# Large lower bounds for the betti numbers of graded modules with low regularity

## TL;DR

This paper establishes new lower bounds on the Betti numbers of finitely-generated graded modules over polynomial rings with low regularity, revealing significant constraints on their algebraic complexity.

## Contribution

It provides the first explicit lower bounds for Betti numbers of modules with bounded regularity and high codimension, extending understanding of their minimal free resolutions.

## Key findings

- Sum of Betti numbers is at least (2^c + 2^{c-1}) times 00(M).
- For codimension c  8, Betti numbers 1 to 00(c/2) are bounded below by 2 times 00(M) times binomial coefficients.
- Results apply to modules with regularity at most 2a-2, where a is the minimal degree of a first syzygy.

## Abstract

Suppose that $M$ is a finitely-generated graded module of codimension $c\geq 3$ over a polynomial ring and that the regularity of $M$ is at most $2a-2$ where $a\geq 2$ is the minimal degree of a first syzygy of $M$. Then we show that the sum of the betti numbers of $M$ is at least $\beta_0(M)(2^c + 2^{c-1})$. In addition, if $c \geq 9$ then for each $1\leq i\leq \lceil c/2\rceil$, we show $\beta_i(M)\geq 2\beta_0(M){c \choose i}$.