Finite complete intersection dimension and vanishing of André-Quillen homology (Q1969501)

Let \(A\) be a noetherian local ring with residue field \(k\), \(I\) an ideal of \(A\), and \(B=A/I\). Denote the corresponding André-Quillen homology functors by \(H_n(A,B,-)\), and let \(E\) be the Koszul complex associated to a set of generators of \(I\). The notion of the complete intersection dimension \(\text{CI-} \dim_A(M)\) of a finite \(A\)-module \(M\) was defined and studied by \textit{L. L. Avramov}, \textit{V. N. Gasharov} and \textit{I. V. Peeva} [Publ. Math., Inst. Hautes Étud. Sci. 86, 67-114 (1997; Zbl 0918.13008)]. It is always finite iff \(A\) is a complete intersection, coincides with \(\text{pd}_AM\) if \(\text{pd}_AM<\infty\), and replaces \(\text{pd}_AM\) in the Auslander-Buchsbaum formula if finite and \(\text{pd}_A M=\infty\). -- The author considers the following conditions: (i) \(H_n(A,B,-)=0\) for all \(n\geq 3\). (ii) \(\text{CI-} \dim_A(B) <\infty\) and \(H_1(E)\) is free \(B\)-module. (iii) The sequence \((\dim_k\text{Tor}^A_n(B,k))_{n\in N}\) has polynomial growth and \(H_1(E)\) is a free \(B\)-module. The author shows that (i) follows from (ii) and implies (iii), and conjectures that (i) and (ii) are equivalent. He also proves that \(\text{CI-}\dim_A(B) <\infty\) implies (i) if \(B\) contains a field of characteristic zero and has a minimal free DG resolution or if \(I\) is generated by a part of a system of parameters of \(A\). In addition to this, he gives several conditions on \(A\) for (i), (ii), and (iii) to be equivalent. Finally, he shows that (ii) implies the formula \(\text{CI-} \dim_A(B)= \mu(I)-\dim_kH_2 (A,B,k)\), where \(\mu(I)\) is the minimal number of generators of \(I\).