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COHOMOLOGY OF SPLIT ALGEBRAS AND OF TRIVIAL EXTENSIONS

Published online by Cambridge University Press:  01 May 2003

CLAUDE CIBILS ,
EDUARDO MARCOS ,
MARÍA JULIA REDONDO  and
ANDREA SOLOTAR
CLAUDE CIBILS
Affiliation:
Département de Mathématiques, Université de Montpellier 2, F-34095 Montpellier cedex 5, France e-mail: Claude.Cibils@math.univ-montp2.fr
EDUARDO MARCOS
Affiliation:
Departamento de Matemática, Universidade de São Paulo, Caixa Postal 66.281 São Paulo – SP, 05315-970, Brasil e-mail: enmarcos@ime.usp.br
MARÍA JULIA REDONDO
Affiliation:
Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca, Argentina e-mail: mredondo@criba.edu.ar
ANDREA SOLOTAR
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellón I – Ciudad Universitaria, 1428 – Buenos Aires, Argentina e-mail: asolotar@dm.uba.ar
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Abstract

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We consider associative algebras $\Lambda$ over a field provided with a direct sum decomposition of a two-sided ideal $M$ and a sub-algebra $A$ – examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of $\Lambda$. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.

Keywords

16E4016D20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 45 , Issue 1 , January 2003 , pp. 21 - 40
Copyright
2003 Glasgow Mathematical Journal Trust

Footnotes

This work has been supported by the projects SECYT-ECOS A98E05 and SECYT-CAPES BR12/99/OG. The first author wishes to thank FAPESP (Brazil) for financial support. The second author has a research scholarship from Cnpq (Brazil). The third and fourth authors are research members of CONICET (Argentina).