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A Note on Quadratic forms Over Arbitrary Semi-Local Rings

Published online by Cambridge University Press:  20 November 2018

K. I. Mandelberg
K. I. Mandelberg*
Affiliation:
Emory University, Atlanta, Georgia
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Let R be a commutative ring. A bilinear space (E, B) over R is â finitely generated projective R-module E together with a symmetric bilinear mapping B:E X E →R which is nondegenerate (i.e. the natural mapping E → HomR(E﹜ R) induced by B is an isomorphism). A quadratic space (E, B, ) is a bilinear space (E, B) together with a quadratic mapping ϕ:E →R such that B(x, y) = ϕ (x + y) — ϕ (x) — ϕ (y) and ϕ (rx) = r2ϕ (x) for all x, y in E and r in R. If 2 is a unit in R, then ϕ (x) = ½. B﹛x,x) and the two types of spaces are in obvious 1 — 1 correspondence.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 3 , 01 June 1975 , pp. 513 - 527
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Arf, C., Untersuchungen iiber quadratische Formen in Kôrpern der Charakteristik 2, J. Reine Angew. Math. 183 (1941), 148–167.Google Scholar
2. Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.Google Scholar
3. Bass, H., Lectures on topics in algebraic K-theory, Tata Institute, 1967.Google Scholar
4. Bourbaki, N., Elements de mathématique, algèbre commutative, Ch. 2 (Hermann, Paris, 1961).Google Scholar
5. Childs, L. N., Garfinkel, G., and Orzech, M., The Brauer group of graded Azumaya algebras, Trans. Amer. Math. Soc. 175 (1973), 299–326.Google Scholar
6. DeMeyer, F., Projective modules over central separable algebras, Can. J. Math. 21 (1969), 39–43.Google Scholar
7. Endo, S. and Watanabe, Y., On separable algebras over a commutative ring, Osaka J. Math. 4 (1967), 233–242.Google Scholar
8. Harrison, D. K., Witt Rings, Lecture Notes, University of Kentucky, 1969.Google Scholar
9. Hsia, J. S. and Peterson, R. D., A note on quadratic forms over semi-local rings (preprint).Google Scholar
10. Kaplansky, I., Linear algebra and geometry, a second course (Allyn and Bacon, Boston, 1969).Google Scholar
11. Knebusch, M., Bemerkungen Théorie der quadratischen Formen fiber semilokalen Ringen, Schriften des Math, Inst, der Univ. des Saarlandes, Saarbriicken, 1969.Google Scholar
12. Knebusch, M., Isometrien iiber semilokalen Ringen, Math. Z. 108 (1969), 255–268.Google Scholar
13. Knebusch, M., Rosenberg, A., and Ware, R., Structure of Witt rings and quotients of abelian group rings, Amer. J. Math. 94 (1972), 119–155.Google Scholar
14. O, O. T.'Meara, Introduction to quadratic forms (Springer-Verlag, Berlin, 1963).Google Scholar
15. Pfister, A., Quadratische Formen in beliebigen Kôrpern, Invent. Math. 1 (1966), 116–132.Google Scholar
16. Small, C., The Brauer-Wall group of a commutative ring, Trans. Amer. Math. Soc. 156 (1971), 455–491.Google Scholar
17. Witt, E., Théorie der quadratischen Formen in beliebigen Kôrpern, J. Reine Angew. Math. 176 (1937), 31–44.Google Scholar