nestedTangent -- calculate first order deformations for nested deformations

Description

The module image F0Y should be contained in image F0X. The output N is a matrix over the same ring as F0X whose columns form a basis for (a graded piece of) the of the submodule of Hom(image F0X,coker F0X)++Hom(image F0Y,coker F0Y) encoding first order deformations. Selection of graded pieces is done in the same manner as with basis. If the selected pieces are infinite dimensional, an error occurs.

By default, trivial first order deformations are killed, and the result is the tangent space to the versal deformation of the pair image F0X,image F0Y. If F0X has more than one row, or the option ModuleDeformation is set true this is treated as a pair of modules, otherwise it is treated as a pair of rings. If the option IncludeTrivial is set to true then the result is instead the tangent space of the appropriate nested Hilbert or Quot scheme.

For example, consider embedded deformations of the fat points in the plane defined by the square and cube of the homogeneous maximal ideal:

i1 : R=QQ[x,y];

i2 : F0Y=basis(3,R);

             1      4
o2 : Matrix R  <-- R

i3 : F0X=basis(2,R);

             1      3
o3 : Matrix R  <-- R

i4 : nestedTangent(F0X,F0Y,IncludeTrivial=>true)

o4 = {-2} | y x 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  |
     {-2} | 0 0 y x 0 0 0  0  0  0  0  0  0  0  0  0  0  0  |
     {-2} | 0 0 0 0 y x 0  0  0  0  0  0  0  0  0  0  0  0  |
     {-3} | 0 0 0 0 0 0 y2 xy x2 0  0  0  0  0  0  0  0  0  |
     {-3} | 0 0 0 0 0 0 0  0  0  y2 xy x2 0  0  0  0  0  0  |
     {-3} | 0 0 0 0 0 0 0  0  0  0  0  0  y2 xy x2 0  0  0  |
     {-3} | 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  y2 xy x2 |

             7      18
o4 : Matrix R  <-- R

The tangent space of the nested Hilbert scheme is 18-dimensional.