# Examples of finite free complexes of small rank and small homology

## TL;DR

This paper constructs finite free complexes over certain rings with smaller total rank and homology length than predicted by existing conjectures, challenging current theoretical expectations.

## Contribution

It provides explicit examples of complexes with minimal rank and homology length, countering conjectural bounds in transformation groups and local algebra.

## Key findings

- Constructed complexes with total rank below conjectured bounds
- Demonstrated complexes with smaller homology length than expected
- Challenged existing conjectures in transformation groups and local algebra

## Abstract

This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying free modules, or the total length of their homology, is less than predicted by various conjectures in the theory of transformation groups and in local algebra.