The linear space of Betti diagrams of multigraded artinian modules

TL;DR

This paper investigates the structure of multigraded Betti diagrams of certain artinian modules, establishing a basis formed by a specific equivariant resolution and its twists, revealing the linear space they generate.

Contribution

It introduces a basis for the linear space of multigraded Betti diagrams of artinian modules, linking equivariant resolutions to this structure.

Findings

The Betti diagrams of the equivariant resolution form a basis for the linear space.

All twists of this equivariant resolution are included in the basis.

The study characterizes the linear space generated by these Betti diagrams.

Abstract

We study the linear space generated by the multigraded Betti diagrams of Z^n-graded artinian modules of codimension n whose resolutions become pure of a given type when taking total degrees. We show that the multigraded Betti diagram of the equivariant resolution constructed by D.Eisenbud, J.Weyman, and the author, and all its twists, form a basis for this linear space.