... Andr´e–Quillen homology modules in dimen-sions 3 and 4, the vanishing of which characterizes complete intersec-tions.Our proofs of these two results follow Andr´e and Avramov’s arguments[An1], [Av1, Av2] ... A-algebra. By (1.4.6) and (1.4.1)we have H2(A, B, M)=H2(A,B,M) and an exact sequence0 → H1(A, B, M) → H1(A,B,M) → H0(A, A,M) and the same isomorphism and exact sequence with ... thatby Andr´e, making essential use of Andr´e–Quillen homology modules inall dimensions.Theorem (V) If f :(A, m,K) → (B,n,L) is a local formally smoothhomomorphism of noetherian local rings and...