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SimplicialModule ** SimplicialModule -- tensor product of simplicial modules

Description

The tensor product is a simplicial module $D$ whose $i$th component is the tensor product of the degree i components of $C1$ and $C2$. The face/degeneracy maps are given by the tensor products of the face and degeneracy maps of the original objects.

As the next example illustrates, the simplicial tensor product in general does not normalize to give an object that is isomorphic to the classically defined tensor product of complexes.

i1 : S = ZZ/101[a..c]

o1 = S

o1 : PolynomialRing
 
i2 : Ca = simplicialModule(complex {matrix{{a}}}, 3)

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

o2 : SimplicialModule
 
i3 : Cb = simplicialModule(complex {matrix{{b}}}, 3)

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

o3 : SimplicialModule
 
i4 : Cc = simplicialModule(complex {matrix{{c}}}, 3)

      1      2      3      4
o4 = S  <-- S  <-- S  <-- S <-- ...
                           
     0      1      2      3

o4 : SimplicialModule
 
i5 : Cab = Cb ** Ca

      1      4      9      16
o5 = S  <-- S  <-- S  <-- S  <-- ...
                           
     0      1      2      3

o5 : SimplicialModule
 
i6 : dd^Cab

               1                    4
o6 = (0, 0) : S  <---------------- S  : (1, 0)
                    | 1 a b ab |

               1                   4
     (0, 1) : S  <--------------- S  : (1, 1)
                    | 1 0 0 0 |

               4                                  9
     (1, 0) : S  <------------------------------ S  : (2, 0)
                    {0} | 1 a 0 b ab 0 0 0 0 |
                    {1} | 0 0 1 0 0  b 0 0 0 |
                    {1} | 0 0 0 0 0  0 1 a 0 |
                    {2} | 0 0 0 0 0  0 0 0 1 |

               4                                 9
     (1, 1) : S  <----------------------------- S  : (2, 1)
                    {0} | 1 0 0 0 0 0 0 0 0 |
                    {1} | 0 1 1 0 0 0 0 0 0 |
                    {1} | 0 0 0 1 0 0 1 0 0 |
                    {2} | 0 0 0 0 1 1 0 1 1 |

               4                                 9
     (1, 2) : S  <----------------------------- S  : (2, 2)
                    {0} | 1 0 0 0 0 0 0 0 0 |
                    {1} | 0 1 0 0 0 0 0 0 0 |
                    {1} | 0 0 0 1 0 0 0 0 0 |
                    {2} | 0 0 0 0 1 0 0 0 0 |

               9                                                16
     (2, 0) : S  <-------------------------------------------- S   : (3, 0)
                    {0} | 1 a 0 0 b ab 0 0 0 0 0 0 0 0 0 0 |
                    {1} | 0 0 1 0 0 0  b 0 0 0 0 0 0 0 0 0 |
                    {1} | 0 0 0 1 0 0  0 b 0 0 0 0 0 0 0 0 |
                    {1} | 0 0 0 0 0 0  0 0 1 a 0 0 0 0 0 0 |
                    {2} | 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 1 0 0 0 0 |
                    {1} | 0 0 0 0 0 0  0 0 0 0 0 0 1 a 0 0 |
                    {2} | 0 0 0 0 0 0  0 0 0 0 0 0 0 0 1 0 |
                    {2} | 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 1 |

               9                                               16
     (2, 1) : S  <------------------------------------------- S   : (3, 1)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {1} | 0 1 1 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 |
                    {1} | 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 |
                    {2} | 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 1 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 |
                    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

               9                                               16
     (2, 2) : S  <------------------------------------------- S   : (3, 2)
                    {0} | 1 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 |
                    {1} | 0 0 1 1 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 |
                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 |
                    {1} | 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
                    {2} | 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 |

               9                                               16
     (2, 3) : S  <------------------------------------------- S   : (3, 3)
                    {0} | 1 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 |
                    {1} | 0 0 1 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 |
                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                    {2} | 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 1 0 0 0 0 0 0 0 |
                    {2} | 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 1 0 0 0 0 0 |

o6 : SimplicialModuleMap
 
i7 : (prune normalize Cab).dd

          1                  3
o7 = 0 : S  <-------------- S  : 1
               | a b ab |

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

o7 : ComplexMap
 
i8 : assert isWellDefined Cab
 
i9 : Cabc = Cc ** Cab

      1      8      27      64
o9 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o9 : SimplicialModule
 
i10 : Cc ** Cb ** Ca

       1      8      27      64
o10 = S  <-- S  <-- S   <-- S  <-- ...
                             
      0      1      2       3

o10 : SimplicialModule
 
i11 : dd^(nC = prune normalize Cabc)

           1                              7
o11 = 0 : S  <-------------------------- S  : 1
                | a b c ab ac bc abc |

           7                                                         12
      1 : S  <----------------------------------------------------- S   : 2
                {1} | -b -b -c -c 0  0  -bc -bc -bc -bc -bc -bc |
                {1} | a  0  0  0  -c -c ac  0   0   0   0   0   |
                {1} | 0  0  a  0  b  0  0   0   ab  0   0   0   |
                {2} | 0  1  0  0  0  0  0   c   0   0   0   0   |
                {2} | 0  0  0  1  0  0  0   0   0   b   0   0   |
                {2} | 0  0  0  0  0  1  0   0   0   0   a   0   |
                {3} | 0  0  0  0  0  0  0   0   0   0   0   1   |

           12                                 6
      2 : S   <----------------------------- S  : 3
                 {2} | -c 0  c  0  0  -c |
                 {2} | 0  -c 0  0  0  0  |
                 {2} | 0  0  -b -b 0  0  |
                 {2} | 0  0  0  0  -b 0  |
                 {2} | 0  0  a  0  0  0  |
                 {2} | 0  0  0  0  0  -a |
                 {3} | 1  0  0  0  0  0  |
                 {3} | 0  1  0  0  0  0  |
                 {3} | 0  0  0  1  0  0  |
                 {3} | 0  0  0  0  1  0  |
                 {3} | 0  0  0  0  0  1  |
                 {3} | 0  0  0  0  0  0  |

o11 : ComplexMap
 
i12 : assert isWellDefined nC

If one of the arguments is a module, it is considered as a complex concentrated in homological degree 0.

i13 : Cabc ** (S^1/(a,b,c));
 
i14 : S^2 ** Cabc

       2      16      54      128
o14 = S  <-- S   <-- S   <-- S   <-- ...
                              
      0      1       2       3

o14 : SimplicialModule

Because the tensor product can be regarded as the total complex of a double complex, each term of the tensor product comes with pairs of indices, labelling the summands.

i15 : indices Cabc_1

o15 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {1, 3}}

o15 : List
 
i16 : components Cabc_1

        1   1   1   1   1   1   1   1
o16 = {S , S , S , S , S , S , S , S }

o16 : List
 
i17 : Cabc_1_[{1,0}]

o17 = {0} | 0 |
      {1} | 0 |
      {1} | 0 |
      {2} | 0 |
      {1} | 1 |
      {2} | 0 |
      {2} | 0 |
      {3} | 0 |

              8      1
o17 : Matrix S  <-- S
 
i18 : indices Cabc_2

o18 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {0, 5}, {0, 6}, {0, 7}, {0,
      -----------------------------------------------------------------------
      8}, {1, 0}, {1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1,
      -----------------------------------------------------------------------
      8}, {2, 0}, {2, 1}, {2, 2}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {2,
      -----------------------------------------------------------------------
      8}}

o18 : List
 
i19 : components Cabc_2

        1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1 
o19 = {S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S ,
      -----------------------------------------------------------------------
       1   1   1   1   1   1   1   1   1   1
      S , S , S , S , S , S , S , S , S , S }

o19 : List
 
i20 : Cabc_2_[{0,2}]

o20 = {0} | 0 |
      {1} | 0 |
      {1} | 1 |
      {1} | 0 |
      {2} | 0 |
      {2} | 0 |
      {1} | 0 |
      {2} | 0 |
      {2} | 0 |
      {1} | 0 |
      {2} | 0 |
      {2} | 0 |
      {2} | 0 |
      {3} | 0 |
      {3} | 0 |
      {2} | 0 |
      {3} | 0 |
      {3} | 0 |
      {1} | 0 |
      {2} | 0 |
      {2} | 0 |
      {2} | 0 |
      {3} | 0 |
      {3} | 0 |
      {2} | 0 |
      {3} | 0 |
      {3} | 0 |

              27      1
o20 : Matrix S   <-- S

See also

Ways to use this method:

  • Complex ** SimplicialModule
  • Module ** SimplicialModule
  • SimplicialModule ** Complex
  • SimplicialModule ** Module
  • SimplicialModule ** SimplicialModule -- tensor product of simplicial modules

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:976:0.