The ultrafilter theorem in real algebraic geometry
G. W. Brumfiel · Rocky Mountain Journal of Mathematics · 1989
Introduction.Let R be a real closed field and let V C R n be a real algebraic set, A(V) = R[x\,... ,x n ]/I{V) the affine coordinate ring of V.The ultrafilter theorem says there is a natural bijective correspondence between ultrafilters of semi-algebraic subsets of V and points of the real spectrum of A(V), [1,2].The real spectrum, Spec R(A), of a commutative ring A can be identified with, or defined as, the collection of prime cones in A: that is, subsets, a, of A which satisfy (i) a + a C a, (ii) a • a C a, (iii) Ei4 2 C a, where Y.A 2 denotes the sums of squares in A, (iv) -1 £ a, (v) a U -a = .A, and (vi) a fl -a -p(a) is a prime ideal of A. Given such an a, the residue ring A/p(a) is totally ordered, with non-negative elements being the image of a. Conversely, given a total ordering on A/p, p C A a prime ideal, the inverse image, a, of its non-negative elements satisfies (i)-(vi).Of course, total orderings of rings must be compatible with the arithmetic operations in the usual way.Note V C Spec ß[j4(V)], since a point of V can be identified with a maximal ideal of A(V) with residue ring R.Given an ultrafilter of semi-algebraic subsets of V, define a C A(V) by / € a if f(x) > 0, all x G C, for some semi-algebraic C Ç V which belongs to the ultrafilter.Then one can show a G Spec^[^4(V r )], and this is the correspondence of the ultrafilter theorem.By a constructible subset of Spec R(A), we mean any member of the smallest family of subsets closed under finite intersections, finite unions, and complements, and containing the setsWe offer the following proof of the ultrafilter theorem, which has certainly been noticed by others, for example, L. van den Dries, M.