Witt Rings in Real Algebraic Geometry

Abstract

In the first section we define the Witt ring W(A) of a commutative ring A and compare it with the group K ₀(A). In particular, if V is an algebraic set over a real closed field R, we show that the Witt ring W(S ⁰(V)) coincides with K ₀(S ⁰(V)) (≃ KO(V) if R = ℝ). The second section is devoted to the result of Mahé concerning the separation of the semi-algebraically connected components of V by the signatures of elements of W(P(V)). In the third section we prove that the morphism W(P(V))[1/2] → W(S ⁰(V))[1/2], induced by the inclusion P(V) → S ⁰(V), is surjective. This is part of the result of Brumfiel, which asserts that this morphism is actually an isomorphism.