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

## Synopsis

• 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 .