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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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 $\mathsf {HH}^*(A) = \mathsf {HH}_0^*(A) = \mathsf {HH}^*(A_0)^{\mathbb {Z}/m \mathbb {Z}}$.
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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