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Let $A$ be a local Noetherian domain of Krull dimension $d$. Heinzer,Rotthaus and Sally have shown that if the generic formal fiber of $A$ hasdimension $d-1$, then $A$ is birationally dominated by a one-dimensionalanalytically ramified local Noetherian ring having residue field finite overthe residue field of $A$. We explore further this correspondence between primeideals in the generic formal fiber and one-dimensional analytically ramifiedlocal rings. Our main focus is on the case where the analytically ramifiedlocal rings are stable, and we show that in this case the embedding dimensionof the stable ring reflects the embedding dimension of a prime ideal maximal inthe generic formal fiber, thus providing a measure of how far the genericformal fiber deviates from regularity. A number of characterizations ofanalytically ramified local stable domains are also given.

Generic formal fibers and analytically ramified stable rings