Finite generation of Hochschild homology algebras
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 \(k\).
scientific article; zbMATH DE number 1443594