Generic fiber rings of mixed power series/polynomial rings

Details Generic fiber rings of mixed power series/polynomial rings Generic fiber rings of mixed power series/polynomial rings

2005-07-16

arxiv.org/abs/math/0507326v1

Mathematics

Commutative Algebra

28 pages

Scientific paper

Let K be a field, m and n positive integers, and X = {x_1,...,x_n}, and Y = {y_1,..., y_m} sets of independent variables over K. Let A be the polynomial ring K[X] localized at (X). We prove that every prime ideal P in A^ = K[[X]] that is maximal with respect to P\cap A = (0) has height n-1. We consider the mixed power series/polynomial rings B := K[[X]][Y]_{(X,Y)} and C := K[Y]_{(Y)}[[X]]. For each prime ideal P of B^ = C that is maximal with respect to either P \cap B = (0) or P \cap C = (0), we prove that P has height n+m-2. We also prove that each prime ideal P of K[[X, Y]] that is maximal with respect to P \cap K[[X]] = (0) is of height either m or n+m-2.

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