## 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
- Proc. Amer. Math. Soc.
**131**(2003), 3387-3393 - DOI: https://doi.org/10.1090/S0002-9939-03-06912-0
- Published electronically: February 24, 2003
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## 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

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## Bibliographic 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