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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

References

1. Buchsbaum, D. A. and Eisenbud, D., What makes a complex exact?, J. of Alg. 25, No. 2 (1973), 259–268.Google Scholar
2. Davis, E. D., and Geramita, A. V., Efficient generation of maximal ideals in polynomial rings, T.A.M.S. 231 (1977), 497–505.Google Scholar
3. Eisenbud, D., and Evans, E. G., Three conjectures about modules over polynomial rings, Conference on Comm. Alg. Lecture Notes in Math. 311 (Springer-Verlag, 1973), 78–89.Google Scholar
4. Eisenbud, D., and Evans, E. G., Every algebraic set in n-space is the intersection of n hyper surfaces, Invent. Math. 19 (1973), 107–112.Google Scholar
5. Estes, D. and Matijevic, J., Matrix factorization over commutative rings, to appear, in J. Alg.Google Scholar
6. Forster, O., Ûber die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring, Math. Z. 84 (1964), 80–87.Google Scholar
7. Geramita, A. V., Decomposability in the exterior algebra of a free module, Thesis, Syracuse Univ., 1968.Google Scholar
8. Kaplansky, I., Commutative rings (Allyn and Bacon, Boston, 1970).Google Scholar
9. Kleiner, G. B., Krull OP-rings are Plucker rings, Math. U.S.S.R.—Ivestija 7 (1973), No. 2.Google Scholar
10. Lissner, D. and Geramita, A. V., Towber rings, J. of Alg. 15, No. 1 (1970), 13–40.Google Scholar
11. Lissner, D. and Geramita, A. V., Remarks on OP and Towber rings, Can. J. Math. 22 (1970), 1109–1117.Google Scholar
12. Mohan Kumar, N., On two conjectures about polynomial rings, Inv. Math. 46 (1978), 225–236.Google Scholar
13. Murthy, M. P. and Swan, R. G., Vector bundles over affine surfaces, Inv. Math. 36 (1976), 125–165.Google Scholar
14. Sathaye, A., On the Forster-Eisenbud-Evan s conjecture, Inv. Math. 46 (1978), 211–224.Google Scholar
15. Serre, J-P., Sur les modules projectif, Sem. Dubreil-Pisot (1960/61), No. 2.Google Scholar
16. Swan, R. G., Conference on Commutative Algebra—1975, Queen's Papers in Pure and Applied Math. No. 42, Queen's University, Kingston, Ontario.Google Scholar
17. Towber, J., Complete reducibility in exterior algebras over free modules, J. Alg. 10 (1968), 299–309.Google Scholar