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Gaps in Hochschild cohomology imply smoothness for commutative algebras. (English) Zbl 1101.13018
Let \(K\) be a commutative noetherian ring, \(S\) a commutative \(K\)-algebra essentially of finite type, flat as a \(K\)-module and \(\Omega_{S/K}\) the \(S\)-module of \(K\)-linear Kähler differentials of \(S\).
The authors show that for a prime ideal \(\mathfrak q\) in \(S\) and a finite \(S\)-module \(M\) with \(M_{\mathfrak q}\not=0\) the following conditions are equivalent: i) The \(K\)-algebra \(S\) is smooth. ii) Each map \(\lambda_n^M:(\bigwedge_S^n\Omega_{S/K}\otimes_S M\to HH_{n}(S/K,M)\) localized in \(\mathfrak q\) is bijective, where \(HH_{*}(S/K,M)\) denotes the Hochschild homology of the \(K\) algebra \(S\) with coefficients in \(M\), and the \(S_{\mathfrak q}\)-module \(\Omega_{S_{\mathfrak q}/K}\) is projective. iii) There exist non-negative integers \(t\) and \(s\) of different parity satisfying \(HH_t(S/K;M)_{\mathfrak q}=0=HH_u(S/K;M)_{\mathfrak q}\). When the \(K\)-module \(S\) is projective they are also equivalent to: ii’) Each map \(\lambda_M^n:HH^n(S/K;M)\to \text{Hom}_S(\bigwedge_S^n\Omega_{S/K}, M)\), where \(HH^{*}(S/K;M)\) denotes the Hochschild cohomology of the \(K\) algebra \(S\) with coefficients in \(M\), localized in \(\mathfrak q\) is bijective. iii’) There exist non-negative integers \(t\) and \(s\) of different parity satisfying \(HH^{t+i}(S/K;M)_{\mathfrak q}=0=HH^{u+i}(S/K;M)_{\mathfrak q}\) for \(0\leq i\leq \dim_{S_{\mathfrak q}}M_{\mathfrak q}\).
This result incorporates G. Hochschild, B. Kostant and A. Rosenberg’s theorem [Trans. Am. Math. Soc. 102, 383–408 (1962; Zbl 0102.27701)] and several other known results relating vanishing of Hochschild (co)homology and smoothness. The use of cohomology and the introduction of coefficients are two new aspects of this paper.

MSC:
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
14M10 Complete intersections
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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