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

Synopsis

Function: sheaf
Usage:

sheaf (X, S)
Inputs:
- X, a normal toric variety,
- S, a ring, the total coordinate ring of X
Outputs:
- a sheaf of rings, the structure sheaf on X

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. For more information, see Subsection 5.3 in Cox-Little-Schenck's Toric Varieties.

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
       PP3

o2 : SheafOfRings

i3 : G = sheaf PP3

o3 = OO
       PP3

o3 : SheafOfRings

i4 : assert (F === G)

i5 : H = OO_PP3

o5 = OO
       PP3

o5 : SheafOfRings

i6 : assert (F === H)

Ways to use this method: