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resolutionMap -- map from a free resolution to the given complex

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 = Hom(freeResolution I, R^1/I)

o3 = cokernel {-3} | bc ab a2 0  0  0  | <-- cokernel {-2} | bc ab a2 0  0  0  0  0  0  | <-- cokernel | bc ab a2 |
              {-3} | 0  0  0  bc ab a2 |              {-2} | 0  0  0  bc ab a2 0  0  0  |      
                                                      {-2} | 0  0  0  0  0  0  bc ab a2 |     0
     -2                                       
                                             -1

o3 : Complex
 
i4 : assert all(min C .. max C, i -> not isFreeModule C_i)
 
i5 : fC = resolutionMap C

                                                                   2
o5 = -2 : cokernel {-3} | bc ab a2 0  0  0  | <------------------ R  : -2
                   {-3} | 0  0  0  bc ab a2 |    {-3} | -1 0  |
                                                 {-3} | 0  -1 |

                                                                                                9
     -1 : cokernel {-2} | bc ab a2 0  0  0  0  0  0  | <-------------------------------------- R  : -1
                   {-2} | 0  0  0  bc ab a2 0  0  0  |    {-2} | 1 0  0  -c -a 0  0  0  0  |
                   {-2} | 0  0  0  0  0  0  bc ab a2 |    {-2} | 0 -1 0  0  0  -b -a 0  0  |
                                                          {-2} | 0 0  -1 0  0  0  -c -b -a |

                                                                      14
     0 : cokernel | bc ab a2 | <------------------------------------ R   : 0
                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 |

o5 : ComplexMap
 
i6 : FC = resolution C

      2      9      14      9      2
o6 = R  <-- R  <-- R   <-- R  <-- R
                                   
     -2     -1     0       1      2

o6 : Complex
 
i7 : prune HH FC

o7 = cokernel {-3} | b a  0 | <-- cokernel {-1} | b a  0 0 0 0 0 0 0 0  0 0 | <-- cokernel | bc ab a2 |
              {-3} | 0 -c a |              {-1} | 0 -c b a 0 0 0 0 0 0  0 0 |      
                                           {-1} | 0 0  0 0 c a 0 0 0 0  0 0 |     0
     -2                                    {-1} | 0 0  0 0 0 0 b 0 0 a  0 0 |
                                           {-1} | 0 0  0 0 0 0 0 c a 0  0 0 |
                                           {-1} | 0 0  0 0 0 0 0 0 0 -c b a |
                                   
                                  -1

o7 : Complex
 
i8 : assert isQuasiIsomorphism fC
 
i9 : assert isFree FC
 
i10 : assert isWellDefined fC
 
i11 : assert(0 == coker fC) -- showing that fC is surjective.

The resolution of a short exact sequence is simply the zero complex.

i12 : J = ideal(a,b)

o12 = ideal (a, b)

o12 : Ideal of R
 
i13 : K = ideal(b^2,c)

              2
o13 = ideal (b , c)

o13 : Ideal of R
 
i14 : g1 = map(R^1/(J+K), R^1/J ++ R^1/K, {{1,-1}})

o14 = | 1 -1 |

o14 : Matrix
 
i15 : g2 = map(R^1/J ++ R^1/K, R^1/(intersect(J,K)), {{1},{1}})

o15 = | 1 |
      | 1 |

o15 : Matrix
 
i16 : D = complex{g1, g2}

o16 = cokernel | a b b2 c | <-- cokernel | a b 0  0 | <-- cokernel | bc ac b2 |
                                         | 0 0 b2 c |      
      0                                                   2
                                1

o16 : Complex
 
i17 : assert isWellDefined D
 
i18 : assert isShortExactSequence(g1,g2)
 
i19 : fD = resolutionMap D

o19 = 0

o19 : ComplexMap
 
i20 : assert isWellDefined fD
 
i21 : assert isQuasiIsomorphism fD
 
i22 : assert(0 == source fD) -- so fD is certainly not surjective!

To avoid recomputation, this method caches its value.

See also

Ways to use resolutionMap:

  • resolutionMap(Complex)

For the programmer

The object resolutionMap is a method function with options.

The source of this document is in Complexes/ChainComplexMapDoc.m2:1791:0.