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

- 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)

## See also

- working with sheaves -- information about coherent sheaves and total coordinate rings (a.k.a. Cox rings)
- ring(NormalToricVariety) -- make the total coordinate ring (a.k.a. Cox ring)
- sheaf(NormalToricVariety,Module) -- make a coherent sheaf

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NormalToricVarieties/SheavesDocumentation.m2:337:0.