# Cox Rings and Algebraic Maps

## TL;DR

This paper explores how algebraic maps between Mori Dream Spaces can be understood through Cox rings, providing a module-theoretic perspective on inverse and direct image sheaves.

## Contribution

It offers a new description of sheaf images under morphisms between Mori Dream Spaces using Cox rings and graded modules, extending the algebraic toolkit.

## Key findings

- Describes inverse image of sheaves via Cox ring modules.
- Describes direct image of sheaves via Cox ring modules.
- Provides algebraic formulas for sheaf transformations.

## Abstract

Given a morphism $F : X \rightarrow Y$ from a Mori Dream Space $X$ to a smooth Mori Dream Space $Y$ and quasicoherent sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$ , we describe the inverse image of $\mathcal{G}$ by $F$ and the direct image of $\mathcal{F}$ by $F$ in terms of the corresponding modules over the Cox rings graded in the class groups.