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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