Corollary

Suppose A commutative. For every flat A-module E, the A-modules T(E), S(E), Λ(E), T^n(E), S^n(E), Λ^n(E) are flat.

Indeed, E is the inductive limit of a filtered system (L_j) of free A-modules of finite type, hence T(E) (resp. S(E), etc.) is the inductive limit of the filtered system of free A-modules T(L_j) (resp. S(L_j), etc.), hence is flat (cf. III, p. 61, prop. 6, p. 62, th. 1, p. 73, prop. 8, p. 75, th. 1, p. 83, prop. 9, and p. 86, th. 1).