Atom spectra of graded rings and sheafification in toric geometry

Abstract

We prove that the atom spectrum, which is a topological space associated to an arbitrary abelian category introduced by Kanda, of the category of finitely presented graded modules over a graded ring $R$ is given as a union of the homogeneous spectrum of $R$ with some additional points, which we call non-standard points. This description of the atom spectrum helps in understanding the sheafification process in toric geometry: if $S$ is the Cox ring of a normal toric variety $X$ without torus factors, then a finitely presented graded $S$-module sheafifies to zero if and only if its atom support consists only of points in the atom spectrum of $S$ which either lie in the vanishing locus of the irrelevant ideal of $X$ or are non-standard.