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# id _ Complex -- the identity map of a chain complex

- Scripted functor: id
- Usage:
    `f = id_C`
- Inputs:
  - C, a complex,
- Outputs:
  - f, a map of complexes, the identity map from $C$ to itself

## Description

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

## See also

- map(Complex,Complex,ZZ) -- make the zero map or identity between chain complexes
- isWellDefined(ComplexMap) -- whether a map of chain complexes is well-defined
- isComplexMorphism(ComplexMap) -- whether a complex map is a morphism of complexes
- Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- homomorphism'(ComplexMap) -- get the element of Hom from a map of complexes

## Ways to use this method:

- id _ Complex -- the identity map of a chain complex

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Complexes/ChainComplexMapDoc.m2:514:0.