homomorphism' -- get the element of Hom from a homomorphism

Synopsis

• Usage:

  homomorphism' f

• Inputs:
  • f, a matrix, of the form $M \to N$
• Optional inputs:
  • DegreeLimit => ..., default value null, an integer or a list, see Hom(...,DegreeLimit=>...)
  • MinimalGenerators => a Boolean value, default value true, see Hom(...,MinimalGenerators=>...)
  • Strategy => a thing, default value null, see Hom(...,Strategy=>...)
• Outputs:
  • a matrix, the map $R^1 \to \mathrm{Hom}(M,N)$, corresponding to the map $f \in \mathrm{Hom}(M,N)$

Description

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : f = vars R ++ vars R

o2 = | x y z 0 0 0 |
     | 0 0 0 x y z |

             2      6
o2 : Matrix R  <-- R

i3 : g = homomorphism' f

o3 = {-1} | x |
     {-1} | 0 |
     {-1} | y |
     {-1} | 0 |
     {-1} | z |
     {-1} | 0 |
     {-1} | 0 |
     {-1} | x |
     {-1} | 0 |
     {-1} | y |
     {-1} | 0 |
     {-1} | z |

             12      1
o3 : Matrix R   <-- R

i4 : target g === Hom(source f, target f)

o4 = true

i5 : f' = homomorphism g

o5 = | x y z 0 0 0 |
     | 0 0 0 x y z |

             2      6
o5 : Matrix R  <-- R

i6 : f === f'

o6 = true

See also

• Hom -- module of homomorphisms
• homomorphism -- get the homomorphism from element of Hom
• adjoint -- the tensor-Hom adjunction maps