#### Vol. 2, No. 3, 2017

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

### Jeremy A. Jacobson

Vol. 2 (2017), No. 3, 357–385
##### 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 $C\left(sperA,ℤ∕2\right)$ of continuous $ℤ∕2$-valued functions on the real spectrum of $A$. Let ${I}^{n}\left(A\right)$ 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 ${\oplus }_{n\ge 0}{I}^{n}\left(A\right)$ with respect to the class $〈〈-1〉〉:=〈1,1〉\in I\left(A\right)$ is isomorphic to the ring $C\left(sperA,ℤ\right)$ 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 ${H}_{Zar}^{\ast }\left(X,{\mathsc{ℐ}}^{n}\right)$ with coefficients in the sheaf ${\mathsc{ℐ}}^{n}$ associated to the $n$-th power of the fundamental ideal in the Witt ring $W\left(X\right)$ and the singular cohomology groups ${H}_{sing}^{\ast }\left(X\left(ℝ\right),ℤ\right)$.

##### Keywords
Witt group, real cohomology, real variety
##### Mathematical Subject Classification 2010
Primary: 11E81, 14F20, 14F25, 19G12