On the free character of the first Koszul homology module (Q1196809)

Let \((A,M,K)\) denote a local noetherian ring \(A\) with maximal ideal \(M\) and residue field \(K\). Let \(I\) be an ideal of \(A\) and \(E\) the Koszul complex associated to an arbitrary system of generators of \(I\). It is shown that if the first Koszul homology module of \(E\) is a free \(A/I\)- module, then the homomorphism \(H_ 3(A/I,K,K)\to H_ 2(A,A/I,K)\) is trivial. This result enables the author to extend results of Gulliksen and of André.