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Proceedings of the American Mathematical Society
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
Published electronically: February 24, 2003
MathSciNet review: 1990627
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-03-06912-0
PII: S 0002-9939(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