Normal graded algebras. III (Q1340000)

[For part I see ibid. 102, 332-352 (1986; Zbl 0593.13014); for part II see Publ. Inst. Rech. Math. Rennes 4, 71-95 (1985; Zbl 0598.13005).] In 1979, M. Demazure described normal \(\mathbb{N}\)-graded \(\kappa\)-algebras by means of Weil divisors with rational coefficients. In the present paper, the author generalizes this work of Demazure to the case of \(\mathbb{N}^ n\)- graduations. To do this, he constructs, for certain \(\mathbb{N}^ n\)-graded rings, a diagram which is a generalization of Proj of \(\mathbb{N}\)-graded rings. In the second part of the paper, the author characterizes those \(\kappa\)- algebras for which the construction previously mentioned is possible. He then establishes, for the constructed diagrams, properties analogous to those which are given for Proj in ``Eléments de géométrie algébrique'' [cf. Publ. Math., Inst. Hautes Étud. Sci. (1960-1967)] by \textit{A. Grothendieck}.