  We give an intrinsic characterization of Cox sheaves on Krull schemes in
terms of their valuative algebraic properties. We also provide a geometric
characterization of their graded relative spectra in terms of good quotients of
graded schemes, extending the work of Arzhantsev, Derenthal, Hausen and Laface
on relative spectra of Cox sheaves on normal varieties. Moreover, we obtain an
irredundant characterization of Cox rings which in turn produces a normality
criterion for certain graded rings.
