Multifiltrations

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.