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# ComplexMap * ComplexMap -- composition of homomorphisms of complexes

- Operator: *
- Usage:
    `f = h * g`
- Inputs:
  - h, a map of complexes, if a ring element or integer, then we multiply the ring element by the appropriate identity map
  - g, a map of complexes,
- Outputs:
  - f, a map of complexes, the composition of $g$ followed by $h$

## Description

If $g_i : C_i \rightarrow D_{d+i}$, and $h_j : D_j \rightarrow E_{e+j}$, then the composition corresponds to $f_i := h_{d+i} * g_i : C_i \rightarrow E_{i+d+e}$. In particular, the degree of the composition $f$ is the sum of the degrees of $g$ and $h$.

    i1 : R = ZZ/101[a..d]

    o1 = R

    o1 : PolynomialRing

    i2 : C = freeResolution coker vars R

          1      4      6      4      1
    o2 = R  <-- R  <-- R  <-- R  <-- R

         0      1      2      3      4

    o2 : Complex

    i3 : 3 * dd^C

              1                       4
    o3 = 0 : R  <------------------- R  : 1
                   | 3a 3b 3c 3d |

              4                                       6
         1 : R  <----------------------------------- R  : 2
                   {1} | -3b -3c 0   -3d 0   0   |
                   {1} | 3a  0   -3c 0   -3d 0   |
                   {1} | 0   3a  3b  0   0   -3d |
                   {1} | 0   0   0   3a  3b  3c  |

              6                               4
         2 : R  <--------------------------- R  : 3
                   {2} | 3c  3d  0   0   |
                   {2} | -3b 0   3d  0   |
                   {2} | 3a  0   0   3d  |
                   {2} | 0   -3b -3c 0   |
                   {2} | 0   3a  0   -3c |
                   {2} | 0   0   3a  3b  |

              4                   1
         3 : R  <--------------- R  : 4
                   {3} | -3d |
                   {3} | 3c  |
                   {3} | -3b |
                   {3} | 3a  |

    o3 : ComplexMap

    i4 : 0 * dd^C

    o4 = 0

    o4 : ComplexMap

    i5 : dd^C * dd^C

    o5 = 0

    o5 : ComplexMap

## See also

- Making maps between chain complexes -- information about the basic constructors
- arithmetic with complex maps -- perform arithmetic operations on complex maps

## Ways to use this method:

- ComplexMap * ComplexMap -- composition of homomorphisms of complexes

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Complexes/ChainComplexMapDoc.m2:928:0.