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Two classes of algebras with infinite Hochschild homology. (English) Zbl 1227.16011

From the introduction: The main purpose of this paper is to prove that higher Hochschild homology groups do not vanish for two large classes of algebras whose global dimension is not finite, without any assumption on the ground field.
In Theorem I, the algebras we consider are generalizations of quantum complete intersections, and they are not assumed to be finite dimensional.
On the other hand, the algebras satisfying the hypotheses of Theorem II are, in some sense, opposite of quantum complete intersections, since we assume that they have two generators \(x\) and \(y\) such that \(xy=yx=0\).
Now we state both main theorems.
Theorem I. Let \(A=k\langle x_1,\dots,x_n\rangle/(f_1,\dots,f_p)\) be a finitely generated \(k\)-algebra, such that \(f_1\) belongs to \(k[x_1]\) and, for \(i\geq 2\), \(f_i\) belongs to the two-sided ideal \((x_2,\dots,x_n)\). If \(B=k[x_1]/(f_1)\) is not smooth, then the Hochschild homology groups \(HH_n(A)\) are not zero for all \(n\in\mathbb N\).
For example Theorem I is valid if \(f_1=x^2_1g_1\), with \(g_1\in k[x_1]\) and \(f_2,\dots,f_p\) satisfying the hypothesis of the theorem.
Theorem II. Let \(A=\bigoplus_{n\geq 0}A_n\) be a finite dimensional graded \(k\)-algebra with \(A_0=k\) and such that \(\overline A=\bigoplus_{n\geq 1}A_n\) is not zero. Assume that there exist two generators \(x\) and \(y\) of the algebra \(A\) verifying \(xy=yx=0\). Then the total Hochschild homology of \(A\) is not finite dimensional.
Remark 1.1. This theorem is valid for very large classes of graded local algebras since relations between the other generators play no role.
The proof of Theorem I follows without any computation from the well known result for commutative algebras.
The methods used in the proof of Theorem II rely on differential homological algebra. In fact, we will work with the cobar construction on the graded coalgebra \(\bigoplus_{n\geq 0}\operatorname{Hom}_k(A^n,k)\). We denote it \((\Omega^*A,d)\). The Hochschild homology groups of the differential graded algebra \((\Omega^*A,d)\) are dual, as vector spaces, to the Hochschild homology groups of the graded \(k\)-algebra \(A\). Since \((\Omega^*A,d)\) is a tensor algebra, a short complex is available to compute its Hochschild homology.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16E10 Homological dimension in associative algebras
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16W50 Graded rings and modules (associative rings and algebras)
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References:

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