Modules over categories and Betti posets of monomial ideals

TL;DR

This paper introduces a categorical approach to multigraded modules, focusing on Betti posets of monomial ideals, and demonstrates how these posets determine minimal free resolutions and Betti numbers.

Contribution

It develops a general theory of modules over categories applied to Betti posets, providing new proofs and characterizations for monomial ideals' resolutions.

Findings

Betti posets support free resolutions of monomial ideals.

Betti numbers can be computed from open intervals of Betti posets.

Betti posets uniquely determine the minimal free resolution structure.

Abstract

We introduce to the context of multigraded modules the methods of modules over categories from algebraic topology and homotopy theory. We develop the basic theory quite generally, with a view toward future applications to a wide class of graded modules over graded rings. The main application in this paper is to study the Betti poset B=B(I,k) of a monomial ideal I in the polynomial ring R=k[x_1,...,x_m] over a field k, which consists of all degrees in Z^m of the homogeneous basis elements of the free modules in the minimal free Z^m-graded resolution of I over R. We show that the order simplicial complex of B supports a free resolution of I over R. We give a formula for the Betti numbers of I in terms of Betti numbers of open intervals of B, and we show that the isomorphism class of B completely determines the structure of the minimal free resolution of I, thus generalizing with new proofs results of Gasharov, Peeva, and Welker. We also characterize the finite posets that are Betti posets of a monomial ideal.