Finite super height of a homogeneous ideal in \({\mathbb{Z}}^ n\)-graded extensions
The purpose of the paper is to consider big ht I, super ht I and finite super ht I for an ideal I of a noetherian ring R, where big\(ht I=\sup \{ht P| \quad P\) is a prime ideal of \(R\) minimal over \(I\},\) super\(ht I=\sup \{ht IS| \quad S\) is a noetherian R-algebra\}, finite super\(ht I=\sup \{ht IS| \quad S\) is a finitely generated R-algebra\}. The main motivation for the work is the direct summand conjecture. The following theorem is shown in section 1. Theorem: The direct summand conjecture is equivalent to the improved monomial conjecture, which asserts: Let I be the ideal generated by the residue classes of \(X_ 1,...,X_ n\) in the ring \(V[X_ 1,...,X_ n,Y_ 1,...,Y_ n]/(X^ t_ 1...X^ t_ n-\sum^{n}_{i=1}Y_ iX_ i^{t+1})\) with \(t\geq 1\), \(n\geq 3\), indeterminates \(X_ i's\) and \(Y_ i's\), and the ring V of Witt vectors over the algebraic closure of \({\mathbb{Z}}/p{\mathbb{Z}}\). Then, finite super\(ht I<n\). - Zariski's main theorem and its graded version, which is proven in section 2, are used.
scientific article; zbMATH DE number 4012775