Krull dimension and Monomial Orders

TL;DR

This paper establishes a new connection between Krull dimension and independent sequences defined via monomial orders, providing novel characterizations and bounds in algebraic structures.

Contribution

It introduces the concept of independent sequences with respect to monomial orders and proves their relation to Krull dimension, leading to new applications in ideal and algebra dimension analysis.

Findings

Krull dimension equals the supremum of lengths of independent sequences.

Characterization of maximum analytically independent elements in ideals.

Dim B is not greater than dim A for subalgebras over a Noetherian Jacobson ring.

Abstract

We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can characterize the maximum number of analytically independent elements in an arbitrary ideal of a local ring and that dim B is not greater than dim A if B is a subalgebra of A and A is a (not necessarily finitely generated) subalgebra of a finitely generated algebra over a Noetherian Jacobson ring.