Hom(CoherentSheaf,CoherentSheaf) -- global Hom

Synopsis

• Function: Hom

• Usage:

  Hom(F,G)

• Inputs:
  • F, a coherent sheaf
  • G, a coherent sheaf, both should be sheaves on a projective variety or scheme X = Proj R
• Outputs:
  • a module, over the coefficient field of R

Description

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

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

See also

• sheafHom -- sheaf Hom
• Ext -- compute an Ext module
• sheafExt -- sheaf Ext of coherent sheaves