Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Hochschild cohomology of Frobenius algebras


Authors: Jorge A. Guccione and Juan J. Guccione
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1241-1250
MSC (2000): Primary 16C40; Secondary 16D20
Published electronically: December 22, 2003
MathSciNet review: 2053327
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $k$ be a field, $A$ a finite-dimensional Frobenius $k$-algebra and $\rho\colon A\to A$, the Nakayama automorphism of $A$ with respect to a Frobenius homomorphism $\varphi\colon A\to k$. Assume that $\rho$has finite order $m$ and that $k$ has a primitive $m$-th root of unity $w$. Consider the decomposition $A = A_0\oplus \cdots\oplus A_{m-1}$ of $A$, obtained by defining $A_i = \{a\in A:\rho(a) = w^i a\}$, and the decomposition $\mathsf{HH}^*(A) = \bigoplus_{i=0}^{m-1} \mathsf{HH}_i^*(A)$ of the Hochschild cohomology of $A$, obtained from the decomposition of $A$. In this paper we prove that $\mathsf{HH}^*(A) = \mathsf{HH}^*_0(A)$ and that if the decomposition of $A$ is strongly $\mathbb{Z}/m\mathbb{Z}$-graded, then $\mathbb{Z}/m\mathbb{Z}$ acts on $\mathsf{HH}^*(A_0)$ and $\mathsf{HH}^*(A) = \mathsf{HH}_0^*(A) = \mathsf{HH}^*(A_0)^{\mathbb{Z}/m \mathbb{Z}}$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16C40, 16D20

Retrieve articles in all journals with MSC (2000): 16C40, 16D20


Additional Information

Jorge A. Guccione
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Pabellón 1 - Ciudad Universitaria, (1428) Buenos Aires, Argentina
Email: vander@dm.uba.ar

Juan J. Guccione
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Pabellón 1 - Ciudad Universitaria, (1428) Buenos Aires, Argentina
Email: jjgucci@dm.uba.ar

DOI: https://doi.org/10.1090/S0002-9939-03-07350-7
Received by editor(s): November 6, 2002
Published electronically: December 22, 2003
Additional Notes: Supported by UBACYT X193 and CONICET
Communicated by: Martin Lorenz
Article copyright: © Copyright 2003 American Mathematical Society