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

## Synopsis

• Usage:
augmentationMap C
• Inputs:
• Outputs:
• , a quasi-isomorphism whose source is $C$ and whose target is the module resolved by $C$

## 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)