Mumford’s Construction

Abstract

In this chapter R denotes a noetherian normal domain complete w.r.t. an ideal I, and we always assume rad(I) = I for convenience. S always denote Spec(R), η = generic point of S, S ₀ = Spec(R/I), K = quotient field of R. As before we assume that for any étale R/I-algebra R ^†₀ its unique lifting R ^† to a formally étale I-adically complete R-algebra is normal. This holds for example if R is regular, or if R is the I-adic completion of a normal excellent ring. In fact everything could be done over a formal scheme which is not necessarily affine whose coordinate rings satisfy the above strong normality condition, but we leave it to the reader to take care of this additional generality. (The reference for relative schemes is [Hak].)