Flat exterior Tor algebras and cotangent complexes (Q1908871)

Let \(A\) be a ring, \(B\) and \(C\) commutative \(A\)-algebras with unit and \(D = B \otimes_AC\). The author relates the vanishing of the André homology modules \(H_n (A,B,C,W)\) with properties of \(\text{Tor}^A (B,C)\). More precisely he proves: Theorem. The following conditions are equivalent: (1) \(H_n (A,B,C,-) = 0\) for \(n \geq 3\). (2) The \(D\)-module \(\text{Tor}_1^A (B,C)\) is flat and \(\gamma : \bigwedge_D \text{Tor}^A_1 (B,C) \to \text{Tor}^A (B,C)\), the canonical homomorphism, is an isomorphism.