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# isQuasiIsomorphism(ComplexMap) -- whether a map of complexes is a quasi-isomorphism

- Function: isQuasiIsomorphism
- Usage:
    `isQuasiIsomorphism f`
- Inputs:
  - f, a map of complexes,
- Optional inputs:
  - Concentration => a sequence, default value (-infinity,infinity), restricts attention to the induced maps indexed by elements in the given interval
- Outputs:
  - a Boolean value, that is true when $f$ is a morphism of complexes such that the induced maps on homology are all isomorphisms

## Description

The cone of a map $f \colon C \to D$ is acyclic exactly when $f$ is a quasi-isomorphism.

    i1 : S = ZZ/32003[x,y,z];

    i2 : C = freeResolution coker vars S

          1      3      3      1
    o2 = S  <-- S  <-- S  <-- S

         0      1      2      3

    o2 : Complex

    i3 : f = augmentationMap C

                                            1
    o3 = 0 : cokernel | x y z | <--------- S  : 0
                                   | 1 |

    o3 : ComplexMap

    i4 : assert isQuasiIsomorphism f

    i5 : assert(0 == prune HH cone f)

    i6 : assert isIsomorphism HH_0 f

    i7 : assert isIsomorphism HH_1 f

XXX TODO. Free resolutions of complexes produce quasi isomorphisms. (use example to doc of (freeResolution, Complex)).

    i8 : D = complex{random(S^2, S^{-3,-3,-4})}

          2      3
    o8 = S  <-- S

         0      1

    o8 : Complex

    i9 : prune HH D

                                                                                                                                                                                                                                                                                             1
    o9 = cokernel | x3-2709x2y-7293xy2+10856y3+3980x2z-11102xyz+7611y2z-4512xz2-2645yz2+11682z3 15989x2y+6239xy2-5484y3-3269x2z-11632xyz+8838y2z+5916xz2+4223yz2+2704z3   10021x2y2+5151xy3-12405y4-10408x2yz+15788xy2z-6266y3z-10577x2z2-9208xyz2-5782y2z2+10728xz3+13597yz3-11490z4 | <-- S
                  | 2794x2y+8580xy2+3768y3-10164x2z+12325xyz+3393y2z-3650xz2-6859yz2+7446z3     x3+13290x2y-12034xy2-2305y3+1291x2z-7317xyz-8020y2z-749xz2-8884yz2+8664z3 x2y2-14582xy3+13590y4-5127x2yz+14886xy2z+4549y3z-7332x2z2-3153xyz2-15321y2z2+1454xz3-6214yz3-3074z4         |
                                                                                                                                                                                                                                                                                            1
         0

    o9 : Complex

## See also

- Basic invariants and properties -- information about accessing basic features
- cone(ComplexMap) -- make the mapping cone of a morphism of chain complexes
- liftMapAlongQuasiIsomorphism -- lift a map of chain complexes along a quasi-isomorphism

## Ways to use this method:

- isQuasiIsomorphism(ComplexMap) -- whether a map of complexes is a quasi-isomorphism

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Complexes/ChainComplexMapDoc.m2:3230:0.