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

## Synopsis

• Function: sheaf
• Usage:
sheaf (X, S)
• Inputs:
• X, ,
• S, a ring, the total coordinate ring of X
• Outputs:
• , 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)