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adjoint -- the tensor-Hom adjunction maps

Synopsis

Description

Recall that tensor(Module,Module) and Hom(Module,Module) form an adjoint pair, meaning that there is a natural isomorphism $$ \mathrm{Hom}(F\otimes G,H) \cong \mathrm{Hom}(F,\mathrm{Hom}(G,H)). $$

Synopsis

  • Usage:
    adjoint(f, F, G)
  • Inputs:
    • f, a matrix, a homomorphism $F \otimes G \to H$
    • F, a module, a free module;
    • G, a module, a free module;
  • Outputs:
    • a matrix, the adjoint homomorphism $F \to \mathrm{Hom}(G,H)$ 
i1 : R = QQ[x_1 .. x_24];
 
i2 : f = genericMatrix(R, 2, 4*3)

o2 = | x_1 x_3 x_5 x_7 x_9  x_11 x_13 x_15 x_17 x_19 x_21 x_23 |
     | x_2 x_4 x_6 x_8 x_10 x_12 x_14 x_16 x_18 x_20 x_22 x_24 |

             2      12
o2 : Matrix R  <-- R
 
i3 : isHomogeneous f

o3 = true
 
i4 : g = adjoint(f, R^4, R^3)

o4 = | x_1 x_7  x_13 x_19 |
     | x_2 x_8  x_14 x_20 |
     | x_3 x_9  x_15 x_21 |
     | x_4 x_10 x_16 x_22 |
     | x_5 x_11 x_17 x_23 |
     | x_6 x_12 x_18 x_24 |

             6      4
o4 : Matrix R  <-- R

If f is homogeneous, and source f === F ** G (including the grading), then the resulting matrix will be homogeneous.

i5 : g = adjoint(f, R^4, R^{-1,-1,-1})

o5 = {-1} | x_1 x_7  x_13 x_19 |
     {-1} | x_2 x_8  x_14 x_20 |
     {-1} | x_3 x_9  x_15 x_21 |
     {-1} | x_4 x_10 x_16 x_22 |
     {-1} | x_5 x_11 x_17 x_23 |
     {-1} | x_6 x_12 x_18 x_24 |

             6      4
o5 : Matrix R  <-- R
 
i6 : isHomogeneous g

o6 = true
 
i7 : f === adjoint'(g, R^{-1,-1,-1}, R^2)

o7 = true

Synopsis

  • Usage:
    adjoint'(g, G, H)
  • Inputs:
    • g, a matrix, a homomorphism $F \to \mathrm{Hom}(G,H)$ between modules
    • G, a module, a free module;
    • H, a module, a free module;
  • Outputs:
    • a matrix, the adjoint homomorphism $F \otimes G \to H$

If g is homogeneous, and target g === Hom(G,H) (including the grading), then the resulting matrix will be homogeneous.

i8 : R = QQ[x_1 .. x_12];
 
i9 : g = genericMatrix(R, 6, 2)

o9 = | x_1 x_7  |
     | x_2 x_8  |
     | x_3 x_9  |
     | x_4 x_10 |
     | x_5 x_11 |
     | x_6 x_12 |

             6      2
o9 : Matrix R  <-- R
 
i10 : f = adjoint'(g, R^2, R^3)

o10 = | x_1 x_4 x_7 x_10 |
      | x_2 x_5 x_8 x_11 |
      | x_3 x_6 x_9 x_12 |

              3      4
o10 : Matrix R  <-- R
 
i11 : isHomogeneous f

o11 = true
 
i12 : g === adjoint(f, R^{-1,-1}, R^2)

o12 = true

See also

Ways to use adjoint :

For the programmer

The object adjoint is a method function with options.