A Note on “Modules, Comodules, and Cotensor Products over Frobenius Algebras”*

Abstract

This is a note on Abrams' paper "Modules, Comodules, and Cotensor Products over Frobenius Algebras, Journal of Algebras" (1999). With the application of Frobenius coordinates developed recently by Kadison, one has a direct proof of Abrams' characterization for Frobenius algebras in terms of comultiplication (see L. Kadison (1999)). For any Frobenius algebra, by using the explicit comultiplication, the explicit correspondence between the category of modules and the category of comodules is obtained. Moreover, with this we give very simplified proofs and improve Abrams' results on the Hom functor description of cotensor functor.

Keywords

Frobenius coordinates / Cotensor / Hochschild cohomology / 17A60 / 18G15

References

[1] Abrams J. Knot Theory and Its Ramifications, 1996, 5: 569

[2] Abrams J. of Algebras, 1999, 291: 201

[3] Abrams Trans. Amer. Math. Soc., 2002, 354: 2173

[4] Curtis, C. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, Interscience Publishers, New York, 1962.

[5] Kadison, L., Frobenius Extensions, University Lecture Series, Vol. 14, A. M. S., Providence, RI, 1999.

[6] Kadison Beiträge Algebra Geom., 2001, 42: 359

[7] Kadison J. Pure Appl. Algebra, 2002, 176: 127

[8] Kock, J., Frobenius Algebras and 2D Topological Quantum Field Theories, Available at http://math1. unice.fr/kock/TQFT.html.

[9] Montgomery, S., Hopf Algebras and Their Actions on Rings, CBMS Lectures, Vol. 82, A. M. S., Providence, RI, 1993.

[10] Wang Tsukuba J. Math., 2004, 28: 215