Multifiltrations

TL;DR

This paper introduces the concept of multifiltrations on vector spaces indexed by finitely generated abelian groups, explores their properties including a new regularity notion, and applies these ideas to toric geometry and group representations.

Contribution

It defines regular multifiltrations, extends classical filtration properties, and provides new insights into toric sheaves and solvable group representations.

Findings

Introduction of regular multifiltrations with no classical analogue

Reinterpretation of torus-equivariant sheaves on toric varieties

Simple formula for pull-backs of such sheaves

Abstract

This paper deals with properties of filtrations on vector spaces indexed by partially ordered finitely generated abelian groups, which we call multifiltrations. We discuss the usual properties of filtrations, like exhaustivity and separatedness in the setting of multifiltrations and we introduce a new notion, regular multifiltrations, that has no analogue for classical filtrations. As applications, we provide a reinterpretation of Klyachko, Perling et al's description of torus-equivariant torsion-free coherent sheaves on a toric variety, and give a simple formula for pull-backs of such sheaves. We also explore the occurrence of such multifiltration among finite dimensional representations of connected, solvable groups.