Algebra & Number Theory

Homology and cohomology of quantum complete intersections

Petter Bergh and Karin Erdmann

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Abstract

We construct a minimal projective bimodule resolution for every finite-dimensional quantum complete intersection of codimension two. Then we use this resolution to compute both the Hochschild cohomology and homology for such an algebra. In particular, we show that the cohomology vanishes in high degrees, while the homology is always nonzero.

Article information

Source
Algebra Number Theory, Volume 2, Number 5 (2008), 501-522.

Dates
Received: 15 October 2007
Revised: 29 May 2008
Accepted: 7 June 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513797286

Digital Object Identifier
doi:10.2140/ant.2008.2.501

Mathematical Reviews number (MathSciNet)
MR2429451

Zentralblatt MATH identifier
1205.16011

Subjects
Primary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Secondary: 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37] 16U80: Generalizations of commutativity 16S80: Deformations of rings [See also 13D10, 14D15]

Keywords
quantum complete intersection Hochschild cohomology Hochschild homology

Citation

Bergh, Petter; Erdmann, Karin. Homology and cohomology of quantum complete intersections. Algebra Number Theory 2 (2008), no. 5, 501--522. doi:10.2140/ant.2008.2.501. https://projecteuclid.org/euclid.ant/1513797286


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References

  • L. L. Avramov, V. N. Gasharov, and I. V. Peeva, “Complete intersection dimension”, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67–114 (1998).
  • D. J. Benson, K. Erdmann, and M. Holloway, “Rank varieties for a class of finite-dimensional local algebras”, J. Pure Appl. Algebra 211:2 (2007), 497–510.
  • R.-O. Buchweitz, E. L. Green, D. Madsen, and Ø. Solberg, “Finite Hochschild cohomology without finite global dimension”, Math. Res. Lett. 12:5-6 (2005), 805–816.
  • H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, Princeton, NJ, 1956.
  • K. Erdmann, M. Holloway, R. Taillefer, N. Snashall, and Ø. Solberg, “Support varieties for selfinjective algebras”, $K$-Theory 33:1 (2004), 67–87. http://www.emis.de/cgi-bin/MATH-item?1116.16007Zbl 1116.16007
  • E. L. Green and N. Snashall, “Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group”, Forum Math. 16:1 (2004), 17–36.
  • Y. Han, “Hochschild (co)homology dimension”, J. London Math. Soc. $(2)$ 73:3 (2006), 657–668.
  • D. Happel, “Hochschild cohomology of finite-dimensional algebras”, pp. 108–126 in Séminaire d'Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988), edited by M.-P. Malliavin, Lecture Notes in Math. 1404, Springer, Berlin, 1989.
  • T. Holm, “Hochschild cohomology rings of algebras $k[X]/(f)$”, Beiträge Algebra Geom. 41:1 (2000), 291–301.
  • Y. I. Manin, “Some remarks on Koszul algebras and quantum groups”, Ann. Inst. Fourier $($Grenoble$)$ 37:4 (1987), 191–205.
  • N. Snashall and Ø. Solberg, “Support varieties and Hochschild cohomology rings”, Proc. London Math. Soc. $(3)$ 88:3 (2004), 705–732. http://www.emis.de/cgi-bin/MATH-item?1067.16010Zbl 1067.16010