Let R =⊕i≧0Ri be
a graded domain and let p be a homogeneous prime ideal in R. Let Rp
be the localization of R at p and R(p)= {ri∕si|risi∈ Ri and si∉p}. If
R1∩ (R − p)≠∅, then Rp is a localization of a transcendental extension of
R(p). Thus Rp is normal (regular) if and only if R(p) is normal (regular).
Let Proj(R) = {p|p is a homogeneous prime ideal and p ⊊⊕i>0Ri}.
Under certain conditions a Noetherian graded domain R is normal if
R(p), is normal for each p ∈Proj(R). If R =⊕i≧0Ri is reduced and
F0= {ri∕ui|ri,ui∈ Ri and ui∈ U where U is the set of all nonzero divisors} is
Noetherian, then the integral closure of R in the total quotient ring of R is also
graded.
