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On Projective Modules and Automorphisms of Central Separable Algebras

Published online by Cambridge University Press:  20 November 2018

L. N. Childs
L. N. Childs*
Affiliation:
Northwestern University, Evanston, Illinois
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This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).

In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).

Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 44 - 53
Copyright
Copyright © Canadian Mathematical Society 1969

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