André-Quillen homology in degree five
The author studies the homology module \(H_5 (A,B,C,B \otimes C)\) for commutative \(A\)-algebras \(B\) and \(C\). As a consequence he obtains some new results in the theory of cotangent complex. In section 1, series of definitions are given and natural homomorphisms \(\varphi (Y)\), \(\psi (Y)\), \(\alpha (Y)\) and \(\beta (Y)\) are defined for a differential \(\Gamma\)-algebra \(Y\). In section 2, the main theorem, theorem 2.1, is stated whose proof is given in sections 3 and 4. The theorem connects the homology modules \(H_4 (A,B,C,D)\), \(H_5 (A,B,C,D)\) and the torsion module \(\text{Tor}^A_3 (B,C)\) where \(D = B \otimes C\). The following results in the theory of cotangent complex are deduced: (i) If \(I\) is an ideal of \(C\) and \(\tau = \{\tau_m\}\) is a system of generators of \(I\), then there is a connection between \(H_5 (C,C/I,C/I)\) and the Koszul homology module \(H_3 (\tau)\). -- In particular if \(I\) is a principal ideal generated by \(\tau\), then \(H_5 (C,C/I,C/I)\) is expressible in terms of the annihilator of \(\tau\). (ii) If \(I\) is an almost complete intersection ideal in a local ring \(C\), then \(H_5 (C,C/I,C/I)\) is \(C/I\)-isomorphic to \(F(H_1)\) where \(H_1\) is the first Koszul homology module of a minimal set of generators of \(I\).
scientific article; zbMATH DE number 612147