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NormalToricVarieties :: sheaf(NormalToricVariety,Ring)

sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings

Synopsis

Description

The category of coherent sheaves on a normal toric variety is equivalent to the quotient category of finitely generated modules over the total coordinate ring by the full subcategory of torsion modules with respect to the irrelevant ideal. In particular, the total coordinate ring corresponds to the structure sheaf.

On projective space, we can make the structure sheaf in a few ways.

i1 : PP3 = toricProjectiveSpace 3;
i2 : F = sheaf (PP3, ring PP3)

o2 = OO
       normalToricVariety((({{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}(,({{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}} )))))

o2 : SheafOfRings
i3 : G = sheaf PP3

o3 = OO
       normalToricVariety((({{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}(,({{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}} )))))

o3 : SheafOfRings
i4 : assert (F === G)
i5 : H = OO_PP3

o5 = OO
       normalToricVariety((({{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}(,({{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}} )))))

o5 : SheafOfRings
i6 : assert (F === H)

See also