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A Hochschild homology criterium for the smoothness of an algebra. (English) Zbl 0824.13009

The subject of this paper originates in a conjecture of A. G. Rodicio [Comment. Math. Helv. 65, No. 3, 474-477 (1990; Zbl 0726.13008)]: “Let \(k\) be a field of characteristic zero. If \(A\) is a finitely generated \(k\)-algebra, and \(HH_ n (A)=0\) for \(n>n_ 0\), then \(A\) is a geometrical regular \(k\)-algebra.”
The authors present here a positive answer to the conjecture as corollary of a more general result. A \(k\)-algebra \(A\) is called essentially of finite type if there exist \(x_ 1, \ldots x_ n\in A\) and a multiplicative system \(S\subset k[x_ 1, \ldots, x_ n]\) such that \(A= S^{-1} (k[ x_ 1, \ldots,x_ n])\). The authors prove that if \(A\) is essentially of finite type and \(HH_ i (A)= HH_ j (A)=0\) for some \(i\) old and \(j\) even, then \(A\) is geometrically regular.

MSC:

13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13E15 Commutative rings and modules of finite generation or presentation; number of generators

Citations:

Zbl 0726.13008
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