Inducing Subgroups on the Brauer and Picard Groups of a Coalgebra via Strong Equivalences
by
Juan Cuadra
Universidad de Almería

In this talk we study strong equivalences for two coalgebras C and D. These are Morita equivalences between C^*-Mod and D^*-Mod which induce, by restriction, equivalences between the categories of comodules M^C and M^D. By considering these equivalences we get a subgroup Pic^s(C) of Pic(C), the Picard group of C. Pic^s(C) is also a subgroup of Pic(C^*), the Picard group of the dual algebra C^*. We give some examples where Pic^s(C) =/= Pic(C) and Pic^s(C) =/= Pic(C^*).

Let Br(C) be the Brauer group of a cocommutative coalgebra C as defined in [5]. Using strong equivalences instead equivalences in the definition of Br(C) one gets a subgroup Br^s(C). There is a duality morphism from Br^s(C) into Br(C^*), the Brauer group of C^*. We study this morphism solving in a positive way the questions formulated in [5, page 568].

References

[1] J. Cuadra, J.R. García Rozas, and B. Torrecillas, Picard Groups and Strongly Graded Coalgebras. To appear in J. Pure Appl. Algebra.

[2] J. Cuadra, J.R. García Rozas, B. Torrecillas, and F. Van Oystaeyen, The Brauer Group of Irreducible Coalgebras. To appear in J. Algebra, 2001.

[3] B. I-Peng Lin, Morita's Theorem for Coalgebras. Comm. in Algebra 1 No. 4 (1974), 311-344.

[4] M. Takeuchi, Morita Theorems for Categories of Comodules. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 629-644.

[5] B. Torrecillas, F. Van Oystaeyen, and Y.H. Zhang, The Brauer Group of a Cocommutative Coalgebra. J. Algebra 177 (1995), 536-568.

Date received: March 12, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cage-15.