maps between chain complexes

maps between chain complexes

One way to make maps between chain complexes is by lifting maps between modules to resolutions of those modules. First we make some modules.

i1 : R = QQ[x,y];

i2 : M = coker vars R



o2 = cokernel | x y |



                            1

o2 : R-module, quotient of R

i3 : N = coker matrix {{x}}



o3 = cokernel | x |



                            1

o3 : R-module, quotient of R

Let's construct the natural map from N to M.

i4 : f = map(M,N)



o4 = | 1 |



o4 : Matrix

Let's lift the map to a map of free resolutions.

i5 : g = res f



o5 = 0



o5 : ChainComplexMap

We can check that it's a map of chain complexes this way.

i6 : g * (source g).dd == (target g).dd * g



o6 = true

We can form the mapping cone of g.

i7 : F = cone g



      1      3      2

o7 = R  <-- R  <-- R  <-- 0

                           

     0      1      2      3



o7 : ChainComplex

Since f is surjective, we know that F is quasi-isomorphic to (kernel f)[-1]. Let's check that.

i8 : prune HH_0 F



o8 = 0



o8 : R-module

i9 : prune HH_1 F



o9 = cokernel {1} | x |



                            1

o9 : R-module, quotient of R

i10 : prune kernel f



o10 = cokernel {1} | x |



                             1

o10 : R-module, quotient of R

There are more elementary ways to make maps between chain complexes. The identity map is available from id.

i11 : C = res M



       1      2      1

o11 = R  <-- R  <-- R  <-- 0

                            

      0      1      2      3



o11 : ChainComplex

i12 : id_C



           1             1

o12 = 0 : R  <--------- R  : 0

                | 1 |



           2                   2

      1 : R  <--------------- R  : 1

                {1} | 1 0 |

                {1} | 0 1 |



           1                 1

      2 : R  <------------- R  : 2

                {2} | 1 |



      3 : 0 <----- 0 : 3

               0



o12 : ChainComplexMap

i13 : x * id_C



           1             1

o13 = 0 : R  <--------- R  : 0

                | x |



           2                   2

      1 : R  <--------------- R  : 1

                {1} | x 0 |

                {1} | 0 x |



           1                 1

      2 : R  <------------- R  : 2

                {2} | x |



      3 : 0 <----- 0 : 3

               0



o13 : ChainComplexMap

We can use map or ** to construct natural maps between chain complexes.

i14 : map(C ** R^1/x,C)



                                     1

o14 = 0 : cokernel | x | <--------- R  : 0

                            | 1 |



                                                 2

      1 : cokernel {1} | x 0 | <--------------- R  : 1

                   {1} | 0 x |    {1} | 1 0 |

                                  {1} | 0 1 |



                                             1

      2 : cokernel {2} | x | <------------- R  : 2

                                {2} | 1 |



o14 : ChainComplexMap

i15 : g ** R^1/x



o15 = 0 : cokernel | x | <--------- cokernel | x | : 0

                            | 1 |



      1 : cokernel {1} | x 0 | <------------- cokernel {1} | x | : 1

                   {1} | 0 x |    {1} | 1 |

                                  {1} | 0 |



o15 : ChainComplexMap

There is a way to make a chain complex map by calling a function for each spot that needs a map.

i16 : q = map(C,C,i -> (i+1) * id_(C_i))



           1             1

o16 = 0 : R  <--------- R  : 0

                | 1 |



           2                   2

      1 : R  <--------------- R  : 1

                {1} | 2 0 |

                {1} | 0 2 |



           1                 1

      2 : R  <------------- R  : 2

                {2} | 3 |



      3 : 0 <----- 0 : 3

               0



o16 : ChainComplexMap

Of course, the formula we used doesn't yield a map of chain complexes.

i17 : C.dd * q == q * C.dd



o17 = false