Some remarks concerning Demazure’s construction of normal graded rings

Abstract

In [1], Demazure showed a new way of constructing normal graded rings using the concept of “rational coefficient Weil divisors” of normal projective varieties and he showed, among other things, the followingTHEOREM ([1], 3.5). If R = ⊕n ≥ 0Rn is a normal graded ring of finite type over a field k and if T is a homogeneous element of degree 1 in the quotient field of R, then there exists unique divisor D ∈ Div (X, Q) (X = Proj (R)), such that for every n ≧ 0.(See (1.1) for the definition of