The total coordinate ring of a normal projective variety
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The total coordinate ring TC(X) of a normal variety is a generalization of the ring introduced and studied by Cox in connection with a toric variety. Consider a normal projective variety X with divisor class group Cl(X), and let us assume that it is a finitely generated free abelian group. We define the total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the sum as above is taken over all Weil divisors of X contained in a fixed complete system of representatives of Cl(X). We prove that for any normal projective variety X, TC(X) is a UFD, this is a corollary of a more general theorem that is proved in the paper. (Berchtold and Haussen proved the unique factorization for a smooth variety independently.) We also prove that for X, the blow up of P^2 along a finite number of collinear points, TC(X) is Noetherian. We also give an example that TC(X) is not Noetherian but oplus_n H^0 (X, O(nD)) is Noetherian for any Weil divisor D.
This is the final version that will appear in the Journal of Algebra. 11 pages. LaTex
This is the final version that will appear in the Journal of Algebra. 11 pages. LaTex