sheaf(NormalToricVariety,Module) -- make a coherent sheaf

Synopsis

Function: sheaf
Usage:

sheaf (X, M)
Inputs:
- X, a normal toric variety,
- M, a module, a graded module over the total coordinate ring
Outputs:
- a coherent sheaf, the coherent sheaf on X corresponding to M

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, each finitely generated module over the total coordinate ring corresponds to coherent sheaf on the normal toric variety and every coherent sheaf arises in this manner. For more information, see Subsection 5.3 in Cox-Little-Schenck's Toric Varieties.

Free modules correspond to reflexive sheaves.

i1 : PP3 = toricProjectiveSpace 3;

i2 : F = sheaf (PP3, (ring PP3)^{{1},{2},{3}})

          1            1            1
o2 = OO    (1) ++ OO    (2) ++ OO    (3)
       PP3          PP3          PP3

o2 : coherent sheaf on PP3, free of rank 3

i3 : FF7 = hirzebruchSurface 7;

i4 : G = sheaf (FF7, (ring FF7)^{{1,0},{0,1}})

          1                1
o4 = OO     (1, 0) ++ OO     (0, 1)
       FF7              FF7

o4 : coherent sheaf on FF7, free of rank 2

Ways to use this method: