components(SimplicialModule) -- list the components of a direct sum

Function: components
Usage:

components C
Inputs:
- C, a Simplicial Module,
Outputs:
- a list, the component simplicial modules of a direct sum (of simplicial modules)

Description

A simplicial module which has been constructed as a direct sum stores its component simplicial modules.

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

i2 : C1 = simplicialModule freeResolution coker vars S

      1      4      10      20
o2 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o2 : SimplicialModule

i3 : C2 = simplicialModule(complex (ideal(a,b,c)), 3)

o3 = image | a b c | <-- image | a b c | <-- image | a b c | <-- image | a b c |<-- ...
                                                                  
     0                   1                   2                   3

o3 : SimplicialModule

i4 : D = C1 ++ C2

o4 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |<-- ...
           | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {0} | 0 0 0 0 a b c |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                           1                                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {0} | 0 0 0 0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                                                           2                                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                                                                                                             {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                                                                                                             {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |
                                                                                                        
                                                                                                       3

o4 : SimplicialModule

i5 : L = components D

       1      4      10      20
o5 = {S  <-- S  <-- S   <-- S  <-- ..., image | a b c | <-- image | a b c |
                                                                           
      0      1      2       3           0                   1              
     ------------------------------------------------------------------------
     <-- image | a b c | <-- image | a b c |<-- ...}
                              
         2                   3

o5 : List

i6 : L_0 === C1

o6 = true

i7 : L_1 === C2

o7 = true

i8 : E = (peanut => C1) ++ (butter => C2)

o8 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |<-- ...
           | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {0} | 0 0 0 0 a b c |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                           1                                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                 {0} | 0 0 0 0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                                                           2                                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                                                                                                             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                                                                                                             {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                                                                                                             {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |
                                                                                                        
                                                                                                       3

o8 : SimplicialModule

i9 : components E

       1      4      10      20
o9 = {S  <-- S  <-- S   <-- S  <-- ..., image | a b c | <-- image | a b c |
                                                                           
      0      1      2       3           0                   1              
     ------------------------------------------------------------------------
     <-- image | a b c | <-- image | a b c |<-- ...}
                              
         2                   3

o9 : List

The names of the component simplicial modules are called indices, and are used to access the relevant inclusion and projection maps.

i10 : indices D

o10 = {0, 1}

o10 : List

i11 : D^[0]

           1
o11 = 0 : S  <--------------- image | 1 0 0 0 | : 0
                | 1 0 0 0 |         | 0 a b c |

           4
      1 : S  <------------------------- image {0} | 1 0 0 0 0 0 0 | : 1
                {0} | 1 0 0 0 0 0 0 |         {1} | 0 1 0 0 0 0 0 |
                {1} | 0 1 0 0 0 0 0 |         {1} | 0 0 1 0 0 0 0 |
                {1} | 0 0 1 0 0 0 0 |         {1} | 0 0 0 1 0 0 0 |
                {1} | 0 0 0 1 0 0 0 |         {0} | 0 0 0 0 a b c |

           10
      2 : S   <------------------------------------- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 | : 2
                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |         {0} | 0 0 0 0 0 0 0 0 0 0 a b c |

           20
      3 : S   <--------------------------------------------------------- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | : 3
                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |         {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |

o11 : SimplicialModuleMap

i12 : indices E

o12 = {peanut, butter}

o12 : List

i13 : E_[butter]

o13 = 0 : image | 1 0 0 0 | <----------------- image | a b c | : 0
                | 0 a b c |    {0} | 0 0 0 |
                               {1} | 1 0 0 |
                               {1} | 0 1 0 |
                               {1} | 0 0 1 |

      1 : image {0} | 1 0 0 0 0 0 0 | <----------------- image | a b c | : 1
                {1} | 0 1 0 0 0 0 0 |    {0} | 0 0 0 |
                {1} | 0 0 1 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 1 0 0 0 |    {1} | 0 0 0 |
                {0} | 0 0 0 0 a b c |    {1} | 0 0 0 |
                                         {1} | 1 0 0 |
                                         {1} | 0 1 0 |
                                         {1} | 0 0 1 |

      2 : image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | : 2
                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |
                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {1} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 |
                {0} | 0 0 0 0 0 0 0 0 0 0 a b c |    {2} | 0 0 0 |
                                                     {1} | 1 0 0 |
                                                     {1} | 0 1 0 |
                                                     {1} | 0 0 1 |

      3 : image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | : 3
                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |
                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {2} | 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 |
                {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 |
                                                                         {1} | 1 0 0 |
                                                                         {1} | 0 1 0 |
                                                                         {1} | 0 0 1 |

o13 : SimplicialModuleMap

Ways to use this method:

components(SimplicialModule) -- list the components of a direct sum

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:913:0.

components(SimplicialModule) -- list the components of a direct sum

Description

See also

Ways to use this method: