  For a Mori dream space X, the Cox ring Cox(X) is a Noetherian Z^n-graded
normal domain for some n > 0. Let C(Cox(X)) be the cone (in R^n) which is
spanned by the vectors a \in Z^n such that Cox(X)_a \neq 0. Then C(Cox(X)) is
decomposed into a union of chambers. Berchtold and Hausen proved the existence
of such decompositions for affine integral domains over an algebraically closed
field. We shall give an elementary algebraic proof to this result in the case
where the homogeneous component of degree 0 is a field.
  Using such decompositions, we develop the Demazure construction for
Z^n-graded Krull domains. That is, under an assumption, we show that a
Z^n-graded Krull domain is isomorphic to the multi-section ring R(X; D_1,
\ldots, D_n) for certain normal projective variety X and Q-divisors D_1,...,D_n
on X.
