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

Abstract

Let $A$ be a local ring in which 2 is invertible. It is known that the localization of the cohomology ring $H_{\text{ ét}}^{\ast }\left(A,\mathbb{Z}∕2\right)$ with respect to the class $\left(-1\right)\in H_{\text{ ét}}^1\left(A,\mathbb{Z}∕2\right)$ is isomorphic to the ring $C\left(sperA,\mathbb{Z}∕2\right)$ of continuous $\mathbb{Z}∕2$-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 $\langle \langle -1\rangle \rangle :=\langle 1,1\rangle \in I\left(A\right)$ is isomorphic to the ring $C\left(sperA,\mathbb{Z}\right)$ of continuous $\mathbb{Z}$-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 $H_{Zar}^{\ast }\left(X,{\mathcal{ℐ}}^n\right)$ with coefficients in the sheaf ${\mathcal{ℐ}}^n$ associated to the $n$-th power of the fundamental ideal in the Witt ring $W\left(X\right)$ and the singular cohomology groups $H_{sing}^{\ast }\left(X\left(ℝ\right),\mathbb{Z}\right)$.