Complete intersection augmented algebras (Q1923268)

In his work in Ann. Math., II. Ser. 141, No. 3, 443-551 (1995; Zbl 0823.11029), \textit{A. Wiles} used a certain criterion in order to prove that a particular Hecke algebra is a one-dimensional complete intersection. This follows by counting the length of the co-normal module of an augmentation ideal that is torsion. The main aim of the paper under review is to give a higher-dimensional generalization of this criterion. To this end let \((R,{\mathfrak m})\) be a local Gorenstein ring and \(A\) an augmented \(R\)-algebra with augmentation ideal \(N_A:= \ker \pi_A\), where \(\pi_A:A\to R\) denotes the augmentation map. Suppose that \(\dim A/{\mathfrak m} A=0\) and that \(N_A/N^2_A\) is torsion as \(R\)-module. Then \(A\) is locally a complete intersection if and only if \(A_{M_A}\), \(M_A={\mathfrak m} +N_A\), is a Gorenstein ring and the congruence ideal \(\eta_A: =\pi_A(0:N_A)\) is a principal ideal with \(F_0(N_A/N^2_A) =\eta_A\), where \(F_0\) denotes the 0-th Fitting ideal. In the particular case of dim \(R=1\) there are a few corollaries, including A. Wiles' result. The main technical tools for the author's proof are some basic facts about linkage and the André-Quillen homology group for showing that a certain ideal is syzygetic.