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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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