Annals of K-Theory

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

Jeremy Jacobson

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Witt group real cohomology real variety


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.

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