The generalized Cox construction associates with an algebraic variety a remarkable invariant—its total coordinate ring, or Cox ring. In this note, we give a new proof of the factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on the notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ring is again factorially graded, but factoriality may be lost.
Key wordstotal coordinate ring Cox ring algebraic variety factorial ring graded factoriality divisor class group torsion Weil divisor Cartier divisor
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- 4.J. Hausen, Cox Rings and Combinatorics, II, arXiv: math. AG/0801.3995.Google Scholar
- 5.D. A. Timashev, Homogeneous Spaces and Equivariant Embeddings, arXiv: math. AG/0602228.Google Scholar
- 6.F. Knop, H. Kraft, D. Luna, and Th. Vust, Algebraische Transformationsgruppen und Invariantentheorie, in DMV Sem. (Birkhäuser, Basel, 1989), Vol. 13, pp. 63–75.Google Scholar
- 7.F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1997), Vol. 1673.Google Scholar
- 8.P. Samuel, Lectures on Unique Factorization Domains, in Notes by M. Pavman Murthy. Tata Inst. Fund. Res. Lectures on Math. (Tata Institute of Fundamental Research, Bombay, 1964), Vol. 30.Google Scholar