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directSum(Complex) -- direct sum of complexes

Description

The direct sum of two complexes is another complex.

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 : C1 ++ complex(S^13)[-2]

      1      3      16      1
o3 = S  <-- S  <-- S   <-- S
                            
     0      1      2       3

o3 : Complex
 
i4 : C2 = complex (ideal(a,b,c))

o4 = image | a b c |
      
     0

o4 : Complex
 
i5 : C1 ++ C2

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

o5 : Complex
 
i6 : assert isWellDefined(C1 ++ C2)

The direct sum of a sequence of complexes can be computed as follows.

i7 : C3 = directSum(C1,C2,C2[-2])

                            3                                    1
o7 = image | 1 0 0 0 | <-- S  <-- image {2} | 1 0 0 0 0 0 | <-- S
           | 0 a b c |                  {2} | 0 1 0 0 0 0 |      
                           1            {2} | 0 0 1 0 0 0 |     3
     0                                  {0} | 0 0 0 a b c |
                                   
                                  2

o7 : Complex
 
i8 : assert isWellDefined C3

The direct sum is an n-ary operator with projection and inclusion maps from each component satisfying appropriate identities.

i9 : C4 = directSum(first => C1, second => C2)

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

o9 : Complex
 
i10 : C4_[first] -- inclusion map C1 --> C4

                                            1
o10 = 0 : image | 1 0 0 0 | <------------- S  : 0
                | 0 a b c |    {0} | 1 |
                               {1} | 0 |
                               {1} | 0 |
                               {1} | 0 |

           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 |

o10 : ComplexMap
 
i11 : C4^[first] -- projection map C4 --> C1

           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 : C4^[first] * C4_[first] == 1

o12 = true
 
i13 : C4^[second] * C4_[second] == 1

o13 = true
 
i14 : C4^[first] * C4_[second] == 0

o14 = true
 
i15 : C4^[second] * C4_[first] == 0

o15 = true
 
i16 : C4_[first] * C4^[first] + C4_[second] * C4^[second] == 1

o16 = true

There are two short exact sequences associated to a direct sum.

i17 : isShortExactSequence(C4^[first], C4_[second])

o17 = true
 
i18 : isShortExactSequence(C4^[second], C4_[first])

o18 = true

Given a complex which is a direct sum, we obtain the component complexes and their names (indices) as follows.

i19 : components C3

        1      3      3      1
o19 = {S  <-- S  <-- S  <-- S , image | a b c |, image | a b c |}
                                                  
       0      1      2      3   0                2

o19 : List
 
i20 : indices C3

o20 = {0, 1, 2}

o20 : List
 
i21 : components C4

        1      3      3      1
o21 = {S  <-- S  <-- S  <-- S , image | a b c |}
                                 
       0      1      2      3   0

o21 : List
 
i22 : indices C4

o22 = {first, second}

o22 : List

See also

Ways to use this method:

The source of this document is in Complexes/ChainComplexDoc.m2:2167:0.