Let
$A$ be a
local ring in which 2 is invertible. It is known that the localization of the cohomology ring
${H}_{\text{\xe9t}}^{\ast}\left(A,\mathbb{Z}\u22152\right)$ with respect to the
class
$\left(1\right)\in {H}_{\text{\xe9t}}^{1}\left(A,\mathbb{Z}\u22152\right)$ is isomorphic to
the ring
$C\left(sperA,\mathbb{Z}\u22152\right)$ of continuous
$\mathbb{Z}\u22152$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
$\mathbb{R}$ and any integer
$n$ strictly greater than
the Krull dimension of
$X$,
we obtain a bijection between the Zariski cohomology groups
${H}_{Zar}^{\ast}\left(X,{\mathcal{\mathcal{I}}}^{n}\right)$ with coefficients
in the sheaf
${\mathcal{\mathcal{I}}}^{n}$
associated to the
$n$th
power of the fundamental ideal in the Witt ring
$W\left(X\right)$ and the singular
cohomology groups
${H}_{sing}^{\ast}\left(X\left(\mathbb{R}\right),\mathbb{Z}\right)$.
