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ChainComplexExtras :: isQuasiIsomorphism

isQuasiIsomorphism -- Test to see if the ChainComplexMap is a quasi-isomorphism.

Synopsis

Description

A quasi-isomorphism is a chain map that is an isomorphism in homology. Mapping cones currently do not work properly for complexes concentrated in one degree, so isQuasiIsomorphism could return bad information in that case.
i1 : R = ZZ/101[a,b,c]

o1 = R

o1 : PolynomialRing
i2 : kRes = res coker vars R

      1      3      3      1
o2 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o2 : ChainComplex
i3 : multBya = extend(kRes,kRes,matrix{{a}})

          1             1
o3 = 0 : R  <--------- R  : 0
               | a |

          3                     3
     1 : R  <----------------- R  : 1
               {1} | a b c |
               {1} | 0 0 0 |
               {1} | 0 0 0 |

          3         3
     2 : R  <----- R  : 2
               0

          1         1
     3 : R  <----- R  : 3
               0

     4 : 0 <----- 0 : 4
              0

o3 : ChainComplexMap
i4 : isQuasiIsomorphism(multBya)

o4 = false
i5 : F = extend(kRes,kRes,matrix{{1_R}})

          1             1
o5 = 0 : R  <--------- R  : 0
               | 1 |

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

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

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

     4 : 0 <----- 0 : 4
              0

o5 : ChainComplexMap
i6 : isQuasiIsomorphism(F)

o6 = true

Ways to use isQuasiIsomorphism :

For the programmer

The object isQuasiIsomorphism is a method function with options.