On the acyclicity of the Tate complex
Let \(A\) be a commutative ring with unit, \(I\) an ideal generated by a set \(t\), \(B = A/I\) and \(E\) the Koszul complex associated to \(t\). Let \(s\) be a set of generators for \(H_1(E)\) as a \(B\)-module. This paper gives a proof that the second stage of the Tate complex of \(B\) over \(A\) is acyclic if and only if \(s\) is a basis of the \(B\)-module \(H_1(E)\) and the canonical map \(\wedge^* H_1(E) \to H_*(E)\) is an isomorphism. A similar theorem for a simplicial resolution of the \(A\)-algebra \(B\) is also proved. These theorems occur also in an earlier paper by the same authors [\textit{A. Blanco, J. Majadas} and \textit{A. G. Rodicio}, Invent. Math. 123, No.~1, 123-140 (1996; Zbl 0856.13009); (theorems 2 and 13)], however the proofs given there rely on results about simplicial rings which are not known to hold in complete generality. The present paper gives alternative proofs based on the spectral sequence associated to a certain bicomplex. As an application a new proof is given of a vanishing theorem for André-Quillen cohomology.
scientific article; zbMATH DE number 1308229
scientific article