André-Quillen Homology and complete intersection dimensions (Q6566451)

The paper studies the relation between the vanishing of André-Quillen homology and complete intersection dimensions. In particular, a characterization of algebra retracts of finite André-Quillen dimension with respect to complete intersection dimensions and complexity. Let \((R,m,k)\) and \((S,n,\ell)\) be local rings and \(\varphi:R\to S\) be a local ring homomorphism. Then the \(n\)-th homology module of \(\mathcal{L}_{\varphi}\otimes N\) is the \(n\)-th André-Quillen homology, where \(\mathcal{L}_{\varphi}\) denotes the cotangent complex of \(\varphi\) is caled the \textit{André-Quillen} homology and is denoted by \(D_n(S\mid R, N)\). The vanishing of André-Quillen homology characterizes important classes of rings and ring homomorphisms and one can define the following André-Quillen dimension\N\[\N\mathrm{AQ}-\dim_R S=\sup\{n\in\mathbb{N}\mid D_n(S\mid R, N)\neq 0\}\N\]\NIn the paper, the relation between André-Quillen dimension and other dimensions such as complete intersection dimension (CI). For example it is proven that \(\mathrm{AQ}-\dim_S T<\infty\) if and only if \(\mathrm{CI}-\dim_S T<\infty\), where \(\sigma: S\to T\) is a local ring homomorphism. Furthermore it is shown that for \(\varphi:R\to S\), where \(S\) is a complete local ring and \(\varphi\) is essentially of finite type and \(\mathrm{CI-fd}_R S<\infty\), then \(\mathrm{AQ}-\dim_R S\leq 2\) if and only if \(H_1(K_.(\varphi))\) is a free \(S\)-module. This extends some results of \textit{J. J. M. Soto} [J. Pure Appl. Algebra 146, No. 2, 197--207 (2000; Zbl 0963.13011)], \textit{R. Takahashi} [Tokyo J. Math. 27, No. 1, 209--219 (2004; Zbl 1058.13009)] and \textit{L. L. Avramov} et al. [Pure Appl. Math. Q. 9, No. 4, 579--612 (2013; Zbl 1310.13020)].