Annals of K-Theory

Real cohomology and the powers of the fundamental ideal in the Witt ring

Jeremy Jacobson

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/akt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let A be a local ring in which 2 is invertible. It is known that the localization of the cohomology ring H ét(A, 2) with respect to the class (1) H ét1(A, 2) is isomorphic to the ring C(sperA, 2) of continuous 2-valued functions on the real spectrum of A. Let In(A) denote the powers of the fundamental ideal in the Witt ring of symmetric bilinear forms over A. The starting point of this article is the “integral” version: the localization of the graded ring n0In(A) with respect to the class 1 := 1,1 I(A) is isomorphic to the ring C(sperA, ) of continuous -valued functions on the real spectrum of A.

This has interesting applications to schemes. For instance, for any algebraic variety X over the field of real numbers and any integer n strictly greater than the Krull dimension of X, we obtain a bijection between the Zariski cohomology groups HZar(X,n) with coefficients in the sheaf n associated to the n-th power of the fundamental ideal in the Witt ring W(X) and the singular cohomology groups Hsing(X(), ).

Article information

Source
Ann. K-Theory, Volume 2, Number 3 (2017), 357-385.

Dates
Received: 16 March 2016
Revised: 4 August 2016
Accepted: 19 September 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.akt/1508431895

Digital Object Identifier
doi:10.2140/akt.2017.2.357

Mathematical Reviews number (MathSciNet)
MR3658988

Zentralblatt MATH identifier
06726476

Subjects
Primary: 11E81: Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24] 14F20: Étale and other Grothendieck topologies and (co)homologies 14F25: Classical real and complex (co)homology 19G12: Witt groups of rings [See also 11E81]

Keywords
Witt group real cohomology real variety

Citation

Jacobson, Jeremy. Real cohomology and the powers of the fundamental ideal in the Witt ring. Ann. K-Theory 2 (2017), no. 3, 357--385. doi:10.2140/akt.2017.2.357. https://projecteuclid.org/euclid.akt/1508431895


Export citation

References

  • J. K. Arason, “Cohomologische invarianten quadratischer Formen”, J. Algebra 36:3 (1975), 448–491.
  • J. K. Arason and M. Knebusch, “Über die Grade quadratischer Formen”, Math. Ann. 234:2 (1978), 167–192.
  • R. Baeza, Quadratic forms over semilocal rings, Lecture Notes in Mathematics 655, Springer, 1978.
  • P. Balmer and C. Walter, “A Gersten–Witt spectral sequence for regular schemes”, Ann. Sci. École Norm. Sup. $(4)$ 35:1 (2002), 127–152.
  • P. Balmer, S. Gille, I. Panin, and C. Walter, “The Gersten conjecture for Witt groups in the equicharacteristic case”, Doc. Math. 7 (2002), 203–217.
  • J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, Ergebnisse der Mathematik $($3$)$ 36, Springer, Berlin, 1998.
  • J. Burési, “Local-global principle for étale cohomology”, $K$-Theory 9:6 (1995), 551–566.
  • J.-L. Colliot-Thélène and R. Parimala, “Real components of algebraic varieties and étale cohomology”, Invent. Math. 101:1 (1990), 81–99.
  • M. Coste and M.-F. Roy, “La topologie du spectre réel”, pp. 27–59 in Ordered fields and real algebraic geometry (San Francisco, 1981), edited by D. W. Dubois and T. Recio, Contemp. Math. 8, American Mathematical Society, Providence, RI, 1982.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, II”, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 5–231.
  • R. Elman and T. Y. Lam, “Quadratic forms over formally real fields and pythagorean fields”, Amer. J. Math. 94 (1972), 1155–1194.
  • R. Elman, N. Karpenko, and A. Merkurjev, The algebraic and geometric theory of quadratic forms, Colloquium Publications 56, American Mathematical Society, Providence, RI, 2008.
  • J. Fasel, “Some remarks on orbit sets of unimodular rows”, Comment. Math. Helv. 86:1 (2011), 13–39.
  • J. Fasel, “The projective bundle theorem for ${\bf I}^j$-cohomology”, J. K-Theory 11:2 (2013), 413–464.
  • O. Gabber, “Affine analog of the proper base change theorem”, Israel J. Math. 87:1 (1994), 325–335.
  • T. Geisser, “Motivic cohomology over Dedekind rings”, Math. Z. 248:4 (2004), 773–794.
  • S. Gille, “A graded Gersten–Witt complex for schemes with a dualizing complex and the Chow group”, J. Pure Appl. Algebra 208:2 (2007), 391–419.
  • S. Gille, “On quadratic forms over semilocal rings”, preprint, 2015, hook http://www.math.ualberta.ca/~gille/QuadFormsLocalRings.pdf \posturlhook.
  • R. T. Hoobler, “The Merkuriev–Suslin theorem for any semi-local ring”, J. Pure Appl. Algebra 207:3 (2006), 537–552.
  • B. Kahn, “$K$-theory of semi-local rings with finite coefficients and étale cohomology”, $K$-Theory 25:2 (2002), 99–138.
  • M. Kerz, “The Gersten conjecture for Milnor $K$-theory”, Invent. Math. 175:1 (2009), 1–33.
  • M. Kerz, “Milnor $K$-theory of local rings with finite residue fields”, J. Algebraic Geom. 19:1 (2010), 173–191.
  • M. Knebusch, “Symmetric bilinear forms over algebraic varieties”, pp. 103–283 in Conference on quadratic forms (Kingston, ON, 1976), edited by G. Orzech, Queen's Papers in Pure and Applied Mathematics 46, Queen's University, Kingston, ON, 1977.
  • M. Knebusch and C. Scheiderer, Einführung in die reelle Algebra, Vieweg Studium: Aufbaukurs Mathematik 63, Friedrich Vieweg & Sohn, Braunschweig, Germany, 1989.
  • L. Mahé, “Signatures et composantes connexes”, Math. Ann. 260:2 (1982), 191–210.
  • L. Mahé, “On the separation of connected components by étale cohomology”, $K$-Theory 9:6 (1995), 545–549.
  • J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer, New York, 1973.
  • A. Pfister, “Quadratische Formen in beliebigen Körpern”, Invent. Math. 1 (1966), 116–132.
  • C. Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics 1588, Springer, Berlin, 1994.
  • C. Scheiderer, “Purity theorems for real spectra and applications”, pp. 229–250 in Real analytic and algebraic geometry (Trento, 1992), edited by F. Broglia et al., de Gruyter, Berlin, 1995.
  • A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (Séminaire de Géométrie Algébrique du Bois Marie 1962), Documents Mathématiques (Paris) 4, Société Mathématique de France, Paris, 2005. Revised reprint of the 1968 French original.
  • M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas, Tome 3: Exposés IX–XIX (Séminaire de Géométrie Algébrique du Bois Marie 1963–1964), Lecture Notes in Mathematics 305, Springer, Berlin, 1973.
  • R. Strano, “On the étale cohomology of Hensel rings”, Comm. Algebra 12:17–18 (1984), 2195–2211.