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.

- ring(NormalToricVariety) -- make the total coordinate ring (a.k.a. Cox ring)
- ideal(NormalToricVariety) -- make the irrelevant ideal
- sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings
- sheaf(NormalToricVariety,Module) -- make a coherent sheaf
- OO ToricDivisor -- make the associated rank-one reflexive sheaf
- cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms
- HH^ZZ(NormalToricVariety,CoherentSheaf) -- compute the cohomology of a coherent sheaf
- intersectionRing(NormalToricVariety) -- make the rational Chow ring
- chern(CoherentSheaf) -- compute the Chern class of a coherent sheaf
- ctop(CoherentSheaf) -- compute the top Chern class of a coherent sheaf
- ch(CoherentSheaf) -- compute the Chern character of a coherent sheaf
- chi(CoherentSheaf) -- compute the Euler characteristic of a coherent sheaf
- todd(CoherentSheaf) -- compute the Todd class of a coherent sheaf
- hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial