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.
References
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.
Department of Mathematics, Southern Illinois University at Carbondale, USA United States
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