operatorname{Hom}_A(P, A)$, $w_1, \ldots, w_n \in E$ such that $u(x) = \sum v_i(x) w_i$ for every $x \in P$; if $v : P \to A^n$ is the homomorphism of components $(v_i)$ and $w : A^n \to E$ the homomorphism $(a_i) \mapsto \sum a_i w_i$, we indeed have $u = w \circ v$. (iii) $\Rightarrow$ (iv) : suppose (iii) is verified, and let $(E_\alpha, \varphi_{\beta, \alpha})$ be an inductive system, relative to a filtered set I, of A-modules of finite presentation, with inductive limit E

(X, p. 11, prop. 7). Replacing I by the lexicographic product I × N, with E_{(α,n)} = E_α for every n, we may suppose that I has no greatest element. For each α ∈ I, let L_α be a free A-module of finite type and let u_α : E_α → L_α, v'_α : L_α → E be homomorphisms such that v'_α ∘ u_α is the canonical mapping φ_α of E_α into E; since L_α is free of finite type and I has no greatest element, there exists an index β > α and a homomorphism v''_α : L_α → E_β such that v'_α = φ_β ∘ v''_α; since φ_β ∘ v''_α ∘ u_α = φ_β ∘ φ_β,α and E_α is of finite presentation, it follows from prop. 8a) of X, p. 12, that there exists γ ≥ β such that φ_{γβ} ∘ v''_α ∘ u_α = φ_{γβ} ∘ φ_{βα} = φ_{γα} ; let γ = f(α) and let v_α be the homomorphism φ_{γβ} ∘ v''_α from L_α into E_{f(α)} ; we have v_α ∘ u_α = φ_{f(α),α}. We can then apply lemma 2, whence (iv).