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 et∗(A, ℤ∕2) with respect to the class (−1) ∈ H et1(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≥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).