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

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- by Edward L. Green, Nicole Snashall and Øyvind Solberg PDF
- Proc. Amer. Math. Soc.
**131**(2003), 3387-3393 Request permission

## 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

- Hideto Asashiba,
*The derived equivalence classification of representation-finite selfinjective algebras*, J. Algebra**214**(1999), no. 1, 182–221. MR**1684880**, DOI 10.1006/jabr.1998.7706 - Maurice Auslander, Idun Reiten, and Sverre O. Smalø,
*Representation theory of Artin algebras*, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR**1314422**, DOI 10.1017/CBO9780511623608 - D. J. Benson,
*Representations and cohomology. II*, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR**1156302** - Brenner, S. and Butler, M.C.R.,
*Almost periodic algebras and pivoted bimodules: resolutions and Yoneda algebras*, preprint 2000. - Karin Erdmann and Thorsten Holm,
*Twisted bimodules and Hochschild cohomology for self-injective algebras of class $A_n$*, Forum Math.**11**(1999), no. 2, 177–201. MR**1680594**, DOI 10.1515/form.1999.002 - 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. - Karin Erdmann and Nicole Snashall,
*On Hochschild cohomology of preprojective algebras. I, II*, J. Algebra**205**(1998), no. 2, 391–412, 413–434. MR**1632808**, DOI 10.1006/jabr.1998.7547 - Karin Erdmann and Nicole Snashall,
*Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology*, Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 183–193. MR**1648626** - 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. MR**1035222**, DOI 10.1007/BFb0084073 - Membrillo-Hernández, F.H.,
*Homological properties of finite-dimensional algebras*, D.Phil. Thesis, University of Oxford (1993). - Christine Riedtmann,
*Representation-finite self-injective algebras of class $A_{n}$*, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 449–520. MR**607169** - Eberhard Scherzler and Josef Waschbüsch,
*A class of self-injective algebras of finite representation type*, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 545–572. MR**607171**

## Additional Information

**Edward L. Green**- Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061–0123
- MR Author ID: 76495
- ORCID: 0000-0003-0281-3489
- 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
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 1990627