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

- Usage:
    `isQuasiIsomorphism(phi)`
- Inputs:
  - phi,
- Optional inputs:
  - LengthLimit => ..., default value infinity, Option to check quasi-isomorphism only up to a certain point
- Outputs:
  - Boolean

## 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:

- `isQuasiIsomorphism(ChainComplexMap)`

## For the programmer

The object isQuasiIsomorphism is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ChainComplexExtras.m2:946:0.