正规簇的全坐标环TC（X）是Cox引入并研究的环在复曲面簇上的推广。考虑一个正规投射簇X，其除子类群为Cl（X），并假设它是一个可生成的自由阿尔贝群.我们定义X的全坐标环为TC（X）= oplus_{D} H ^0（X，O_X（D）），其中上述和是对包含在Cl（X）的一个固定的完备表示系中的X的所有Weil因子取的。本文证明了对任意正规投射簇X，TC（X）是一个UFD，这是本文所证明的一个更一般的定理的推论。（Berchtold和Haussen独立地证明了光滑簇的唯一因子分解。我们还证明了对于X，P^2沿有限个共线点的爆破，TC（X）是Noether的。我们还给出了TC（X）不是Noetherian的例子，但对任意Weil因子D，oplus_nH^0（X，O（nD））是Noetherian的.
英文摘要：
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.
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