Super height of an ideal in a Noetherian ring
The aim of this paper is to prove the following two results: Theorem 1: Let R be a commutative finitely generated algebra over a field and let P be a prime ideal of R. Then: super ht IR\(_ P=\) finite super ht IR\(_ P\) for all ideals I of R contained in P. - Theorem 2: Let p be a prime integer, \(F_ p\) the algebraic closure of \({\mathbb{Z}}/p{\mathbb{Z}}\), \(V_ p\) the ring of Witt vectors over \(F_ p\), and R a commutative finitely generated algebra over \(V_ p\). Then: super ht I=finite super ht I for all ideals I of R. Recall that if I is an arbitrary ideal in a Noetherian ring R, then super ht I (resp. finite super ht I)\(=\) \(\sup \{ht(IS)| \quad IS\neq S,\) S is a Noetherian (resp. S is a finitely generated) R-algebra\}.
scientific article; zbMATH DE number 4069038