isQuasiIsomorphism(DGAlgebraMap) -- Determine whether a DG algebra map induces an isomorphism on homology

• Function: isQuasiIsomorphism

• Usage:

  b = isQuasiIsomorphism phi

• Inputs:
  • phi, an instance of the type DGAlgebraMap,
• Optional inputs:
  • Concentration => ..., default value (-infinity,infinity),
• Outputs:
  • b, a Boolean value, True when the induced chain map toComplexMap phi is a quasi-isomorphism on the underlying complexes.

i1 : R = QQ[x, y, z]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,3
      Differential => {x, y, z}

o2 : DGAlgebra

i3 : phi = identityDGAlgebraMap A

o3 = map (R[T   ..T   ], R[T   ..T   ], {T   , T   , T   , x, y, z})
             1,1   1,3      1,1   1,3     1,1   1,2   1,3

o3 : DGAlgebraMap

i4 : isQuasiIsomorphism phi

o4 = true