2003/05/31 by E. Javier Elizondo, Kazuhiko Kurano, Kei-ichi Watanabe
Mathematics · #math.AG #math.AC #msc:14 #msc:13
published as J. Algebra. Vol. 276, Issue 2 , June 2004, Pages 625-637 · This is the final version that will appear in the Journal of Algebra. 11 pages. LaTex
arxiv created 2003/08/12 · arxiv updated 2009/11/30
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) = oplusD H0 (X, OX (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 P2 along a finite number of collinear points, TC(X) is Noetherian. We also give an example that TC(X) is not Noetherian but oplusn H0 (X, O(nD)) is Noetherian for any Weil divisor D.