Eakin-Sathaye type theorems for joint reductions and good filtrations of ideals

TL;DR

This paper extends Eakin-Sathaye theorems to joint reductions and good filtrations of ideals, providing bounds and examples demonstrating their effectiveness in various algebraic contexts.

Contribution

It introduces analogues of Eakin-Sathaye theorems for ${ m extbf{N}}^s$-graded good filtrations, offering new bounds on joint reduction vectors and reduction numbers.

Findings

Established bounds on joint reduction vectors for families of ideals.

Derived bounds on reduction numbers for $ m extbf{N}$-graded filtrations.

Provided examples illustrating the bounds' effectiveness in specific algebraic settings.

Abstract

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in $2$-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.