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

## Synopsis

• Function: isQuasiIsomorphism
• Usage:
isQuasiIsomorphism f
• Inputs:
• f, ,
• Optional inputs:
• Concentration => , default value (-infinity,infinity), restricts attention to the induced maps indexed by elements in the given interval
• Outputs:
• , 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

TODO. Free resolutions of complexes produce quasi isomorphisms. (use example to doc of (resolution, 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