Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case

TL;DR

This paper proves the Multiplicity Conjecture for non-Cohen-Macaulay modules by expressing Betti diagrams as positive combinations of pure resolutions, using combinatorial and algebraic techniques.

Contribution

It extends the validity of the Multiplicity Conjecture to a broader class of modules by decomposing Betti diagrams into pure components.

Findings

Betti diagrams can be expressed as positive linear combinations of pure diagrams.

The convexity of the simplicial fan spanned by pure diagrams is established.

The Multiplicity Conjecture holds for non-Cohen-Macaulay modules.

Abstract

We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams.