Postulation and reduction vectors of multigraded filtrations of ideals

TL;DR

This paper explores the relationship between postulation and reduction vectors in multigraded ideal filtrations within Cohen-Macaulay rings, using a generalized complex and homological analysis to extend fundamental lemmas.

Contribution

It introduces a generalized Kirby-Mehran Complex and an analogue of Huneke's Fundamental Lemma for multigraded filtrations, linking Cohen-Macaulay properties to reduction vectors.

Findings

Established a new relationship between Cohen-Macaulay properties and reduction vectors.

Extended fundamental lemmas to multigraded filtrations.

Provided tools for analyzing multigraded Rees algebras.

Abstract

We study relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal F= \{\mathcal F (\underline n)\}_{\underline n\in\mathbb Z^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalisation of the Kirby-Mehran Complex. An analysis of its homology leads to an analogue of Huneke's Fundamental Lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal F$ and reduction vectors with respect to complete reductions of $\mathcal F.$