Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Unramified cohomology and Witt groups of anisotropic Pfister quadrics

Author: R. Sujatha
Journal: Trans. Amer. Math. Soc. 349 (1997), 2341-2358
MSC (1991): Primary 11E70; Secondary 13K05, 12G05
MathSciNet review: 1422911
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The unramified Witt group of an anisotropic conic over a field $k$, with $char~k \neq 2$, defined by the form $\langle 1,-a,-b\rangle $ is known to be a quotient of the Witt group $W(k)$ of $k$ and isomorphic to $W( {k})/\langle 1,-a,-b,ab \rangle W( {k})$. We compute the unramified cohomology group $H^{3}_{nr}{k({C})}$, where $C$ is the three dimensional anisotropic quadric defined by the quadratic form $\langle 1,-a,-b,ab,-c\rangle $ over $k$. We use these computations to study the unramified Witt group of $C$.

References [Enhancements On Off] (What's this?)

  • [A] Arason, J. Kr., Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), 448-491. MR 52:10592
  • [A2] Arason, J. Kr., A proof of Merkurjev's Theorem, Canadian Math. Soc. Conf. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 1984, pp. 121-130. MR 86f:11029
  • [AEJ] Arason, J. Kr., Elman, R., Jacob, B., Graded Witt ring and Galois cohomology, Canadian Math. Soc. Conf. Proc. vol. 4, Amer. Math. Soc., Providence, RI, 1984, pp. 17-50. MR 86g:11020
  • [AEJ2] Arason, J. Kr., Elman, R., Jacob, B., Fields of cohomological 2-dimension three, Math. Ann. 274 (1986), 649-657. MR 87m:12006
  • [AEJ3] Arason, J. Kr., Elman, R., Jacob, B., Graded Witt ring and Galois cohomology II, Transactions Amer. Math. Soc. 314 (1989), 745-780. MR 90a:11043
  • [B-O] Bloch, S., Ogus, A., Gersten's conjecture and the homology of schemes, Ann. Sc. Éc. Norm. Supér (4) (1974), 181-202. MR 54:318
  • [CT] Colliot-Thélène, J.-L., Birational invariants, purity and the Gersten conjecture, Proc. Sym. Pure Math 58 (I) (1995), 1-64. MR 96c:14016
  • [CT1] Colliot-Thélène, J.-L., Cycles algébriques de torsion et K-théorie algébrique, Lecture notes in Mathematics 1553 (1992), 1-49. MR 96h:14006
  • [CT-O] Colliot-Thélène, J.-L., Ojanguren, M., Variétés unirationnelles non rationnelles: au-delà de l'exemple d'Artin et Mumford, Invent. Math. 97 (1989), 141-158. MR 90m:14012
  • [CT-S] Colliot-Thélène, J.-L., Sujatha, R., Unramified Witt groups of real anisotropic quadrics, Proc. Sym. Pure Math 58 (II) (1995), 127-147. MR 96e:19009
  • [E-L] Elman, R., Lam, T.-Y., Pfister forms and K-theory of fields, Jour. Alg 23 (1972), 181-213. MR 46:1882
  • [ELW] Elman, R., Lam, T. Y., Wadsworth, A. R., Function fields of Pfister forms, Invent. Math. 51 (1979), 61-75. MR 80m:10017
  • [J-R] Jacob, B., Rost, M., Degree four cohomological invariants for quadratic forms, Invent. Math 96 (1989), 551-570. MR 90g:11044
  • [Ja] Jannsen, U., Cohomological Hasse principles, Handwritten notes.
  • [Ka] Kahn, B., Lower ${\mathcal {H}}$-cohomology of higher dimensional quadrics, Arch. Math. (Basel) 65 (1995), 244-250. MR 97a:14017
  • [K-Me] Karpenko, N.A., Merkurjev, A.S., Chow groups of projective quadrics, Leningrad Math. Jour. 2 (1991), 655-671. MR 91i:14003
  • [K] Kato, K., Milnor K-theory and the Chow group of zero cycles, Contemp. Math. 55 (1986), 241-253. MR 88c:14012
  • [Me] Merkurjev, A. S., On the norm residue symbol of degree 2, Soviet Math. Dokl. 24 (1981), 546-551. MR 83h:12015
  • [Me-S1] Merkurjev, A. S., Suslin, A. A., The norm-residue homomorphism in degree three, Math. USSR. Izv. 36 (1991), 349-367. MR 91f:11083
  • [Mi] Milne, J. S., Étale Cohomology, Princeton University Press, 1980. MR 81j:14002
  • [O] Ojanguren, M., The Witt group and the problem of Lüroth, ETS Editrice Pisa, Pisa, 1991. MR 91m:11031
  • [P] Parimala, R., Witt groups of conics, elliptic and hyperelliptic curves, Jour. Number Theory 28 (1988), 69-93. MR 89a:14028
  • [P2] Parimala, R., Witt groups of affine three-folds, Duke Math. J. 57 (1988), 947-954. MR 90a:14006
  • [Pf] Pfister, A., Multiplikative quadratische Formen, Arch. Math. 16 (1965), 363-370. MR 32:2408
  • [Ra] Raskind, W., Abelian class field theory of arithmetic schemes, Proc. Sym. Pure Math 58 (I) (1995), 85-187. MR 96b:11089
  • [R] Rost, M., Hilbert Theorem 90 for $K_{3}$ for degree two extensions, preprint, Regensburg, 1986.
  • [R1] Rost, M., Talk at the Ascona Conference on Quadratic Forms, 1991.
  • [Sc] Scharlau, W., Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften 270, Berlin, Heidelberg, New York: Springer, 1985. MR 86k:11022
  • [Su] Suslin, A. A., Torsion in $K_{2}$ of fields, $K$-theory 1 (1987), 5-29. MR 89a:11123
  • [Sz] Shyevski [Szyjewski], M., The fifth invariant of quadratic forms, Leningrad Math Jour. 2 (1991), 179-198. MR 91d:11040
  • [Sz1] Shyevski [Szyjewski], M., Algebraic K-theory and quadratic forms (in Russian), Dissertation, Leningrad State University (1989).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11E70, 13K05, 12G05

Retrieve articles in all journals with MSC (1991): 11E70, 13K05, 12G05

Additional Information

R. Sujatha
Affiliation: Department of Mathematics, Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210 \indent{Permanent address}: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

Keywords: Pfister forms, unramified cohomology, ├ętale cohomology
Received by editor(s): November 7, 1995
Dedicated: Dedicated to my father on his sixtieth birthday
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society