minimalPresentation(Complex) -- minimal presentation of all terms in a complex

Synopsis

Function: minimalPresentation
Usage:

D = minimalPresentation C

D = prune C

h = minimalPresentation f

h = prune f
Inputs:
- C, a complex, or $f$ a map of complexes
Optional inputs:
- Exclude => ..., default value {}, unused
Outputs:
- D, a complex, isomorphic to the input, where each term is replaced by a minimally presented model, or $h$ a map of complexes where the source and target are minimally presented
Consequences:
- The isomorphism $g : D \to C$ is available as g = D.cache.pruningMap. The inverse isomorphism can be obtained as g^-1

Description

This is frequently useful to make the output of certain operations readable or understandable. This operation is functorial, applying both to complexes and complex maps.

In particular, homology often needs to be pruned to be understood. For instance, this is useful for recognizing when terms given by subquotient modules are actually zero.

i1 : S = ZZ/101[a,b,c,d,e];

i2 : I = ideal(a,b) * ideal(c,d,e)

o2 = ideal (a*c, a*d, a*e, b*c, b*d, b*e)

o2 : Ideal of S

i3 : F = dual freeResolution I

      1      5      9      6      1
o3 = S  <-- S  <-- S  <-- S  <-- S
                                  
     -4     -3     -2     -1     0

o3 : Complex

i4 : C = HH F

o4 = cokernel {-5} | -e d -c -b a | <-- subquotient ({-4} | d c  0 b  0 0  a 0  0 0 |, {-4} | d -c -b a 0  0  0  0  0 |) <-- subquotient ({-3} | c b a  0  0  0  0 |, {-3} | -b a  0  0  0  0 |) <-- subquotient ({-2} | ac |, {-2} | ac |) <-- image 0
                                                     {-4} | e 0  c 0  b 0  0 a  0 0 |  {-4} | e 0  0  0 -c -b a  0  0 |                   {-3} | d 0 0  b  0  a  0 |  {-3} | 0  0  -b a  0  0 |                   {-2} | bc |  {-2} | bc |       
     -4                                              {-4} | 0 -e d 0  0 b  0 0  a 0 |  {-4} | 0 e  0  0 -d 0  0  -b a |                   {-3} | 0 d 0  -c 0  0  0 |  {-3} | -d 0  c  0  0  0 |                   {-2} | ad |  {-2} | ad |      0
                                                     {-4} | 0 0  0 -e d -c 0 0  0 a |  {-4} | 0 0  e  0 0  -d 0  c  0 |                   {-3} | 0 0 -d 0  0  c  0 |  {-3} | 0  -d 0  c  0  0 |                   {-2} | bd |  {-2} | bd |
                                                     {-4} | 0 0  0 0  0 0  e -d c b |  {-4} | 0 0  0  e 0  0  -d 0  c |                   {-3} | e 0 0  0  b  0  a |  {-3} | 0  0  0  0  -b a |                   {-2} | ae |  {-2} | ae |
                                                                                                                                          {-3} | 0 e 0  0  -c 0  0 |  {-3} | -e 0  0  0  c  0 |                   {-2} | be |  {-2} | be |
                                        -3                                                                                                {-3} | 0 0 -e 0  0  0  c |  {-3} | 0  -e 0  0  0  c |       
                                                                                                                                          {-3} | 0 0 0  e  -d 0  0 |  {-3} | 0  0  -e 0  d  0 |      -1
                                                                                                                                          {-3} | 0 0 0  0  0  -e d |  {-3} | 0  0  0  -e 0  d |
                                                                                                                              
                                                                                                                             -2

o4 : Complex

i5 : D = prune C

o5 = cokernel {-5} | e d c b a | <-- cokernel {-3} | e d c | <-- cokernel {-2} | b a |
                                                                  
     -4                              -3                          -2

o5 : Complex

i6 : g = D.cache.pruningMap

o6 = -4 : cokernel {-5} | -e d -c -b a | <-------------- cokernel {-5} | e d c b a | : -4
                                            {-5} | 1 |

     -3 : subquotient ({-4} | d c  0 b  0 0  a 0  0 0 |, {-4} | d -c -b a 0  0  0  0  0 |) <-------------- cokernel {-3} | e d c | : -3
                       {-4} | e 0  c 0  b 0  0 a  0 0 |  {-4} | e 0  0  0 -c -b a  0  0 |     {-3} | 0 |
                       {-4} | 0 -e d 0  0 b  0 0  a 0 |  {-4} | 0 e  0  0 -d 0  0  -b a |     {-3} | 0 |
                       {-4} | 0 0  0 -e d -c 0 0  0 a |  {-4} | 0 0  e  0 0  -d 0  c  0 |     {-3} | 0 |
                       {-4} | 0 0  0 0  0 0  e -d c b |  {-4} | 0 0  0  e 0  0  -d 0  c |     {-3} | 0 |
                                                                                              {-3} | 0 |
                                                                                              {-3} | 0 |
                                                                                              {-3} | 0 |
                                                                                              {-3} | 0 |
                                                                                              {-3} | 0 |
                                                                                              {-3} | 1 |

     -2 : subquotient ({-3} | c b a  0  0  0  0 |, {-3} | -b a  0  0  0  0 |) <-------------- cokernel {-2} | b a | : -2
                       {-3} | d 0 0  b  0  a  0 |  {-3} | 0  0  -b a  0  0 |     {-2} | 1 |
                       {-3} | 0 d 0  -c 0  0  0 |  {-3} | -d 0  c  0  0  0 |     {-2} | 0 |
                       {-3} | 0 0 -d 0  0  c  0 |  {-3} | 0  -d 0  c  0  0 |     {-2} | 0 |
                       {-3} | e 0 0  0  b  0  a |  {-3} | 0  0  0  0  -b a |     {-2} | 0 |
                       {-3} | 0 e 0  0  -c 0  0 |  {-3} | -e 0  0  0  c  0 |     {-2} | 0 |
                       {-3} | 0 0 -e 0  0  0  c |  {-3} | 0  -e 0  0  0  c |     {-2} | 0 |
                       {-3} | 0 0 0  e  -d 0  0 |  {-3} | 0  0  -e 0  d  0 |     {-2} | 0 |
                       {-3} | 0 0 0  0  0  -e d |  {-3} | 0  0  0  -e 0  d |

o6 : ComplexMap

i7 : assert isWellDefined g

i8 : assert isComplexMorphism g

i9 : assert (target g == C)

i10 : assert (source g == D)

i11 : g^-1

o11 = -4 : cokernel {-5} | e d c b a | <-------------- cokernel {-5} | -e d -c -b a | : -4
                                          {-5} | 1 |

      -3 : cokernel {-3} | e d c | <-------------------------------- subquotient ({-4} | d c  0 b  0 0  a 0  0 0 |, {-4} | d -c -b a 0  0  0  0  0 |) : -3
                                      {-3} | 0 0 0 0 0 0 0 0 0 1 |                {-4} | e 0  c 0  b 0  0 a  0 0 |  {-4} | e 0  0  0 -c -b a  0  0 |
                                                                                  {-4} | 0 -e d 0  0 b  0 0  a 0 |  {-4} | 0 e  0  0 -d 0  0  -b a |
                                                                                  {-4} | 0 0  0 -e d -c 0 0  0 a |  {-4} | 0 0  e  0 0  -d 0  c  0 |
                                                                                  {-4} | 0 0  0 0  0 0  e -d c b |  {-4} | 0 0  0  e 0  0  -d 0  c |

      -2 : cokernel {-2} | b a | <-------------------------- subquotient ({-3} | c b a  0  0  0  0 |, {-3} | -b a  0  0  0  0 |) : -2
                                    {-2} | 1 0 0 0 0 0 0 |                {-3} | d 0 0  b  0  a  0 |  {-3} | 0  0  -b a  0  0 |
                                                                          {-3} | 0 d 0  -c 0  0  0 |  {-3} | -d 0  c  0  0  0 |
                                                                          {-3} | 0 0 -d 0  0  c  0 |  {-3} | 0  -d 0  c  0  0 |
                                                                          {-3} | e 0 0  0  b  0  a |  {-3} | 0  0  0  0  -b a |
                                                                          {-3} | 0 e 0  0  -c 0  0 |  {-3} | -e 0  0  0  c  0 |
                                                                          {-3} | 0 0 -e 0  0  0  c |  {-3} | 0  -e 0  0  0  c |
                                                                          {-3} | 0 0 0  e  -d 0  0 |  {-3} | 0  0  -e 0  d  0 |
                                                                          {-3} | 0 0 0  0  0  -e d |  {-3} | 0  0  0  -e 0  d |

o11 : ComplexMap

i12 : assert(g*g^-1 == 1 and g^-1*g == 1)

The image of a map of complexes also becomes more understandable via pruning.

i13 : S = ZZ/101[a,b,c];

i14 : I = ideal(a^2,b^2,c^2);

o14 : Ideal of S

i15 : J = I + ideal(a*b*c);

o15 : Ideal of S

i16 : FI = freeResolution I

       1      3      3      1
o16 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o16 : Complex

i17 : FJ = freeResolution J

       1      4      6      3
o17 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o17 : Complex

i18 : f = randomComplexMap(FJ, FI ** S^{-1}, Cycle => true)

           1                       1
o18 = 0 : S  <------------------- S  : 0
                | 32a-36b-30c |

           4                                                   3
      1 : S  <----------------------------------------------- S  : 1
                {2} | 32a-36b-30c 0           0           |
                {2} | 0           32a-36b-30c 0           |
                {2} | 0           0           32a-36b-30c |
                {3} | 0           0           0           |

           6                                                  3
      2 : S  <---------------------------------------------- S  : 2
                {4} | 32a-36b+42c -19c       19c         |
                {4} | 29b+10c     19b-29c    -19b+22c    |
                {4} | -29a+29c    -19a-24c   19a+8c      |
                {4} | -10b        32a-7b-30c -22b        |
                {4} | -10a-29b    29a+24b    -22a-8b     |
                {4} | -29a        24a        24a-36b-30c |

           3                                        1
      3 : S  <------------------------------------ S  : 3
                {5} | 19a2+19b2+32ac-36bc+42c2 |
                {5} | -22a2-32ab+7b2+30bc-10c2 |
                {5} | 24a2-36ab-24b2-30ac-29c2 |

o18 : ComplexMap

i19 : C = image f

o19 = image | 32a-36b-30c | <-- image {2} | 32a-36b-30c 0           0           | <-- image {4} | 32a-36b+42c -19c       19c         | <-- image {5} | 19a2+19b2+32ac-36bc+42c2 |
                                      {2} | 0           32a-36b-30c 0           |           {4} | 29b+10c     19b-29c    -19b+22c    |           {5} | -22a2-32ab+7b2+30bc-10c2 |
      0                               {2} | 0           0           32a-36b-30c |           {4} | -29a+29c    -19a-24c   19a+8c      |           {5} | 24a2-36ab-24b2-30ac-29c2 |
                                      {3} | 0           0           0           |           {4} | -10b        32a-7b-30c -22b        |      
                                                                                            {4} | -10a-29b    29a+24b    -22a-8b     |     3
                                1                                                           {4} | -29a        24a        24a-36b-30c |
                                                                                       
                                                                                      2

o19 : Complex

i20 : D = prune C

       1      3      3      1
o20 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o20 : Complex

i21 : g = D.cache.pruningMap

                                                1
o21 = 0 : image | 32a-36b-30c | <------------- S  : 0
                                   {1} | 1 |

                                                                                3
      1 : image {2} | 32a-36b-30c 0           0           | <----------------- S  : 1
                {2} | 0           32a-36b-30c 0           |    {3} | 1 0 0 |
                {2} | 0           0           32a-36b-30c |    {3} | 0 1 0 |
                {3} | 0           0           0           |    {3} | 0 0 1 |

                                                                               3
      2 : image {4} | 32a-36b+42c -19c       19c         | <----------------- S  : 2
                {4} | 29b+10c     19b-29c    -19b+22c    |    {5} | 1 0 0 |
                {4} | -29a+29c    -19a-24c   19a+8c      |    {5} | 0 1 0 |
                {4} | -10b        32a-7b-30c -22b        |    {5} | 0 0 1 |
                {4} | -10a-29b    29a+24b    -22a-8b     |
                {4} | -29a        24a        24a-36b-30c |

                                                                 1
      3 : image {5} | 19a2+19b2+32ac-36bc+42c2 | <------------- S  : 3
                {5} | -22a2-32ab+7b2+30bc-10c2 |    {7} | 1 |
                {5} | 24a2-36ab-24b2-30ac-29c2 |

o21 : ComplexMap

i22 : assert isWellDefined g

i23 : assert isComplexMorphism g

i24 : assert (target g == C)

i25 : assert (source g == D)

i26 : g^-1

           1
o26 = 0 : S  <------------- image | 32a-36b-30c | : 0
                {1} | 1 |

           3
      1 : S  <----------------- image {2} | 32a-36b-30c 0           0           | : 1
                {3} | 1 0 0 |         {2} | 0           32a-36b-30c 0           |
                {3} | 0 1 0 |         {2} | 0           0           32a-36b-30c |
                {3} | 0 0 1 |         {3} | 0           0           0           |

           3
      2 : S  <----------------- image {4} | 32a-36b+42c -19c       19c         | : 2
                {5} | 1 0 0 |         {4} | 29b+10c     19b-29c    -19b+22c    |
                {5} | 0 1 0 |         {4} | -29a+29c    -19a-24c   19a+8c      |
                {5} | 0 0 1 |         {4} | -10b        32a-7b-30c -22b        |
                                      {4} | -10a-29b    29a+24b    -22a-8b     |
                                      {4} | -29a        24a        24a-36b-30c |

           1
      3 : S  <------------- image {5} | 19a2+19b2+32ac-36bc+42c2 | : 3
                {7} | 1 |         {5} | -22a2-32ab+7b2+30bc-10c2 |
                                  {5} | 24a2-36ab-24b2-30ac-29c2 |

o26 : ComplexMap

i27 : assert(g*g^-1 == 1 and g^-1*g == 1)

One can directly prune the map of complexes $f$.

i28 : h = prune f

           1                       1
o28 = 0 : S  <------------------- S  : 0
                | 32a-36b-30c |

           4                                                   3
      1 : S  <----------------------------------------------- S  : 1
                {2} | 32a-36b-30c 0           0           |
                {2} | 0           32a-36b-30c 0           |
                {2} | 0           0           32a-36b-30c |
                {3} | 0           0           0           |

           6                                                  3
      2 : S  <---------------------------------------------- S  : 2
                {4} | 32a-36b+42c -19c       19c         |
                {4} | 29b+10c     19b-29c    -19b+22c    |
                {4} | -29a+29c    -19a-24c   19a+8c      |
                {4} | -10b        32a-7b-30c -22b        |
                {4} | -10a-29b    29a+24b    -22a-8b     |
                {4} | -29a        24a        24a-36b-30c |

           3                                        1
      3 : S  <------------------------------------ S  : 3
                {5} | 19a2+19b2+32ac-36bc+42c2 |
                {5} | -22a2-32ab+7b2+30bc-10c2 |
                {5} | 24a2-36ab-24b2-30ac-29c2 |

o28 : ComplexMap

i29 : assert(source h === prune source f)

i30 : assert(target h === prune target f)

Ways to use this method: