On the vanishing of the homology of the exterior powers of the cotangent complex. (English) Zbl 0820.13013
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 M. André, “Homologie des algèbres commutatives” (Berlin 1974; Zbl 0284.18009) or 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 \(\bigwedge^ n_ B \Omega_{B | A} \neq 0\).
(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 M. André, “Homologie des algèbres commutatives” (Berlin 1974; Zbl 0284.18009) or 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 \(\bigwedge^ n_ B \Omega_{B | A} \neq 0\).
Reviewer: J.Majadas (Vigo)
MSC:
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
| 13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |
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