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

Authors:
Ada Maria de Souza Doering and Yves Lequain

Journal:
Trans. Amer. Math. Soc. **260** (1980), 583-593

MSC:
Primary 13E05

DOI:
https://doi.org/10.1090/S0002-9947-1980-0574801-5

MathSciNet review:
574801

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Abstract | References | Similar Articles | Additional Information

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.

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Keywords:
Gluing of maximal ideals,
integral extension,
altitude formula,
finite ordered set,
spectrum of a Noetherian ring,
Going Up,
Going Down,
unibranchness,
polynomial ring

Article copyright:
© Copyright 1980
American Mathematical Society