The chain complexes together with complex morphisms forms a category. In particular, every chain complex has an identity map.
i1 : R = ZZ/101[x,y]/(x^3, y^3) o1 = R o1 : QuotientRing |
i2 : C = freeResolution(coker vars R, LengthLimit=>6) 1 2 3 4 5 6 7 o2 = R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 o2 : Complex |
i3 : f = id_C 1 1 o3 = 0 : R <--------- R : 0 | 1 | 2 2 1 : R <--------------- R : 1 {1} | 1 0 | {1} | 0 1 | 3 3 2 : R <----------------- R : 2 {2} | 1 0 0 | {3} | 0 1 0 | {3} | 0 0 1 | 4 4 3 : R <------------------- R : 3 {4} | 1 0 0 0 | {4} | 0 1 0 0 | {4} | 0 0 1 0 | {4} | 0 0 0 1 | 5 5 4 : R <--------------------- R : 4 {5} | 1 0 0 0 0 | {5} | 0 1 0 0 0 | {6} | 0 0 1 0 0 | {6} | 0 0 0 1 0 | {6} | 0 0 0 0 1 | 6 6 5 : R <----------------------- R : 5 {7} | 1 0 0 0 0 0 | {7} | 0 1 0 0 0 0 | {7} | 0 0 1 0 0 0 | {7} | 0 0 0 1 0 0 | {7} | 0 0 0 0 1 0 | {7} | 0 0 0 0 0 1 | 7 7 6 : R <------------------------- R : 6 {8} | 1 0 0 0 0 0 0 | {8} | 0 1 0 0 0 0 0 | {8} | 0 0 1 0 0 0 0 | {9} | 0 0 0 1 0 0 0 | {9} | 0 0 0 0 1 0 0 | {9} | 0 0 0 0 0 1 0 | {9} | 0 0 0 0 0 0 1 | o3 : ComplexMap |
i4 : assert isWellDefined f |
i5 : assert isComplexMorphism f |
The identity map corresponds to an element of the Hom complex.
i6 : R = ZZ/101[a,b,c] o6 = R o6 : PolynomialRing |
i7 : I = ideal(a^2, b^2, b*c, c^3) 2 2 3 o7 = ideal (a , b , b*c, c ) o7 : Ideal of R |
i8 : C = freeResolution I 1 4 5 2 o8 = R <-- R <-- R <-- R 0 1 2 3 o8 : Complex |
i9 : D = Hom(C, C) 2 13 34 46 34 13 2 o9 = R <-- R <-- R <-- R <-- R <-- R <-- R -3 -2 -1 0 1 2 3 o9 : Complex |
i10 : homomorphism' id_C 46 1 o10 = 0 : R <-------------- R : 0 {0} | 1 | {0} | 1 | {0} | 0 | {0} | 0 | {1} | 0 | {0} | 0 | {0} | 1 | {0} | 0 | {1} | 0 | {0} | 0 | {0} | 0 | {0} | 1 | {1} | 0 | {-1} | 0 | {-1} | 0 | {-1} | 0 | {0} | 1 | {0} | 1 | {1} | 0 | {1} | 0 | {1} | 0 | {2} | 0 | {-1} | 0 | {0} | 1 | {0} | 0 | {0} | 0 | {1} | 0 | {-1} | 0 | {0} | 0 | {0} | 1 | {0} | 0 | {1} | 0 | {-1} | 0 | {0} | 0 | {0} | 0 | {0} | 1 | {1} | 0 | {-2} | 0 | {-1} | 0 | {-1} | 0 | {-1} | 0 | {0} | 1 | {0} | 1 | {1} | 0 | {-1} | 0 | {0} | 1 | o10 : ComplexMap |