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On Diagrams of Vector Spaces

Published online by Cambridge University Press:  09 April 2009

Sheila Brenner  and
M. C. R. Butler
Sheila Brenner
Affiliation:
Department of Mathematics, Monash University
M. C. R. Butler
Affiliation:
Department of Mathematics, University of Liverpool
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We record here two further remarks about the systems, studied in [1] and [2], consisting of a vector space U and a set K of subspaces of U. In § 1, we show that such a system may be viewed as a module over a suitable artinian ring; the results of [1] and [2] thus serve to illustrate the complexity of structure of these modules. The main idea, a little wider than one introduced by Mitchell in Chapter IX of [3], is to view a diagram of vector spaces, with a small category as the scheme of the diagram, as a module over the ‘category ring’ of the category.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 3-4 , May 1969 , pp. 445 - 448
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Brenner, Sheila, ‘Endomorphism algebras of vector spaces with distinguished sets of subspaces’, J. Algebra, 6, (1967) 100–114.CrossRefGoogle Scholar
[2]Corner, A. L. S., ‘Endomorphism algebras of large modules with distinguished sub-modules’ J. Algebra (to appear).Google Scholar
[3]Mitchell, B., Theory of Categories (Academic Press, 1965).Google Scholar