## Annals of K-Theory

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

Jeremy Jacobson

#### Abstract

Let $A$ be a local ring in which 2 is invertible. It is known that the localization of the cohomology ring with respect to the class is isomorphic to the ring $C(sperA, ℤ∕2)$ of continuous $ℤ∕2$-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 $⊕ n≥0In(A)$ with respect to the class $〈〈−1〉〉 := 〈1,1〉∈ I(A)$ is isomorphic to the ring $C(sperA, ℤ)$ of continuous $ℤ$-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 $ℝ$ and any integer $n$ strictly greater than the Krull dimension of $X$, we obtain a bijection between the Zariski cohomology groups $HZar∗(X,ℐn)$ with coefficients in the sheaf $ℐn$ associated to the $n$-th power of the fundamental ideal in the Witt ring $W(X)$ and the singular cohomology groups $Hsing∗(X(ℝ), ℤ)$.

#### Article information

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

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

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

#### 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

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