Controlling the generic formal fiber of local domains and their polynomial rings

Abstract

Let $T$ be a complete local ring with maximal ideal $M$, $C$ a countable set of incomparable prime ideals of $T$, and $B_1$ and $B_2$ sets of prime ideals of $T[[x_1,\ldots,x_n]]$ with cardinality less than that of $T$. We present necessary and sufficient conditions for the existence of a local domain $A$ with completion $T$, such that the generic formal fiber of $A$ has maximal elements equal to the ideals in $C$ and the generic formal fiber of $A[x_1,\ldots,x_n]_{(M\cap A,x_1,\ldots,x_n)}$ contains every element of $B_1$ but no element of $B_2$. If $T$ has characteristic $0$, we present necessary and sufficient conditions for the existence of an excellent local domain $A$ with the above properties.