Betti numbers of multigraded modules of generic type

TL;DR

This paper introduces a new concept of generic multidegrees and modules of generic type, providing a combinatorial formula for multigraded Betti numbers of such modules using simplicial homology and matroid invariants.

Contribution

It defines the notion of generic multidegrees and modules of generic type, and derives a Hochster-type formula for Betti numbers in this setting, linking algebraic invariants to combinatorial structures.

Findings

Hochster-type formula for Betti numbers at generic multidegrees

Unique non-zero homological degree for Betti numbers of generic modules

Betti numbers expressed via matroid minors and simplicial homology

Abstract

Let $R=\Bbbk[x_1,...,x_m]$ be the polynomial ring over a field $\Bbbk$ with the standard $\mathbb Z^m$-grading (multigrading), let $L$ be a Noetherian multigraded $R$-module, let $\beta_{i,\alpha}(L)$ the $i$th (multigraded) Betti number of $L$ of multidegree $\a$. We introduce the notion of a generic (relative to $L$) multidegree, and the notion of multigraded module of generic type. When the multidegree $\a$ is generic (relative to $L$) we provide a Hochster-type formula for $\beta_{i,\alpha}(L)$ as the dimension of the reduced homology of a certain simplicial complex associated with $L$. This allows us to show that there is precisely one homological degree $i\ge 1$ in which $\beta_{i,\alpha}(L)$ is non-zero and in this homological degree the Betti number is the $\beta$-invariant of a certain minor of a matroid associated to $L$. In particular, this provides a precise combinatorial description of all multigraded Betti numbers of $L$ when it is a multigraded module of generic type.