working with sheaves -- information about coherent sheaves and total coordinate rings (a.k.a. Cox rings)

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 categorical 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.

Sheaf-theoretic methods

• 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)
• ctop(CoherentSheaf)
• ch(CoherentSheaf)
• chi(CoherentSheaf)
• todd(CoherentSheaf)
• hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial