Corollary

Every flat module of finite presentation is projective.

Indeed, let $E$ be a flat and of finite presentation $A$-module. Applying (b) to the case $B = A, G = {}_sA_d, F = E$, we see that the canonical homomorphism

is surjective. This implies that $E$ is projective (II, p. 77, remark 1).

According to the preceding corollary and prop. 5 of X, p. 10, there is an identity between flat modules of finite presentation and projective modules of finite type. On the other hand, there exist flat modules of finite type which are not of finite presentation, hence which are not projective (cf. X, p. 170, exercise 17, see however X, p. 169, exercises 13 and 14).