On the vanishing of the homology of the exterior powers of the cotangent complex
Let \(A\) be a commutative noetherian ring and \(B\) a flat commutative \(A\)- algebra essentially of finite type. The following are equivalent: (i) \(B\) is a smooth \(A\)-algebra, (ii) \(H_ p (\bigwedge^ q_ B \mathbb{L}_{B | A}) = 0\) for all \(p \geq 1\), \(q \geq 0\), (iii) There exists some \(p \geq 1\) such that \(H_ p (\bigwedge^ q_ B \mathbb{L}_{B | A}) = 0\) for all \(q \in [p + 1, p + \text{ext.rk.} (\Omega_{B | A})]\). Here \(\mathbb{L}_{B | A}\) is the cotangent complex [see \textit{M. André}, ``Homologie des algèbres commutatives'' (Berlin 1974; Zbl 0284.18009) or \textit{D. Quillen}, in: Appl. categorical algebra, Proc. Symp. Pure Math. 17, 65-87 (1970; Zbl 0234.18010)], \(\bigwedge^ q_ B \mathbb{L}_{B | A}\) is the simplicial \(B\)-module resulting from applying the \(q\)-th exterior power functor \(\bigwedge^ q_ B\) to the simplicial \(B\)-module \(\mathbb{L}_{B | A}\), and \(\text{ext.rk.} (\Omega_{B | A})\) is the maximum \(n\) such that \(\bigwedge^ n_ B \Omega_{B | A} \neq 0\).
scientific article; zbMATH DE number 769535