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

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper describes the Hochschild cohomology ring of a selfinjective algebra of finite representation type over an algebraically closed field , showing that the quotient of the Hochschild cohomology ring by the ideal generated by all homogeneous nilpotent elements is isomorphic to either or , 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 -modules are -periodic, then is selfinjective.

**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*, Cambridge Studies in Advanced Mathematics*II*: Cohomology of groups and modules**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*, 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**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*, 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**

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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:
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