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

Let A be a local ring with 2 invertible. It is known that the localization of the cohomology ring H∗ ét(A,Z/2) with respect to the class (−1) ∈ H1 ét(A,Z/2) 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≥0 I n(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 ofX , we obtain a bijection between the Zariski cohomology groups H∗ Zar(X, In) with coefficients in the sheaf In associated to the nth power of the fundamental ideal in the Witt ring W (X) and the singular cohomology groups H∗ sing(X(R),Z).