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) |