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Finite generation of Hochschild homology algebras. (English) Zbl 0966.13009

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 \(k\).

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)
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