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2017 Real cohomology and the powers of the fundamental ideal in the Witt ring
Jeremy Jacobson
Ann. K-Theory 2(3): 357-385 (2017). DOI: 10.2140/akt.2017.2.357

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(), ).

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Jeremy Jacobson. "Real cohomology and the powers of the fundamental ideal in the Witt ring." Ann. K-Theory 2 (3) 357 - 385, 2017. https://doi.org/10.2140/akt.2017.2.357

Information

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

zbMATH: 06726476
MathSciNet: MR3658988
Digital Object Identifier: 10.2140/akt.2017.2.357

Subjects:
Primary: 11E81 , 14F20 , 14F25 , 19G12

Keywords: real cohomology , real variety , Witt group

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.2 • No. 3 • 2017
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