# On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

## TL;DR

This paper provides an alternative proof of the derived equivalence for the Abuaf-Ueda flop using tilting bundles and establishes the existence of a non-commutative crepant resolution, along with results on related moduli spaces.

## Contribution

It introduces a new proof method for the Abuaf-Ueda flop's derived equivalence and demonstrates the existence of a non-commutative crepant resolution.

## Key findings

- Derived equivalence via tilting bundles
- Existence of non-commutative crepant resolution
- Results on moduli spaces of finite-length modules

## Abstract

The Abuaf-Ueda flop is a 7-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.