Hochschild cohomology of Frobenius algebras
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- by Jorge A. Guccione and Juan J. Guccione
- Proc. Amer. Math. Soc. 132 (2004), 1241-1250
- DOI: https://doi.org/10.1090/S0002-9939-03-07350-7
- Published electronically: December 22, 2003
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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
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Bibliographic 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
- Received by editor(s): November 6, 2002
- Published electronically: December 22, 2003
- Additional Notes: Supported by UBACYT X193 and CONICET
- Communicated by: Martin Lorenz
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1241-1250
- MSC (2000): Primary 16C40; Secondary 16D20
- DOI: https://doi.org/10.1090/S0002-9939-03-07350-7
- MathSciNet review: 2053327