SimplicialModule ** Matrix -- create the tensor product of a simplicial module and a map of modules

Operator: **
Usage:

h = C ** f

h = f ** C
Inputs:
- C, a Simplicial Module, 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 Simplicial Modules, from $C \otimes M$ to $C \otimes N$

Description

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

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 = simplicialModule(koszulComplex vars R, Degeneracy => true)

      1      5      15      35      70
o4 = R  <-- R  <-- R   <-- R   <-- R  <-- ...
                                    
     0      1      2       3       4

o4 : SimplicialModule

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

o5 = | 1 |

o5 : Matrix

i6 : C ** f;

i7 : isSimplicialMorphism oo

o7 = true

i8 : f ** C;

i9 : isSimplicialMorphism oo

o9 = true

i10 : f' = random(R^2, R^{-1, -1, -1})

o10 = | 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
o10 : Matrix R  <-- R

i11 : C ** f';

i12 : f' ** C;

i13 : assert isWellDefined(C ** f')

i14 : assert isWellDefined(f' ** C)

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

i15 : f'' = random(source f', R^{-2,-2})

o15 = {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
o15 : Matrix R  <-- R

i16 : assert((C ** f') * (C ** f'') == C ** (f' * f''))

i17 : assert(C ** id_(R^{-1,-2,-3}) == id_(C ** R^{-1,-2,-3}))

Ways to use this method:

Matrix ** SimplicialModule
SimplicialModule ** Matrix -- create the tensor product of a simplicial module and a map of modules

The source of this document is in SimplicialModules/SimplicialModuleDOC.m2:2229:0.

SimplicialModule ** Matrix -- create the tensor product of a simplicial module and a map of modules

Description

See also

Ways to use this method: