Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Hochschild cohomology ring of a selfinjective algebra of finite representation type


Authors: Edward L. Green, Nicole Snashall and Øyvind Solberg
Journal: Proc. Amer. Math. Soc. 131 (2003), 3387-3393
MSC (2000): Primary 16E40, 16G10, 16G60
DOI: https://doi.org/10.1090/S0002-9939-03-06912-0
Published electronically: February 24, 2003
MathSciNet review: 1990627
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper describes the Hochschild cohomology ring of a selfinjective algebra $\Lambda$ of finite representation type over an algebraically closed field $K$, showing that the quotient $\operatorname{HH}^*(\Lambda)/\mathcal{N}$ of the Hochschild cohomology ring by the ideal ${\mathcal N}$ generated by all homogeneous nilpotent elements is isomorphic to either $K$ or $K[x]$, and is thus finitely generated as an algebra. We also consider more generally the property of a finite dimensional algebra being selfinjective, and as a consequence show that if all simple $\Lambda$-modules are $\Omega$-periodic, then $\Lambda$ is selfinjective.


References [Enhancements On Off] (What's this?)

  • 1. Asashiba, H., The derived equivalence classification of representation-finite self-injective algebras, J. Algebra 214 (1999), 182-221. MR 2000g:16019
  • 2. Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995. MR 96c:16015
  • 3. Benson, D., Representation theory and cohomology II: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics 31, CUP, 1991. MR 93g:20099
  • 4. Brenner, S. and Butler, M.C.R., Almost periodic algebras and pivoted bimodules: resolutions and Yoneda algebras, preprint 2000.
  • 5. Erdmann, K. and Holm, T., Twisted bimodules and Hochschild cohomology for self-injective algebras of type $A_n$, Forum Math. 11 (1999), 177-201. MR 2001c:16018
  • 6. Erdmann, K., Holm, T. and Snashall, N., Twisted bimodules and Hochschild cohomology for self-injective algebras of type $A_n$ II, Algebras and Representation Theory 5 (2002), 457-482.
  • 7. Erdmann, K. and Snashall, N., On the Hochschild cohomology of preprojective algebras I, II, J. Algebra 205 (1998), 391-412, 413-434. MR 99e:16013
  • 8. Erdmann, K. and Snashall, N., Preprojective algebras of Dynkin type: periodicity and the second Hochschild cohomology, Canad. Math. Soc. Conference Proceedings 24 (1998), 183-193. MR 99h:16016
  • 9. Happel, D., Hochschild cohomology of finite-dimensional algebras, Springer Lecture Notes in Mathematics 1404 (1989), 108-126. MR 91b:16012
  • 10. Membrillo-Hernández, F.H., Homological properties of finite-dimensional algebras, D.Phil. Thesis, University of Oxford (1993).
  • 11. Riedtmann, C., Representation-finite self-injective algebras of class $A_n$, In: Representation Theory II, Proc. Second Internat. Conf., Carleton Univ., Ottawa, Springer Lecture Notes in Mathematics 832 (1979), 449-520. MR 82k:16040
  • 12. Scherzler, E. and Waschbüsch, J., A class of self-injective algebras of finite representation type, In: Representation Theory II, Proc. Second Internat. Conf., Carleton Univ., Ottawa, Springer Lecture Notes in Mathematics 832 (1979), 545-572. MR 82i:16034

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16E40, 16G10, 16G60

Retrieve articles in all journals with MSC (2000): 16E40, 16G10, 16G60


Additional Information

Edward L. Green
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061–0123
Email: green@math.vt.edu

Nicole Snashall
Affiliation: Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, England
Email: N.Snashall@mcs.le.ac.uk

Øyvind Solberg
Affiliation: Institutt for matematiske fag, NTNU, N–7491 Trondheim, Norway
Email: oyvinso@math.ntnu.no

DOI: https://doi.org/10.1090/S0002-9939-03-06912-0
Received by editor(s): December 5, 2001
Received by editor(s) in revised form: June 17, 2002
Published electronically: February 24, 2003
Communicated by: Martin Lorenz
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society