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If Mt,. . ., M, 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 Mx n D = .. . = M, n D can be constructed. We use this gluing process to prove the following result: Given any finite ordered set ®, there exists a reduced Noetherian ring B and an embedding i//: ® -» Spec B such that i// establishes a bijection between the maximal (respectively minimal) elements of % and the maximal (respectively minimal) prime ideals of B and such that given any elements s', s" of ®, there exists a saturated chain of prime ideals of length r between uV(/3') and >K/3") if and only i there exists a saturated chain of length r between s' and s". 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 "^ B possesses the Going Up and the Going Down properties, A(X)c* B(X) is unibranched and A(X) «-» B(X) possesses neither the Going Up nor the Going Down properties. 1. Introduction and definitions. In (11), Nagata constructed his famous example of a local domain D that does not satisfy the altitude formula. The basic idea of the construction is to consider two maximal ideals AT, and M2 of different height of a domain R and to "glue" them in the sense of constructing a subring D of R that satisfies the following two properties: R is integral over D and Mx n D = M2 n D. Nagata noted that if 7? contains a field K that is a field of representatives for both RM and RM, then Mx and M2 can indeed be glued by taking D = K + (Mx n M2); he also noted that D is Noetherian if R is Noetherian. The main difficulty that Nagata encountered was to come up with a Noetherian domain R having two maximal ideals Mx and M2 of different height on which his gluing process could be performed. It is clear that such a ring R cannot be a ring of quotients of a finite Tv-algebra, and it is working in a very clever way with formal power series that Nagata constructed one.

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