Hom(CoherentSheaf,CoherentSheaf) -- global Hom

Function: Hom
Usage:

Hom(F, G)
Inputs:
- F, a sheaf of rings or a coherent sheaf,
- G, a sheaf of rings or a coherent sheaf,
Optional inputs:
- DegreeLimit => ..., default value null,
- MinimalGenerators => ..., default value true,
- Strategy => ..., default value null,
Outputs:
- a module, representing a vector space over the coefficient field of $X$

Description

If F or G is a sheaf of rings, it is regarded as a sheaf of modules in the evident way.

Both F and G must be coherent sheaves on the same projective variety or scheme $X$.

The result is the sheaf associated to the graded module Hom(module F, module G).

i1 : R = QQ[a..d];

i2 : P3 = Proj R

o2 = P3

o2 : ProjectiveVariety

i3 : I = monomialCurveIdeal(R, {1,3,4})

                        3      2     2    2    3    2
o3 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o3 : Ideal of R

i4 : G = sheaf module I

o4 = image | bc-ad c3-bd2 ac2-b2d b3-a2c |

                                           1
o4 : coherent sheaf on P3, subsheaf of OO
                                         P3

i5 : Hom(OO_P3, G(3))

       7
o5 = QQ

o5 : QQ-module, free

i6 : HH^0(G(3))

       7
o6 = QQ

o6 : QQ-module, free

Ways to use this method:

Hom(CoherentSheaf,CoherentSheaf) -- global Hom
Hom(CoherentSheaf,SheafOfRings)
Hom(SheafOfRings,CoherentSheaf)
Hom(SheafOfRings,SheafOfRings)

The source of this document is in Varieties/doc-functors.m2:600:0.

Hom(CoherentSheaf,CoherentSheaf) -- global Hom

Description

See also

Ways to use this method: