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Complexes :: augmentationMap

augmentationMap -- map from a free resolution to a module regarded as a complex

Synopsis

Description

Given a complex $C$, this method produces the natural quasi-isomorphism from a complex $F$ all of whose terms are free modules to the complex $C$. The algorithm used minimizes the ranks of the free modules in $F$.

i1 : R = ZZ/101[a,b,c];
i2 : I = ideal(a^2, a*b, b*c)

             2
o2 = ideal (a , a*b, b*c)

o2 : Ideal of R
i3 : C = freeResolution I

      1      3      2
o3 = R  <-- R  <-- R
                    
     0      1      2

o3 : Complex
i4 : f = augmentationMap C

                                           1
o4 = 0 : cokernel | a2 ab bc | <--------- R  : 0
                                  | 1 |

o4 : ComplexMap
i5 : assert isWellDefined f
i6 : assert isComplexMorphism f
i7 : assert isQuasiIsomorphism f
i8 : g = resolutionMap complex comodule I

                                           1
o8 = 0 : cokernel | a2 ab bc | <--------- R  : 0
                                  | 1 |

o8 : ComplexMap
i9 : assert(f == g)

See also

Ways to use augmentationMap :

For the programmer

The object augmentationMap is a method function.