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Complete Decomposability in the Exterior Algebra of a Free Module

Published online by Cambridge University Press:  20 November 2018

M. Boratynski ,
E. D. Davis  and
A. V. Geramita
M. Boratynski
Affiliation:
Polish Academy of Sciences, ul. Sniadeckich, Warsaw, Poland
E. D. Davis
Affiliation:
State University of New York at Albany, Albany, New York
A. V. Geramita
Affiliation:
Queen's University, Kingston, Ontario
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Recall the classical criterion for the complete decomposability of exterior vectors: the completely decomposable vectors in ∧pRn, R a field, are precisely the “Plücker vectors,” i.e. those whose coordinates (relative to the standard bases for ∧pRn) satisfy the Plücker equations. For R an arbitrary commutative ring, completely decomposable exterior vectors are still Plücker vectors, but the converse is not generally true. Rings for which the converse is true (for all 1 ≤ p ≤ n) are called Towber rings. Noetherian Towber rings are regular and, in fact, are characterized by the property that every Plücker vector in ∧2R4 is completely decomposable. (See [10] for these two results as well as for the above mentioned facts.) The present note develops a new characterization of Towber rings, combining it with results of Kleiner [9] and Estes-Matijevic [5] in (1) below.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 27 - 33
Copyright
Copyright © Canadian Mathematical Society 1980

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