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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.

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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