directSum(SimplicialModule) -- direct sum of simplicial modules

Function: directSum
Usage:

D = C1 ++ C2

D = directSum(C1,C2,...)

D = directSum(name1 => C1, name2 => C2, ...)
Inputs:
- Ci, a Simplicial Module,
Outputs:
- D, a Simplicial Module, the direct sum of the input simplicial modules

Description

The direct sum of two simplicial modules is another simplicial module.

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

i2 : C1 = simplicialModule(freeResolution coker vars S, Degeneracy => true)

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

o2 : SimplicialModule

i3 : D1 = C1 ++ simplicialModule(complex(S^13)[-2], 3, Degeneracy => true)

      1      4      23      59
o3 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o3 : SimplicialModule

i4 : isWellDefined D1

o4 = true

i5 : D1.?ss --knows to cache degeneracy maps if inputs have degeneracy maps

o5 = true

i6 : C2 = simplicialModule(ideal(a,b,c), 3, Degeneracy => true)

o6 = image | a b c | <-- image | a b c | <-- image | a b c | <-- image | a b c |<-- ...
                                                                  
     0                   1                   2                   3

o6 : SimplicialModule

i7 : C1 ++ C2

o7 = 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

o7 : SimplicialModule

i8 : assert isWellDefined(C1 ++ C2)

The direct sum of a sequence of simplicial modules can be computed as follows.

i9 : C3 = directSum(C1,C2,C2[-2])

o9 = 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 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 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 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 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 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 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 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 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 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 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 |           {1} | 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 0 0 |
                           1                                     {1} | 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 |
                                                                 {2} | 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 1 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 a b c 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 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 0 1 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 |
                                                           2                                                       {2} | 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 |
                                                                                                                   {2} | 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 |
                                                                                                                   {2} | 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 |
                                                                                                                   {2} | 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 |
                                                                                                                   {2} | 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 |
                                                                                                                   {2} | 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 |
                                                                                                                   {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 0 0 0 0 0 0 0 0 0 a b c 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 0 0 0 0 0 0 0 0 a b c 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 0 0 0 0 0 0 0 0 a b c 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 0 0 0 0 0 0 0 0 a b c |
                                                                                                              
                                                                                                             3

o9 : SimplicialModule

i10 : assert isWellDefined C3

The direct sum is an n-ary operator with projection and inclusion maps from each component satisfying appropriate identities.

i11 : C4 = directSum(first => C1, second => C2)

o11 = 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

o11 : SimplicialModule

i12 : C4_[first] -- inclusion map C1 --> C4

                                            1
o12 = 0 : image | 1 0 0 0 | <------------- S  : 0
                | 0 a b c |    {0} | 1 |
                               {1} | 0 |
                               {1} | 0 |
                               {1} | 0 |

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

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

                                                                                                                            20
      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 | <--------------------------------------------------- S   : 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} | 1 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 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 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 |
                {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 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 1 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 1 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 1 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 1 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 1 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 |    {1} | 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 1 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 |
                {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 1 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 1 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 1 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 1 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 1 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 1 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 1 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 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                         {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 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 |

o12 : SimplicialModuleMap

i13 : C4^[first] -- projection map C4 --> C1

           1
o13 = 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 |

o13 : SimplicialModuleMap

i14 : C4^[first] * C4_[first] == 1

o14 = true

i15 : C4^[second] * C4_[second] == 1

o15 = true

i16 : C4^[first] * C4_[second] == 0

o16 = true

i17 : C4^[second] * C4_[first] == 0

o17 = true

i18 : C4_[first] * C4^[first] + C4_[second] * C4^[second] == 1

o18 = true

There are two short exact sequences associated to a direct sum.

i19 : isShortExactSequence(C4^[first], C4_[second])

o19 = true

i20 : isShortExactSequence(C4^[second], C4_[first])

o20 = true

Given a simplicial module which is a direct sum, we obtain the component simplicial modules and their names (indices) as follows.

i21 : components C3

        1      4      10      20
o21 = {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 |<-- ..., 0 <-- 0 <-- image | a b
                                                                             
          2                   3                       0     1     2          
                                                                             
                                                                             
      -----------------------------------------------------------------------
      c | <-- image | a b c 0 0 0 0 0 0 |<-- ...}
                    | 0 0 0 a b c 0 0 0 |
                    | 0 0 0 0 0 0 a b c |
               
              3

o21 : List

i22 : indices C3

o22 = {0, 1, 2}

o22 : List

i23 : components C4

        1      4      10      20
o23 = {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

o23 : List

i24 : indices C4

o24 = {first, second}

o24 : List

Ways to use this method:

directSum(SimplicialModule) -- direct sum of simplicial modules

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:3215:0.

directSum(SimplicialModule) -- direct sum of simplicial modules

Description

See also

Ways to use this method: