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

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.

19 pages, minor corrections done