The gluing of maximal ideals—spectrum of a Noetherian ring—going up and going down in polynomial rings

Abstract:

If ${M_1}, ... , {M_s}$ are maximal ideals of a ring R that have isomorphic residue fields, then they can be “glued” in the sense that a subring D of R with R is integral over D and ${M_1} \cap D = ... = {M_s} \cap D$ can be constructed. We use this gluing process to prove the following result: Given any finite ordered set $\mathcal {B}$, there exists a reduced Noetherian ring B and an embedding $\psi : \mathcal {B} \to Spec B$ such that $\psi$ establishes a bijection between the maximal (respectively minimal) elements of $\mathcal {B}$ and the maximal (respectively minimal) prime ideals of B and such that given any elements $\beta ’$, $\beta ''$ of $\mathcal {B}$, there exists a saturated chain of prime ideals of length r between $\psi (\beta ’)$ and $\psi (\beta '')$ if and only if there exists a saturated chain of length r between $\beta ’$ and $\beta ''$. We also use the gluing process to construct a Noetherian domain A with quotient field L and a Noetherian domain B between A and L such that: $A \hookrightarrow B$ possesses the Going Up and the Going Down properties, $A[X] \hookrightarrow B[X]$ is unibranched and $A[X] \hookrightarrow B[X]$ possesses neither the Going Up nor the Going Down properties.