ComplexMap * ComplexMap -- composition of homomorphisms of complexes

Synopsis

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

Ways to use this method:

ComplexMap * ComplexMap -- composition of homomorphisms of complexes

ComplexMap * ComplexMap -- composition of homomorphisms of complexes

Synopsis

Description

See also

Ways to use this method: