components(Complex) -- list the components of a direct sum

Synopsis

Function: components
Usage:

components C
Inputs:
- C, a complex,
Outputs:
- a list, the component complexes of a direct sum (of complexes)

Description

A complex which has been constructed as a direct sum stores its component complexes.

i1 : S = ZZ/101[a,b,c];

i2 : C1 = freeResolution coker vars S

      1      3      3      1
o2 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o2 : Complex

i3 : C2 = complex (ideal(a,b,c))

o3 = image | a b c |
      
     0

o3 : Complex

i4 : D = C1 ++ C2

                            3      3      1
o4 = image | 1 0 0 0 | <-- S  <-- S  <-- S
           | 0 a b c |                    
                           1      2      3
     0

o4 : Complex

i5 : L = components D

       1      3      3      1
o5 = {S  <-- S  <-- S  <-- S , image | a b c |}
                                
      0      1      2      3   0

o5 : List

i6 : L_0 === C1

o6 = true

i7 : L_1 === C2

o7 = true

i8 : E = (mike => C1) ++ (greg => C2)

                            3      3      1
o8 = image | 1 0 0 0 | <-- S  <-- S  <-- S
           | 0 a b c |                    
                           1      2      3
     0

o8 : Complex

i9 : components E

       1      3      3      1
o9 = {S  <-- S  <-- S  <-- S , image | a b c |}
                                
      0      1      2      3   0

o9 : List

The names of the component complexes are called indices, and are used to access the relevant inclusion and projection maps.

i10 : indices D

o10 = {0, 1}

o10 : List

i11 : D^[0]

           1
o11 = 0 : S  <--------------- image | 1 0 0 0 | : 0
                | 1 0 0 0 |         | 0 a b c |

           3                     3
      1 : S  <----------------- S  : 1
                {1} | 1 0 0 |
                {1} | 0 1 0 |
                {1} | 0 0 1 |

           3                     3
      2 : S  <----------------- S  : 2
                {2} | 1 0 0 |
                {2} | 0 1 0 |
                {2} | 0 0 1 |

           1                 1
      3 : S  <------------- S  : 3
                {3} | 1 |

o11 : ComplexMap

i12 : indices E

o12 = {mike, greg}

o12 : List

i13 : E_[greg]

o13 = 0 : image | 1 0 0 0 | <----------------- image | a b c | : 0
                | 0 a b c |    {0} | 0 0 0 |
                               {1} | 1 0 0 |
                               {1} | 0 1 0 |
                               {1} | 0 0 1 |

o13 : ComplexMap

Ways to use this method: