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 noetherian 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.