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NormalToricVarieties :: Total coordinate rings and coherent sheaves

Total coordinate rings and coherent sheaves

David A. Cox introduced the total coordinate ring S of a normal toric variety X and the irrelevant ideal B. The polynomial ring S has one variable for each ray in the associated fan and a natural grading by the class group. The monomial ideal B encodes the maximal cones. The following results of Cox indicate the significance of the pair (S,B).

  • The variety X is a good categorial quotient of Spec(S) - V(B) by a suitable group action.
  • The category of coherent sheaves on X is equivalent to the quotient of the category of finitely generated graded S-modules by the full subcategory of B-torsion modules.

In particular, we may represent any coherent sheaf on X by giving a finitely generated graded S-module.

The following methods allow one to make and manipulate coherent sheaves on normal toric varieties.

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