Avramov, Luchezar L.; Iyengar, Srikanth Finite generation of Hochschild homology algebras. (English) Zbl 0966.13009 Invent. Math. 140, No. 1, 143-170 (2000). The authors, in this paper, attempt to prove the converse of the following classical result due to Hochschild, Kostant and Rosenberg: Let \(S\) be a commutative algebra over a commutative Noetherian ring \(k\). If \(S\) is smooth over \(k\), then the \(S\)-module \(\Omega^1_{S/k}\) of Kähler differentials is projective and the canonical homomorphism \(W^*_{S/k}: \wedge^*_S \Omega^1_{S/k} \to\text{HH}_*(S/k)\) of graded algebras is bijective. The authors prove the following converses of the theorem quoted above.(i) If \(S\) is a flat \(k\)-algebra essentially of finite type over \(k\) and the \(S\)-algebra HH\(_*(S/k)\) is finitely generated, then \(S\) is smooth over \(k\).(ii) If \(S\) is a flat \(k\)-algebra essentially of finite type over \(k\) and if HH\(_{2i-1} (S/k)= 0=\text{HH}_{2j}(S/k)\) for some positive integers \(i,j\), then \(S\) is smooth over \(k\). Reviewer: V.A.Hiremath (Dharwad) Cited in 2 ReviewsCited in 4 Documents MSC: 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 13N05 Modules of differentials 13C40 Linkage, complete intersections and determinantal ideals 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) Keywords:\(DG\)-algebra; Hochschild homology; module of Kähler differentials; smoothness of flat algebra PDFBibTeX XMLCite \textit{L. L. Avramov} and \textit{S. Iyengar}, Invent. Math. 140, No. 1, 143--170 (2000; Zbl 0966.13009) Full Text: DOI