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.


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)
Full Text: DOI arXiv


[1] Luchezar L. Avramov and Micheline Vigué-Poirrier, Hochschild homology criteria for smoothness, Internat. Math. Res. Notices 1 (1992), 17 – 25. · Zbl 0755.13006 · doi:10.1155/S1073792892000035
[2] Vladimir Bavula, Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120 (1996), no. 3, 293 – 335. · Zbl 0855.16010
[3] Petter Andreas Bergh and Karin Erdmann, Homology and cohomology of quantum complete intersections, Algebra Number Theory 2 (2008), no. 5, 501 – 522. · Zbl 1205.16011 · doi:10.2140/ant.2008.2.501
[4] Petter Andreas Bergh and Dag Madsen, Hochschild homology and global dimension, Bull. Lond. Math. Soc. 41 (2009), no. 3, 473 – 482. · Zbl 1207.16006 · doi:10.1112/blms/bdp018
[5] Ragnar-Olaf Buchweitz, Edward L. Green, Dag Madsen, and Øyvind Solberg, Finite Hochschild cohomology without finite global dimension, Math. Res. Lett. 12 (2005), no. 5-6, 805 – 816. · Zbl 1138.16003 · doi:10.4310/MRL.2005.v12.n6.a2
[6] Buenos Aires Cyclic Homology Group, Cyclic homology of algebras with one generator, \?-Theory 5 (1991), no. 1, 51 – 69. Jorge A. Guccione, Juan José Guccione, María Julia Redondo, Andrea Solotar and Orlando E. Villamayor participated in this research. · Zbl 0743.13008 · doi:10.1007/BF00538879
[7] Yves Félix, Steve Halperin, and Jean-Claude Thomas, Differential graded algebras in topology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 829 – 865. · Zbl 0868.55016 · doi:10.1016/B978-044481779-2/50017-1
[8] Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. · Zbl 0961.55002
[9] Yves Felix, Jean-Claude Thomas, and Micheline Vigué-Poirrier, The Hochschild cohomology of a closed manifold, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 235 – 252. · Zbl 1060.57019 · doi:10.1007/s10240-004-0021-y
[10] Stephen Halperin and Micheline Vigué-Poirrier, The homology of a free loop space, Pacific J. Math. 147 (1991), no. 2, 311 – 324. · Zbl 0666.55011
[11] Yang Han, Hochschild (co)homology dimension, J. London Math. Soc. (2) 73 (2006), no. 3, 657 – 668. · Zbl 1139.16010 · doi:10.1112/S002461070602299X
[12] Dieter Happel, Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108 – 126. · doi:10.1007/BFb0084073
[13] Jean-Louis Loday, Cyclic homology, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998. Appendix E by María O. Ronco; Chapter 13 by the author in collaboration with Teimuraz Pirashvili. · Zbl 0885.18007
[14] Saunders MacLane, Homology, 1st ed., Springer-Verlag, Berlin-New York, 1967. Die Grundlehren der mathematischen Wissenschaften, Band 114. · Zbl 0059.16405
[15] Jean-Pierre Serre, Algèbre locale. Multiplicités, Cours au Collège de France, 1957 – 1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). · Zbl 0142.28603
[16] Andréa Solotar, Cyclic homology of a free loop space, Comm. Algebra 21 (1993), no. 2, 575 – 582. · Zbl 0772.55006 · doi:10.1080/00927879308824580
[17] Micheline Vigué-Poirrier, Homologie de Hochschild et homologie cyclique des algèbres différentielles graduées, Astérisque 191 (1990), 7, 255 – 267 (French). International Conference on Homotopy Theory (Marseille-Luminy, 1988). · Zbl 0728.19003
[18] Micheline Vigué-Poirrier, Critères de nullité pour l’homologie des algèbres graduées, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 7, 647 – 649 (French, with English and French summaries). · Zbl 0788.13007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.