by Fottner, Hubert

It is due to Thévenaz that a large part of Puig”s theory of pointed groups carries over to the context of Green functors for finite groups, where here maximal ideals play the role of points in the G-algebra situation. The objective of this paper is to generalize the results further to a situation where one replaces maximal ideals by prime ideals. Moreover, we show that also Puig”s version of Sylow”s first theorem for local pointed groups can be extended to this situation, and we demonstrate that Dress”s induction theorem is a consequence of this result.

DOI: 10.1023/A:1009977503773
Print publication date: 12/1/1999
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by Guil-Asensio, Francisco; Saorín, Manuel

The primary goal of this work is to develop a strategy to compute Pic_K(A) and Out_K(A) where A is a finite-dimensional algebra over a field K. The basic idea is to put the normal subgroup Inn^*(A) of the inner automorphisms of A induced by elements of 1 + J(A), as a common denominator in the ‘fraction’ Aut_K(A) / Inn(A). The new numerator Aut_K(A)/Inn^*(A) and the new denominator Inn(A)/Inn^*(A) are much easier to deal with than the original Aut_K(A) and Inn(A). The main ingredient to study Out(A) is now the appearance of an Abelian group Ch(Γ, K), the group of acyclic characters of the quiver Γ of A, that can be completely calculated. We show how to apply these results to compute the Picard group of a split finite-dimensional algebra in several cases.

DOI: 10.1023/A:1009973319703
Print publication date: 12/1/1999
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by Kadison, Lars

This paper begins with an introduction to β-Frobenius structure on a finite-dimensional Hopf subalgebra pair. In Section 2 a study is made of a generalization of Frobenius bimodules and β-Frobenius extensions. Also a special type of twisted Frobenius bimodule which gives an endomorphism ring theorem and converse is studied. Section 3 brings together material on separable bimodules, the dual definitions of split and separable extension, and a theorem of Sugano on endomorphism rings of separable bimodules. In Section 4, separable twisted Frobenius bimodules are characterized in terms of data that generalizes a Frobenius homomorphism and a dual base. In the style of duality, two corollaries characterizing split β-Frobenius and separable β-Frobenius extensions are proven. Sugano”s theorem is extended to β-Frobenius extensions and their endomorphism rings. In Section 5, the problem of when separable extensions are Frobenius extensions is discussed. A Hopf algebra example and a matrix example are given of finite rank free separable β-Frobenius extensions which are not Frobenius in the ordinary sense.

DOI: 10.1023/A:1009974918794
Print publication date: 12/1/1999
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DOI: 10.1023/A:1017129929255
Print publication date: 12/1/1999
View article on SpringerLink