RingMap SimplicialModule -- apply a ring map to a simplicial module

Operator: SPACE
Usage:

phi C
Inputs:
- phi, a ring map, whose source is a ring $R$, and whose target is a ring $S$
- C, a Simplicial Module, over the ring $R$
Outputs:
- a Simplicial Module, over the ring $S$

Description

We illustrate the image of a simplicial module under a ring map.

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : S = QQ[s,t]

o2 = S

o2 : PolynomialRing

i3 : phi = map(S, R, {s, s+t, t})

o3 = map (S, R, {s, s + t, t})

o3 : RingMap S <-- R

i4 : I = ideal(x^3, x^2*y, x*y^4, y*z^5)

             3   2      4     5
o4 = ideal (x , x y, x*y , y*z )

o4 : Ideal of R

i5 : C = simplicialModule(freeResolution I, 3, Degeneracy => true)

      1      5      13      26
o5 = R  <-- R  <-- R   <-- R  <-- ...
                            
     0      1      2       3

o5 : SimplicialModule

i6 : D = phi C

      1      5      13      26
o6 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o6 : SimplicialModule

i7 : isWellDefined D

o7 = true

i8 : dd^D

               1                                                      5
o8 = (0, 0) : S  <-------------------------------------------------- S  : (1, 0)
                    | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 |

               1                     5
     (0, 1) : S  <----------------- S  : (1, 1)
                    | 1 0 0 0 0 |

               5                                                                                                              13
     (1, 0) : S  <---------------------------------------------------------------------------------------------------------- S   : (2, 0)
                    {0} | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 0 0 0 0 0    0                0   0                 |
                    {3} | 0 0  0      0                       0      1 0 0 0 -s-t 0                0   0                 |
                    {3} | 0 0  0      0                       0      0 1 0 0 s    -s3-3s2t-3st2-t3 -t5 0                 |
                    {5} | 0 0  0      0                       0      0 0 1 0 0    s                0   -t5               |
                    {6} | 0 0  0      0                       0      0 0 0 1 0    0                s2  s4+3s3t+3s2t2+st3 |

               5                                         13
     (1, 1) : S  <------------------------------------- S   : (2, 1)
                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                    {3} | 0 1 0 0 0 1 0 0 0 0 0 0 0 |
                    {3} | 0 0 1 0 0 0 1 0 0 0 0 0 0 |
                    {5} | 0 0 0 1 0 0 0 1 0 0 0 0 0 |
                    {6} | 0 0 0 0 1 0 0 0 1 0 0 0 0 |

               5                                         13
     (1, 2) : S  <------------------------------------- S   : (2, 2)
                    {0} | 1 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 |
                    {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                    {5} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                    {6} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |

               13                                                                                                                                                                                            26
     (2, 0) : S   <---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : (3, 0)
                     {0}  | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 0 0 0 0 0 0 0 0 0    0                0   0                 0    0                0   0                 0 0 0 0 0                |
                     {3}  | 0 0  0      0                       0      1 0 0 0 0 0 0 0 -s-t 0                0   0                 0    0                0   0                 0 0 0 0 0                |
                     {3}  | 0 0  0      0                       0      0 1 0 0 0 0 0 0 s    -s3-3s2t-3st2-t3 -t5 0                 0    0                0   0                 0 0 0 0 0                |
                     {5}  | 0 0  0      0                       0      0 0 1 0 0 0 0 0 0    s                0   -t5               0    0                0   0                 0 0 0 0 0                |
                     {6}  | 0 0  0      0                       0      0 0 0 1 0 0 0 0 0    0                s2  s4+3s3t+3s2t2+st3 0    0                0   0                 0 0 0 0 0                |
                     {3}  | 0 0  0      0                       0      0 0 0 0 1 0 0 0 0    0                0   0                 -s-t 0                0   0                 0 0 0 0 0                |
                     {3}  | 0 0  0      0                       0      0 0 0 0 0 1 0 0 0    0                0   0                 s    -s3-3s2t-3st2-t3 -t5 0                 0 0 0 0 0                |
                     {5}  | 0 0  0      0                       0      0 0 0 0 0 0 1 0 0    0                0   0                 0    s                0   -t5               0 0 0 0 0                |
                     {6}  | 0 0  0      0                       0      0 0 0 0 0 0 0 1 0    0                0   0                 0    0                s2  s4+3s3t+3s2t2+st3 0 0 0 0 0                |
                     {4}  | 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                |
                     {6}  | 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 t5               |
                     {8}  | 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 -s3-3s2t-3st2-t3 |
                     {10} | 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 s                |

               13                                                                    26
     (2, 1) : S   <---------------------------------------------------------------- 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 0 0 0 |
                     {3}  | 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 |
                     {3}  | 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 |
                     {5}  | 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 |
                     {6}  | 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 |
                     {3}  | 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 |
                     {3}  | 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 |
                     {5}  | 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 |
                     {6}  | 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 |
                     {4}  | 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 |
                     {6}  | 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 |
                     {8}  | 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 |
                     {10} | 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 |

               13                                                                    26
     (2, 2) : S   <---------------------------------------------------------------- S   : (3, 2)
                     {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 |
                     {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 0 0 |
                     {3}  | 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 |
                     {5}  | 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 |
                     {6}  | 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 |
                     {3}  | 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 |
                     {3}  | 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 |
                     {5}  | 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 |
                     {6}  | 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 |
                     {4}  | 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 |
                     {6}  | 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 |
                     {8}  | 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 |
                     {10} | 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 |

               13                                                                    26
     (2, 3) : S   <---------------------------------------------------------------- S   : (3, 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 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 0 0 |
                     {3}  | 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 |
                     {5}  | 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 |
                     {6}  | 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 |
                     {3}  | 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 |
                     {3}  | 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 |
                     {5}  | 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 |
                     {6}  | 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 |
                     {4}  | 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 |
                     {6}  | 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 |
                     {8}  | 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 |
                     {10} | 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 |

o8 : SimplicialModuleMap

When the ring map doesn't preserve homogeneity, the DegreeMap option is needed to determine the degrees of the image free modules in the simplicial module.

i9 : R = ZZ/101[a..d]

o9 = R

o9 : PolynomialRing

i10 : S = ZZ/101[s,t]

o10 = S

o10 : PolynomialRing

i11 : phi = map(S, R, {s^4, s^3*t, s*t^3, t^4}, DegreeMap => i -> 4*i)

                   4   3      3   4
o11 = map (S, R, {s , s t, s*t , t })

o11 : RingMap S <-- R

i12 : C = simplicialModule(freeResolution coker vars R, 4, Degeneracy => true)

       1      5      15      35      70
o12 = R  <-- R  <-- R   <-- R   <-- R  <-- ...
                                     
      0      1      2       3       4

o12 : SimplicialModule

i13 : D = phi C

       1      5      15      35      70
o13 = S  <-- S  <-- S   <-- S   <-- S  <-- ...
                                     
      0      1      2       3       4

o13 : SimplicialModule

i14 : assert isWellDefined D

Caveat

Every term in the simplicial module must be free or a submodule of a free module. Otherwise, use tensor(RingMap,SimplicialModule).

Ways to use this method:

RingMap SimplicialModule -- apply a ring map to a simplicial module

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:1082:0.

RingMap SimplicialModule -- apply a ring map to a simplicial module

Description

Caveat

See also

Ways to use this method: