The Compactness of the Riemann Manifold of an Abstract Field of Algebraic Functions

1. The existence of finite resolving systems. In an earlier paper we have announced the result that the existence of a resolving system of the Riemann manifold of an abstract field of algebraic functions (in any number of variables) or—what is the same—the local uniformization theorem implies the existence of finite resolving systems of the Riemann manifold. We have proved this result for algebraic surfaces by arithmetic considerations. The proof for the general case of varieties, which at that time was in our possession, and which we have promised to publish in a subsequent paper, was of similar nature, that is, it was based upon considerations involving the structure of certain infinite sequences of quotient rings. However, we have succeeded lately in finding a much simpler proof which is based on topological considerations. Let S be a field of algebraic functions of several variables, over an arbitrary ground field k. By the Riemann manifold M of S we mean the totality of places of 2 , that is, the totality of zero-dimensional valuations v of 2J/&. If F is a projective model of 2 /£ , and if H is any subset of V, we denote by N(H) the subset of M consisting of those valuations t> which have center in H. By a resolving system of M we mean a collection 33 = { Va} of projective models (finite or infinite in number) with the property that for any vin M there exists a Va in S3 such that the center of v on Va is a simple point. The topology which we introduce in M is simply this: we choose as a basis for the closed sets of M the sets N(W), where W is any algebraic subvariety of any projective model of 2 . We prove that if topologized in this fashion, the set Mis a compact* topological space. From this the result announced above follows immediately. For if { Va} is a resolving system, and if we denote by Sa the singular locus of Va, then N( Va — So) is an open set and {N( Va — Sa)} is an open covering of M. Received by the editors April 10, 1944. 1 A simplified proof for the resolution of singularities of an algebraic surface^ Ann. of Math. vol. 43 (1942) p. 583. 2 See loc. cit. footnote 1. 8 That proof was presented by us at a seminar in algebraic geometry at Johns Hopkins in 1942. 4 We use the term compact in the same sense as it is used by S. Lefschetz in his Algebraic topology (Amer. Math. Soc. Colloquium Publications, vol. 27, 1942). The old term is bicompact.