Complex ** Matrix -- create the tensor product of a complex and a map of modules

Synopsis

Operator: **
Usage:

h = C ** f

h = f ** C
Inputs:
- C, a complex, over a ring $R$
- f, a matrix, defining a homomorphism from the $R$-module $M$ to the $R$-module $N$
Outputs:
- h, a map of complexes, from $C \otimes M$ to $C \otimes N$

Description

For any chain complex $C$, a map $f \colon M \to N$ of $R$-modules induces a morphism $C \otimes f$ of chain complexes from $C \otimes M$ to $C \otimes N$. This method returns this map of chain complexes.

i1 : R = ZZ/101[a..d];

i2 : I = ideal(c^2-b*d, b*c-a*d, b^2-a*c)

             2                    2
o2 = ideal (c  - b*d, b*c - a*d, b  - a*c)

o2 : Ideal of R

i3 : J = ideal(I_0, I_1)

             2
o3 = ideal (c  - b*d, b*c - a*d)

o3 : Ideal of R

i4 : C = koszulComplex vars R

      1      4      6      4      1
o4 = R  <-- R  <-- R  <-- R  <-- R
                                  
     0      1      2      3      4

o4 : Complex

i5 : f = map(R^1/I, R^1/J, 1)

o5 = | 1 |

o5 : Matrix

i6 : C ** f

o6 = 0 : cokernel | c2-bd bc-ad b2-ac | <--------- cokernel | c2-bd bc-ad | : 0
                                           | 1 |

     1 : cokernel {1} | c2-bd bc-ad b2-ac 0     0     0     0     0     0     0     0     0     | <------------------- cokernel {1} | c2-bd bc-ad 0     0     0     0     0     0     | : 1
                  {1} | 0     0     0     c2-bd bc-ad b2-ac 0     0     0     0     0     0     |    {1} | 1 0 0 0 |            {1} | 0     0     c2-bd bc-ad 0     0     0     0     |
                  {1} | 0     0     0     0     0     0     c2-bd bc-ad b2-ac 0     0     0     |    {1} | 0 1 0 0 |            {1} | 0     0     0     0     c2-bd bc-ad 0     0     |
                  {1} | 0     0     0     0     0     0     0     0     0     c2-bd bc-ad b2-ac |    {1} | 0 0 1 0 |            {1} | 0     0     0     0     0     0     c2-bd bc-ad |
                                                                                                     {1} | 0 0 0 1 |

     2 : cokernel {2} | c2-bd bc-ad b2-ac 0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     | <----------------------- cokernel {2} | c2-bd bc-ad 0     0     0     0     0     0     0     0     0     0     | : 2
                  {2} | 0     0     0     c2-bd bc-ad b2-ac 0     0     0     0     0     0     0     0     0     0     0     0     |    {2} | 1 0 0 0 0 0 |            {2} | 0     0     c2-bd bc-ad 0     0     0     0     0     0     0     0     |
                  {2} | 0     0     0     0     0     0     c2-bd bc-ad b2-ac 0     0     0     0     0     0     0     0     0     |    {2} | 0 1 0 0 0 0 |            {2} | 0     0     0     0     c2-bd bc-ad 0     0     0     0     0     0     |
                  {2} | 0     0     0     0     0     0     0     0     0     c2-bd bc-ad b2-ac 0     0     0     0     0     0     |    {2} | 0 0 1 0 0 0 |            {2} | 0     0     0     0     0     0     c2-bd bc-ad 0     0     0     0     |
                  {2} | 0     0     0     0     0     0     0     0     0     0     0     0     c2-bd bc-ad b2-ac 0     0     0     |    {2} | 0 0 0 1 0 0 |            {2} | 0     0     0     0     0     0     0     0     c2-bd bc-ad 0     0     |
                  {2} | 0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     c2-bd bc-ad b2-ac |    {2} | 0 0 0 0 1 0 |            {2} | 0     0     0     0     0     0     0     0     0     0     c2-bd bc-ad |
                                                                                                                                         {2} | 0 0 0 0 0 1 |

     3 : cokernel {3} | c2-bd bc-ad b2-ac 0     0     0     0     0     0     0     0     0     | <------------------- cokernel {3} | c2-bd bc-ad 0     0     0     0     0     0     | : 3
                  {3} | 0     0     0     c2-bd bc-ad b2-ac 0     0     0     0     0     0     |    {3} | 1 0 0 0 |            {3} | 0     0     c2-bd bc-ad 0     0     0     0     |
                  {3} | 0     0     0     0     0     0     c2-bd bc-ad b2-ac 0     0     0     |    {3} | 0 1 0 0 |            {3} | 0     0     0     0     c2-bd bc-ad 0     0     |
                  {3} | 0     0     0     0     0     0     0     0     0     c2-bd bc-ad b2-ac |    {3} | 0 0 1 0 |            {3} | 0     0     0     0     0     0     c2-bd bc-ad |
                                                                                                     {3} | 0 0 0 1 |

     4 : cokernel {4} | c2-bd bc-ad b2-ac | <------------- cokernel {4} | c2-bd bc-ad | : 4
                                               {4} | 1 |

o6 : ComplexMap

i7 : f ** C

o7 = 0 : cokernel | c2-bd bc-ad b2-ac | <--------- cokernel | c2-bd bc-ad | : 0
                                           | 1 |

     1 : cokernel {1} | c2-bd 0     0     0     bc-ad 0     0     0     b2-ac 0     0     0     | <------------------- cokernel {1} | c2-bd 0     0     0     bc-ad 0     0     0     | : 1
                  {1} | 0     c2-bd 0     0     0     bc-ad 0     0     0     b2-ac 0     0     |    {1} | 1 0 0 0 |            {1} | 0     c2-bd 0     0     0     bc-ad 0     0     |
                  {1} | 0     0     c2-bd 0     0     0     bc-ad 0     0     0     b2-ac 0     |    {1} | 0 1 0 0 |            {1} | 0     0     c2-bd 0     0     0     bc-ad 0     |
                  {1} | 0     0     0     c2-bd 0     0     0     bc-ad 0     0     0     b2-ac |    {1} | 0 0 1 0 |            {1} | 0     0     0     c2-bd 0     0     0     bc-ad |
                                                                                                     {1} | 0 0 0 1 |

     2 : cokernel {2} | c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     0     b2-ac 0     0     0     0     0     | <----------------------- cokernel {2} | c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     0     | : 2
                  {2} | 0     c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     0     b2-ac 0     0     0     0     |    {2} | 1 0 0 0 0 0 |            {2} | 0     c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     |
                  {2} | 0     0     c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     0     b2-ac 0     0     0     |    {2} | 0 1 0 0 0 0 |            {2} | 0     0     c2-bd 0     0     0     0     0     bc-ad 0     0     0     |
                  {2} | 0     0     0     c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     0     b2-ac 0     0     |    {2} | 0 0 1 0 0 0 |            {2} | 0     0     0     c2-bd 0     0     0     0     0     bc-ad 0     0     |
                  {2} | 0     0     0     0     c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     0     b2-ac 0     |    {2} | 0 0 0 1 0 0 |            {2} | 0     0     0     0     c2-bd 0     0     0     0     0     bc-ad 0     |
                  {2} | 0     0     0     0     0     c2-bd 0     0     0     0     0     bc-ad 0     0     0     0     0     b2-ac |    {2} | 0 0 0 0 1 0 |            {2} | 0     0     0     0     0     c2-bd 0     0     0     0     0     bc-ad |
                                                                                                                                         {2} | 0 0 0 0 0 1 |

     3 : cokernel {3} | c2-bd 0     0     0     bc-ad 0     0     0     b2-ac 0     0     0     | <------------------- cokernel {3} | c2-bd 0     0     0     bc-ad 0     0     0     | : 3
                  {3} | 0     c2-bd 0     0     0     bc-ad 0     0     0     b2-ac 0     0     |    {3} | 1 0 0 0 |            {3} | 0     c2-bd 0     0     0     bc-ad 0     0     |
                  {3} | 0     0     c2-bd 0     0     0     bc-ad 0     0     0     b2-ac 0     |    {3} | 0 1 0 0 |            {3} | 0     0     c2-bd 0     0     0     bc-ad 0     |
                  {3} | 0     0     0     c2-bd 0     0     0     bc-ad 0     0     0     b2-ac |    {3} | 0 0 1 0 |            {3} | 0     0     0     c2-bd 0     0     0     bc-ad |
                                                                                                     {3} | 0 0 0 1 |

     4 : cokernel {4} | c2-bd bc-ad b2-ac | <------------- cokernel {4} | c2-bd bc-ad | : 4
                                               {4} | 1 |

o7 : ComplexMap

i8 : f' = random(R^2, R^{-1, -1, -1})

o8 = | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
     | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

             2       3
o8 : Matrix R  <--- R

i9 : C ** f'

          2                                                             3
o9 = 0 : R  <--------------------------------------------------------- R  : 0
               | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
               | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

          8                                                                                                                                                                                                                       12
     1 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 1
               {1} | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                0               0                0                |
               {1} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                0               0                0                |
               {1} | 0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                |
               {1} | 0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                |
               {1} | 0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                |
               {1} | 0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                |
               {1} | 0               0                0                0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
               {1} | 0               0                0                0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

          12                                                                                                                                                                                                                                                                                                                           18
     2 : R   <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 2
                {2} | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                0               0                0                0               0                0                0               0                0                |
                {2} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                0               0                0                0               0                0                0               0                0                |
                {2} | 0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                0               0                0                0               0                0                |
                {2} | 0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                0               0                0                0               0                0                |
                {2} | 0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                0               0                0                |
                {2} | 0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                0               0                0                |
                {2} | 0               0                0                0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                |
                {2} | 0               0                0                0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                |
                {2} | 0               0                0                0               0                0                0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                |
                {2} | 0               0                0                0               0                0                0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                |
                {2} | 0               0                0                0               0                0                0               0                0                0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
                {2} | 0               0                0                0               0                0                0               0                0                0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

          8                                                                                                                                                                                                                       12
     3 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 3
               {3} | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                0               0                0                |
               {3} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                0               0                0                |
               {3} | 0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                0               0                0                |
               {3} | 0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                0               0                0                |
               {3} | 0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  0               0                0                |
               {3} | 0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0               0                0                |
               {3} | 0               0                0                0               0                0                0               0                0                24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
               {3} | 0               0                0                0               0                0                0               0                0                19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

          2                                                                 3
     4 : R  <------------------------------------------------------------- R  : 4
               {4} | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
               {4} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

o9 : ComplexMap

i10 : f' ** C

           2                                                             3
o10 = 0 : R  <--------------------------------------------------------- R  : 0
                | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
                | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

           8                                                                                                                                                                                                                       12
      1 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 1
                {1} | 24a-36b-30c-29d 0               0               0               -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  0                0                0                |
                {1} | 0               24a-36b-30c-29d 0               0               0                -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  0                0                |
                {1} | 0               0               24a-36b-30c-29d 0               0                0                -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  0                |
                {1} | 0               0               0               24a-36b-30c-29d 0                0                0                -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  |
                {1} | 19a+19b-10c-29d 0               0               0               -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d 0                0                0                |
                {1} | 0               19a+19b-10c-29d 0               0               0                -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d 0                0                |
                {1} | 0               0               19a+19b-10c-29d 0               0                0                -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d 0                |
                {1} | 0               0               0               19a+19b-10c-29d 0                0                0                -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d |

           12                                                                                                                                                                                                                                                                                                                           18
      2 : R   <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 2
                 {2} | 24a-36b-30c-29d 0               0               0               0               0               -8a-22b-29c-24d  0                0                0                0                0                34a+19b-47c-39d  0                0                0                0                0                |
                 {2} | 0               24a-36b-30c-29d 0               0               0               0               0                -8a-22b-29c-24d  0                0                0                0                0                34a+19b-47c-39d  0                0                0                0                |
                 {2} | 0               0               24a-36b-30c-29d 0               0               0               0                0                -8a-22b-29c-24d  0                0                0                0                0                34a+19b-47c-39d  0                0                0                |
                 {2} | 0               0               0               24a-36b-30c-29d 0               0               0                0                0                -8a-22b-29c-24d  0                0                0                0                0                34a+19b-47c-39d  0                0                |
                 {2} | 0               0               0               0               24a-36b-30c-29d 0               0                0                0                0                -8a-22b-29c-24d  0                0                0                0                0                34a+19b-47c-39d  0                |
                 {2} | 0               0               0               0               0               24a-36b-30c-29d 0                0                0                0                0                -8a-22b-29c-24d  0                0                0                0                0                34a+19b-47c-39d  |
                 {2} | 19a+19b-10c-29d 0               0               0               0               0               -38a-16b+39c+21d 0                0                0                0                0                -18a-13b-43c-15d 0                0                0                0                0                |
                 {2} | 0               19a+19b-10c-29d 0               0               0               0               0                -38a-16b+39c+21d 0                0                0                0                0                -18a-13b-43c-15d 0                0                0                0                |
                 {2} | 0               0               19a+19b-10c-29d 0               0               0               0                0                -38a-16b+39c+21d 0                0                0                0                0                -18a-13b-43c-15d 0                0                0                |
                 {2} | 0               0               0               19a+19b-10c-29d 0               0               0                0                0                -38a-16b+39c+21d 0                0                0                0                0                -18a-13b-43c-15d 0                0                |
                 {2} | 0               0               0               0               19a+19b-10c-29d 0               0                0                0                0                -38a-16b+39c+21d 0                0                0                0                0                -18a-13b-43c-15d 0                |
                 {2} | 0               0               0               0               0               19a+19b-10c-29d 0                0                0                0                0                -38a-16b+39c+21d 0                0                0                0                0                -18a-13b-43c-15d |

           8                                                                                                                                                                                                                       12
      3 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R   : 3
                {3} | 24a-36b-30c-29d 0               0               0               -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  0                0                0                |
                {3} | 0               24a-36b-30c-29d 0               0               0                -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  0                0                |
                {3} | 0               0               24a-36b-30c-29d 0               0                0                -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  0                |
                {3} | 0               0               0               24a-36b-30c-29d 0                0                0                -8a-22b-29c-24d  0                0                0                34a+19b-47c-39d  |
                {3} | 19a+19b-10c-29d 0               0               0               -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d 0                0                0                |
                {3} | 0               19a+19b-10c-29d 0               0               0                -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d 0                0                |
                {3} | 0               0               19a+19b-10c-29d 0               0                0                -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d 0                |
                {3} | 0               0               0               19a+19b-10c-29d 0                0                0                -38a-16b+39c+21d 0                0                0                -18a-13b-43c-15d |

           2                                                                 3
      4 : R  <------------------------------------------------------------- R  : 4
                {4} | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |
                {4} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |

o10 : ComplexMap

i11 : assert isWellDefined(C ** f')

i12 : assert isWellDefined(f' ** C)

Tensoring with a complex defines a functor from the category of $R$-modules to the category of complexes over $R$.

i13 : f'' = random(source f', R^{-2,-2})

o13 = {1} | -28a-47b+38c+2d  -16a+7b+15c-23d |
      {1} | 16a+22b+45c-34d  39a+43b-17c-11d |
      {1} | -48a-47b+47c+19d 48a+36b+35c+11d |

              3       2
o13 : Matrix R  <--- R

i14 : assert((C ** f') * (C ** f'') == C ** (f' * f''))

i15 : assert(C ** id_(R^{-1,-2,-3}) == id_(C ** R^{-1,-2,-3}))

Complex ** Matrix -- create the tensor product of a complex and a map of modules

Synopsis

Description

See also