# What makes a complex a virtual resolution?

## TL;DR

This paper characterizes when chain complexes over Cox rings of toric varieties are virtual resolutions and explores properties of Fitting ideals in this context.

## Contribution

It introduces two algebraic conditions that determine virtual resolutions and analyzes the saturation of Fitting ideals by the irrelevant ideal.

## Key findings

- Two algebraic conditions characterize virtual resolutions.
- Saturation of Fitting ideals by the irrelevant ideal is studied.
- Results mirror classical Fitting ideal theory for Noetherian rings.

## Abstract

Virtual resolutions are homological representations of finitely generated $\text{Pic}(X)$-graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.