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

## Synopsis

• Usage:
sheaf (X, M)
• Function: sheaf
• Inputs:
• M, , a graded module over the total coordinate ring
• Outputs:
• , 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.

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) 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}} ))))) 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}} ))))) 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 : coherent sheaf on 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}} )))))``` `i3 : FF7 = hirzebruchSurface 7;` ```i4 : G = sheaf (FF7, (ring FF7)^{{1,0},{0,1}}) 1 1 o4 = OO (1, 0) ++ OO (0, 1) normalToricVariety((({{1, 0}, {0, 1}, {-1, 7}, {0, -1}}(,({{0, 1}, {0, 3}, {1, 2}, {2, 3}} ))))) normalToricVariety((({{1, 0}, {0, 1}, {-1, 7}, {0, -1}}(,({{0, 1}, {0, 3}, {1, 2}, {2, 3}} ))))) o4 : coherent sheaf on normalToricVariety((({{1, 0}, {0, 1}, {-1, 7}, {0, -1}}(,({{0, 1}, {0, 3}, {1, 2}, {2, 3}} )))))```