Multigraded regularity, reduction vectors and postulation vectors

TL;DR

This paper explores the relationship between multigraded regularity, reduction vectors, and postulation vectors in the context of $Z^s$-graded filtrations of ideals, establishing equalities and connections under cohomological conditions.

Contribution

It introduces new links between reduction vectors and multigraded regularities, and relates reduction vectors to postulation vectors under specific conditions.

Findings

reg$(G(F))$ equals reg$( F)$

Established a relation between reduction vectors and postulation vectors

Connected multigraded regularity with reduction vectors

Abstract

We relate the set of complete reduction vectors of a $\mathbb{Z}^s$-graded admissible filtration of ideals $\mathcal{F}$ with the set of multigraded regularities of $G(\mathcal{F}).$ We prove reg$(G(\mathcal{F}))$=reg$(\mathcal{R}(\mathcal{F})).$ We establish a relation between the sets of complete reduction vectors of $\mathcal{F}$ and postulation vectors of $\mathcal{F}$ under some cohomological conditions.