For the identity map on a normal toric variety, the underlying map of lattices is given by the identity matrix. For more information on this correspondence, see Theorem 3.3.4 in Cox-Little-Schenck's Toric Varieties.
i1 : X = hirzebruchSurface 2; |
i2 : f = id_X
o2 = | 1 0 |
| 0 1 |
o2 : ToricMap X <--- X
|
i3 : assert (isWellDefined f and source f === X and
target f === X and matrix f === id_(ZZ^(dim X)))
|
Identity maps also arise as edge cases of the canonical projections and inclusions associated to Cartesian products.
i4 : X2 = X ** X; |
i5 : X2^[0,1]
o5 = | 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
o5 : ToricMap X2 <--- X2
|
i6 : X2_[0,1]
o6 = | 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
o6 : ToricMap X2 <--- X2
|
i7 : assert (X2^[0,1] == id_X2 and X2_[0,1] == id_X2) |