Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-03T20:36:20.251Z Has data issue: false hasContentIssue false
  • English
  • Français

Signatures and Semi Signatures of Abstract Witt Rings and Witt Rings of Semilocal Rings

Published online by Cambridge University Press:  20 November 2018

Jerrold L. Kleinstein  and
Alex Rosenberg
Jerrold L. Kleinstein
Affiliation:
SUNY at Stony Brook, Stony Brook, New York;
Alex Rosenberg
Affiliation:
SUNY at Stony Brook, Stony Brook, New York;
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper originated in an attempt to carry over the results of [3] from the case of a field of characteristic different from two to that of semilocal rings. To carry this out, we reverse the point of view of [3] and do assume a full knowledge of the theory of Witt rings of classes of nondegenerate symmetric bilinear forms over semilocal rings as given, for example, in [10; 11]. It turns out that the rings WT of [3] are just the residue class rings of W(C), the Witt ring of a semilocal ring C, modulo certain intersections of prime ideals.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 4 , 01 August 1978 , pp. 872 - 895
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Beaza, R., Quadratische Formen iiber semilokalen Ringen, Habilitationschrift, Saarbrucken (1975).Google Scholar
2. Baeza, R. and Knebusch, M., Annulatoren von Pfisterformen iiber semilokalen Ringen, Math. Z. 140 (1974), 41–62.Google Scholar
3. Becker, E. and E. Kôpping, Reduzierte quadratische Formen und Semiordnungen reeler Korper, Abh. Math. Sem. Hamburg 46 (1977), 143–177.Google Scholar
4. Elman, R., Lam, T. Y., and Prestel, A., On some Hasse principles over formally real fields, Math. Z. 134 (1973), 291–301.Google Scholar
5. Knebusch, M., Isometrien iiber semilokalen Ringen, Math. Z. 108 (1969), 255–268.Google Scholar
6. Knebusch, M., Grothendieck-und Wittringe von nichtausgearteten symmetrischen Bilinearjormen, Sitzber. Heidelberg Akad. Wiss. (1969/70), 93–157.Google Scholar
7. Knebusch, M., Runde Formen iiber semilokalen Ringen, Math. Ann. 193 (1971), 21–34.Google Scholar
8. Knebusch, M., Generalization of a theorem of Artin-Pfister, J. of Alg. 36 (1975), 46–67.Google Scholar
9. Knebusch, M., Remarks on the paper “Equivalent topological properties of the space of signatures of a semilocal ring” by Rosenberg, A. and Ware, R., Pub. Math. 24 (1977), 181–188.Google Scholar
10. 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
11. Knebusch, M., Rosenberg, A., and Ware, R., Signatures on semilocal rings, J. of Alg. 26 (1973), 208–250.Google Scholar
12. Milnor, J. and Husemoller, D., Symmetric bilinear forms, Ergeb. d. Math. 73 (Springer Verlag, New York-Heidelberg-Berlin, 1973).Google Scholar
13. Prestel, A., Lectures on formally real fields, Instituto de Matemâtica Pura e Aplicada, Brasilia, 1975.Google Scholar
14. Roy, A., Cancellation of quadratic forms over commutative rings, J. of Alg. 10 (1968), 286–298.Google Scholar
15. Rosenberg, A. and Ware, R., Equivalent topological properties of the space of signatures of a semilocal ring, Pub. Math. 23 (1976), 283–289.Google Scholar