The topological and algebraic Picard groups
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Date
2024-06
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University of New Brunswick
Abstract
The Picard group of a compact Hausdorff space is the group of isomorphism classes of line bundles over that space. The Picard group of an algebra is the group of isomorphism classes of line modules over that algebra. In this thesis, we show that in the case of C(X) the algebra of continuous complex-valued functions over a compact Hausdorff space X, isomorphism classes of balanced line modules over C(X) are in bijection with isomorphism classes of line bundles over X, showing the relationship between the two types of Picard groups. In the second part of this thesis, we prove that, in general, the Picard group of a finite-dimensional semisimple complex algebra is isomorphic to the symmetric group on the number of components of the algebra’s Wedderburn decomposition.