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Complexes :: source(ComplexMap)

source(ComplexMap) -- get the source of a map of chain complexes

Synopsis

Description

Given a complex map $f : C \to D$ this method returns the chain complex $C$.

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

o1 = R

o1 : PolynomialRing
i2 : I = ideal(a^2, b^2, c^2)

             2   2   2
o2 = ideal (a , b , c )

o2 : Ideal of R
i3 : J = I + ideal(a*b*c)

             2   2   2
o3 = ideal (a , b , c , a*b*c)

o3 : Ideal of R
i4 : FI = freeResolution I

      1      3      3      1
o4 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o4 : Complex
i5 : FJ = freeResolution J

      1      4      6      3
o5 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o5 : Complex
i6 : f = randomComplexMap(FJ, FI, Cycle=>true)

          1              1
o6 = 0 : R  <---------- R  : 0
               | 24 |

          4                        3
     1 : R  <-------------------- R  : 1
               {2} | 24 0  0  |
               {2} | 0  24 0  |
               {2} | 0  0  24 |
               {3} | 0  0  0  |

          6                        3
     2 : R  <-------------------- R  : 2
               {4} | 24 0  0  |
               {4} | 0  0  0  |
               {4} | 0  0  0  |
               {4} | 0  24 0  |
               {4} | 0  0  0  |
               {4} | 0  0  24 |

          3                    1
     3 : R  <---------------- R  : 3
               {5} | 24c  |
               {5} | -24b |
               {5} | 24a  |

o6 : ComplexMap
i7 : source f

      1      3      3      1
o7 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o7 : Complex
i8 : assert isWellDefined f
i9 : assert isComplexMorphism f
i10 : assert(source f == FI)
i11 : assert(target f == FJ)

The differential in a complex is a map of chain complexes.

i12 : kk = coker vars R

o12 = cokernel | a b c d |

                             1
o12 : R-module, quotient of R
i13 : F = freeResolution kk

       1      4      6      4      1
o13 = R  <-- R  <-- R  <-- R  <-- R
                                   
      0      1      2      3      4

o13 : Complex
i14 : source dd^F == F

o14 = true
i15 : target dd^F == F

o15 = true
i16 : degree dd^F == -1

o16 = true

See also