A lifting functor for toric sheaves

TL;DR

This paper introduces a new functor for toric sheaves that connects quasi-coherent sheaves on a variety with graded modules over its Cox ring, preserving key properties and providing a combinatorial characterization.

Contribution

It defines a lifting functor for toric sheaves that is right-adjoint to sheafification and characterizes its derived functors combinatorially.

Findings

Functor is right-adjoint to sheafification

Preserves torsion-freeness and reflexivity

Provides combinatorial description of derived functors

Abstract

For a variety X which admits a Cox ring we introduce a functor from the category of quasi-coherent sheaves on $X$ to the category of graded modules over the homogeneous coordinate ring of $X$. We show that this functor is right-adjoint to the sheafification functor and therefore left-exact. Moreover, we show that this functor preserves torsion-freeness and reflexivity. For the case of toric sheaves we give a combinatorial characterization of its right-derived functors in terms of certain right-derived limit functors.