A classification theorem for quadratic forms over semi-local rings



Abstract

Let R be a semi-local ring with 2∈U(R) and such that all residue class fields of R contain more than 3 elements. It is proved here that bilinear spaces over R are classified by dimension, determinant, Hasse invariant and total signature if and only if the third power of the fundamental ideal of Witt ring W(R) is torsion free. This is a generalization of the same result when R is a field due to Elman and Lam.


1. R. Baeza, Über die Torsion der Witt-Gruppe W(A) eines semi-localen Ringes, Math. Ann. 207 (1974), 121-131.
2. R. Baeza, Quadratic forms over semilocal rings, Lecture notes in Math., 655, Springer-Verlag, 1978.
3. R. Baeza, M. Knebusch, Annulatoren von Pfister formen über semilokalen Ringen, Math. Z. 140 (1974), 41-62.
4. R. Elman, T.Y. Lam, Quadratic forms over formally real and pythagorean fields, Amer. J. Math. 94 (1972), 1155-1194.
5. R. Elman, T.Y. Lam, Quadratic forms and the u-invariant 1, Math. Z. 131 (1973), 283-304.
6. R. Elman, T.Y. Lam, On the quaternion symbol homomorphism g_{F}: k_{2}F→B(F), Proc. of Seattle Algebraic K-Theory Conference, Springer Lecture Notes in Math. 342 (1973), 447-463.
7. R. Elman, T.Y. Lam, Classification Theorems for Quadratic Forms over Fields, Comment. Math. Helv. 49 (1974), 373-381.
8. M. Knebusch, A. Rosenberg, R. Ware, Signatures on semi-local rings, J. Algebra 26 (1973), 208-250.
9. M. Knebusch, A. Rosenberg, R. Ware, Structure of Witt Rings and Quotients of Abelian Group Rings, Amer. J. Math. 274 (1975), 61-89.
10. T.Y. Lam, The algebraic theory of quadratic forms, W.A. Benjamin, Reading, Massachusetts, 1973.
11. K. Mandelberg, On the classification of quadratic forms over semilocal rings, J. Algebra 33 (1975), 463-471.
Download

Published : 1986-09-30


YucasJ. L. (1986). A classification theorem for quadratic forms over semi-local rings. Annales Mathematicae Silesianae, 2, 7-12. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14313

Joseph L. Yucas 
Department of Mathematics, Southern Illinois University at Carbondale, USA  United States



The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.