Lemma 10.119.1. Let R be a local Noetherian domain with fraction field K. Assume R is not a field. Then there exist R \subset R' \subset K with
R' local Noetherian of dimension 1,
R \to R' a local ring map, i.e., R' dominates R, and
R \to R' essentially of finite type.
Proof. Choose any valuation ring A \subset K dominating R (which exist by Lemma 10.50.2). Denote v the corresponding valuation. Let x_1, \ldots , x_ r be a minimal set of generators of the maximal ideal \mathfrak m of R. We may and do assume that v(x_ r) = \min \{ v(x_1), \ldots , v(x_ r)\} . Consider the ring
Note that \mathfrak mS = x_ rS is a principal ideal. Note that S \subset A and that v(x_ r) > 0, hence we see that x_ rS \not= S. Choose a minimal prime \mathfrak q over x_ rS. Then \text{height}(\mathfrak q) = 1 by Lemma 10.60.11 and \mathfrak q lies over \mathfrak m. Hence we see that R' = S_{\mathfrak q} is a solution. \square
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