# Decomposing moduli of representations of finite-dimensional algebras

## TL;DR

This paper establishes a decomposition theorem for moduli spaces of representations of finite-dimensional algebras, linking complex components to simpler products and proving rationality for tame algebra cases.

## Contribution

It introduces a decomposition theorem connecting irreducible components of moduli spaces to products of simpler spaces, with conditions for isomorphism and rationality.

## Key findings

- Decomposition theorem relating irreducible components to product spaces
- Isomorphism condition when components are normal
- All moduli spaces for tame algebras are rational varieties

## Abstract

Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ to a product of simpler moduli spaces via a finite and birational map. Furthermore, this morphism is an isomorphism when the irreducible component is normal. As an example application, we show that the irreducible components of all moduli spaces associated to tame (or even Schur-tame) algebras are rational varieties.