Valuative and geometric characterizations of Cox sheaves (Q1753158)

The context is that of Cox sheaves on Krull schemes. If \(X\) is an appropriate Krull scheme, \(G\) an abelian group and \(\mathcal R\) is a \(G\)-graded sheaf of \(\mathcal O_X\)-algebras, for an appropriate Krull sheaf \(\mathcal O_X\), then Theorem 0.1 gives necessary and sufficient conditions for \(\mathcal R\) to be isomorphic to a Cox sheaf, in terms of \(G\)-valuations, and other algebraic conditions. Further details on Cox sheaves are given in Theorem 0.2; those involve essential \(G\)-valuations and \(G\)-Krull rings and specific grading properties. Theorem 0.4 gives a geometric characterization of graded characteristic spaces, relative spectra, in terms of well-behaved quotients of graded schemes; this extends previous geometric characterization of relative spectra of Cox sheaves of finite type on normal prevarieties given by \textit{I. Arzhantsev} et al. [Cox rings. Cambridge: Cambridge University Press (2015; Zbl 1360.14001)]. The author also generalizes, in Theorem 0.5, a characterization of Cox rings, given by the same authors. This is an irredundant characterization of Cox rings which generates, in turn, a normality criterion for certain graded rings.