ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations

Operator: **
Usage:

h = f ** g

h = tensor(f, g)
Inputs:
- f, a map of ZZ/d-graded factorizations,
- g, a map of ZZ/d-graded factorizations,
Outputs:
- h, a map of ZZ/d-graded factorizations,

Description

The maps $f : C \to D$ and $g : E \to F$ of ZZ/d-graded factorizations induce the map $h = f \otimes g : C \otimes E \to D \otimes F$ defined by $c \otimes e \mapsto f(c) \otimes g(e)$.

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

i2 : R = S/(a^3+b^3);

i3 : m = ideal vars R

o3 = ideal (a, b)

o3 : Ideal of R

i4 : C = tailMF m

      2      2      2
o4 = S  <-- S  <-- S
                    
     0      1      0

o4 : ZZdFactorization

i5 : D = tailMF (m^2)

      3      3      3
o5 = S  <-- S  <-- S
                    
     0      1      0

o5 : ZZdFactorization

i6 : f = randomFactorizationMap(C,D)

          2                                         3
o6 = 0 : S  <------------------------------------- S  : 0
               {2} | 24a-36b -29a+19b -10a-29b |
               {3} | -30     19       -8       |

          2                                          3
     1 : S  <-------------------------------------- S  : 1
               {4} | -22a-29b -16a+39b 19a-47b  |
               {4} | -24a-38b 21a+34b  -39a-18b |

o6 : ZZdFactorizationMap

i7 : E = (dual C)

      2      2      2
o7 = S  <-- S  <-- S
                    
     0      1      0

o7 : ZZdFactorization

i8 : F = (dual D)

      3      3      3
o8 = S  <-- S  <-- S
                    
     0      1      0

o8 : ZZdFactorization

i9 : g = randomFactorizationMap(F,E)

          3                            2
o9 = 0 : S  <------------------------ S  : 0
               {-3} | -13a-43b 2  |
               {-3} | -15a-28b 16 |
               {-3} | -47a+38b 22 |

          3                                 2
     1 : S  <----------------------------- S  : 1
               {-5} | 45a-34b  -16a+7b |
               {-5} | -48a-47b 15a-23b |
               {-5} | 47a+19b  39a+43b |

o9 : ZZdFactorizationMap

i10 : h = f ** g

           12                                                                                                                                                                                         12
o10 = 0 : S   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 0
                 {-1} | -9a2+42ab+33b2  48a+29b  -27a2-10ab-9b2 43a+38b  29a2-ab+35b2   -20a+43b 0               0               0               0               0               0               |
                 {-1} | 44a2-31ab-2b2   -20a+30b 31a2+22ab-27b2 41a+b    49a2+8ab+4b2   42a+41b  0               0               0               0               0               0               |
                 {-1} | -17a2-22ab+46b2 23a+16b  50a2+25ab+15b2 -32a+14b -35a2-27ab+9b2 -18a-32b 0               0               0               0               0               0               |
                 {0}  | -14a-23b        41       -45a-9b        38       3a+41b         -16      0               0               0               0               0               0               |
                 {0}  | 46a+32b         25       18a-27b        1        19a+22b        -27      0               0               0               0               0               0               |
                 {0}  | -4a-29b         47       16a+15b        14       -28a-b         26       0               0               0               0               0               0               |
                 {-1} | 0               0        0              0        0              0        20a2+49ab-24b2  49a2+7ab-b2     -13a2-24ab-13b2 -47a2-29ab-30b2 47a2-34ab-18b2  -a2-24ab-26b2   |
                 {-1} | 0               0        0              0        0              0        46a2+2ab+50b2   -27a2-30ab-40b2 -40a2-9ab-15b2  -38a2+44ab+12b2 -3a2+50ab-13b2  -18a2-31ab-30b2 |
                 {-1} | 0               0        0              0        0              0        -24a2+37ab-46b2 -50a2+44ab-35b2 -45a2+14ab+34b2 -18a2+25ab-40b2 -16a2-30ab+16b2 34a2-6ab-b2     |
                 {-1} | 0               0        0              0        0              0        31a2+15ab-21b2  -20a2+36ab+37b2 36a2+8ab-45b2   -33a2+7ab+36b2  -38a2+11ab+6b2  18a2+15ab-25b2  |
                 {-1} | 0               0        0              0        0              0        41a2+23ab-32b2  44a2-18ab-35b2  2a2+7ab+18b2    12a2+27ab+26b2  -47a2-30ab+38b2 21a2+21ab+10b2  |
                 {-1} | 0               0        0              0        0              0        -17a2-20ab-15b2 -27a2+11ab-18b2 -23a2-23ab+40b2 11a2+7ab+48b2   -15a2+29ab-39b2 -6a2+45ab+34b2  |

           12                                                                                                                                                                                         12
      1 : S   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 1
                 {-3} | -31a2-12ab+12b2 20a2+37ab-50b2  8a2+23ab-40b2   -41a2-2ab+32b2  -46a2+45ab-24b2 -42a2-10ab-b2  0               0        0               0        0              0        |
                 {-3} | -41a2-6ab-25b2  -44a2+19ab+20b2 -22a2+47ab+16b2 -31a2+43ab-33b2 -25a2+44ab+50b2 -49a2-3ab-40b2 0               0        0               0        0              0        |
                 {-3} | 17a2-24ab+23b2  27a2+32ab-33b2  -50a2+39ab-43b2 -20a2-ab+9b2    35a2-38ab-46b2  14a2-46ab-35b2 0               0        0               0        0              0        |
                 {-2} | -37a+10b        -25a-8b         47a-40b         -a+32b          44a-31b         27a+45b        0               0        0               0        0              0        |
                 {-2} | 26a-4b          -46a-17b        -3a+16b         -18a-33b        -20a-28b        -19a-18b       0               0        0               0        0              0        |
                 {-2} | 4a+36b          42a+23b         -16a-43b        34a+9b          28a+50b         -9a-41b        0               0        0               0        0              0        |
                 {1}  | 0               0               0               0               0               0              -17a2+10ab+35b2 -44a+43b 6a2-21ab+40b2   -32a-23b -45a2-4ab+b2   38a+7b   |
                 {1}  | 0               0               0               0               0               0              27a2+41ab+4b2   -49a+41b 38a2-36ab+19b2  47a+18b  18a2-29ab+3b2  a-45b    |
                 {1}  | 0               0               0               0               0               0              24a2+22ab+9b2   21a-32b  45a2-17ab-33b2  -49a+50b 16a2+2ab+32b2  14a-24b  |
                 {1}  | 0               0               0               0               0               0              9a2+11ab+18b2   -48a+25b 30a2-32ab-48b2  42a-33b  2a2-8ab-34b2   23a-36b  |
                 {1}  | 0               0               0               0               0               0              -44a2+30ab-47b2 20a-2b   -12a2+13ab-43b2 33a+39b  -21a2+49ab-b2  -18a+15b |
                 {1}  | 0               0               0               0               0               0              17a2-35ab-30b2  -23a-28b 23a2+8ab-21b2   -43a+41b 15a2-30ab+23b2 -50a+8b  |

o10 : ZZdFactorizationMap

i11 : assert isWellDefined h

i12 : assert(source h === D ** E)

i13 : assert(target h === C ** F)

If one argument is a ZZdFactorization or Module, then the identity map of the corresponding factorization is used.

i14 : fE = f ** E

           8                                                                                                                          12
o14 = 0 : S  <---------------------------------------------------------------------------------------------------------------------- S   : 0
                {0}  | 24a-36b 0       -29a+19b 0        -10a-29b 0        0        0        0        0        0        0        |
                {-1} | 0       24a-36b 0        -29a+19b 0        -10a-29b 0        0        0        0        0        0        |
                {1}  | -30     0       19       0        -8       0        0        0        0        0        0        0        |
                {0}  | 0       -30     0        19       0        -8       0        0        0        0        0        0        |
                {0}  | 0       0       0        0        0        0        -22a-29b 0        -16a+39b 0        19a-47b  0        |
                {0}  | 0       0       0        0        0        0        0        -22a-29b 0        -16a+39b 0        19a-47b  |
                {0}  | 0       0       0        0        0        0        -24a-38b 0        21a+34b  0        -39a-18b 0        |
                {0}  | 0       0       0        0        0        0        0        -24a-38b 0        21a+34b  0        -39a-18b |

           8                                                                                                                          12
      1 : S  <---------------------------------------------------------------------------------------------------------------------- S   : 1
                {-2} | 24a-36b 0       -29a+19b 0        -10a-29b 0        0        0        0        0        0        0        |
                {-2} | 0       24a-36b 0        -29a+19b 0        -10a-29b 0        0        0        0        0        0        |
                {-1} | -30     0       19       0        -8       0        0        0        0        0        0        0        |
                {-1} | 0       -30     0        19       0        -8       0        0        0        0        0        0        |
                {2}  | 0       0       0        0        0        0        -22a-29b 0        -16a+39b 0        19a-47b  0        |
                {1}  | 0       0       0        0        0        0        0        -22a-29b 0        -16a+39b 0        19a-47b  |
                {2}  | 0       0       0        0        0        0        -24a-38b 0        21a+34b  0        -39a-18b 0        |
                {1}  | 0       0       0        0        0        0        0        -24a-38b 0        21a+34b  0        -39a-18b |

o14 : ZZdFactorizationMap

i15 : assert(fE === f ** id_E)

i16 : k = coker vars S

o16 = cokernel | a b |

                             1
o16 : S-module, quotient of S

i17 : gk = g ** k

o17 = 0 : cokernel {-3} | a b 0 0 0 0 | <----------------- cokernel {-2} | a b 0 0 | : 0
                   {-3} | 0 0 a b 0 0 |    {-3} | 0 2  |            {-3} | 0 0 a b |
                   {-3} | 0 0 0 0 a b |    {-3} | 0 16 |
                                           {-3} | 0 22 |

      1 : cokernel {-5} | a b 0 0 0 0 | <----- cokernel {-4} | a b 0 0 | : 1
                   {-5} | 0 0 a b 0 0 |    0            {-4} | 0 0 a b |
                   {-5} | 0 0 0 0 a b |

o17 : ZZdFactorizationMap

i18 : assert(gk == g ** id_(complex k))

This routine is functorial.

i19 : use S;

i20 : D' = randomTailMF(a^3+b^3)

       10      10      10
o20 = S   <-- S   <-- S
                       
      0       1       0

o20 : ZZdFactorization

i21 : f' = randomFactorizationMap(D, D',InternalDegree => 3)

           3                                                       10
o21 = 0 : S  <--------------------------------------------------- S   : 0
                {3} | -16 -18 23  44  19  -28 6   28  5   -28 |
                {3} | -46 27  -37 -39 0   47  -9  -29 -37 42  |
                {3} | 12  -21 -23 20  -47 -28 -33 26  -33 44  |

           3                                           10
      1 : S  <--------------------------------------- S   : 1
                {5} | 0 0 0 0 0 30 5   -29 12 -2  |
                {5} | 0 0 0 0 0 4  -20 15  3  20  |
                {5} | 0 0 0 0 0 22 -13 -4  9  -26 |

o21 : ZZdFactorizationMap

i22 : (f * f') ** g == (f ** g) * (f' ** id_E)

o22 = true

i23 : (f * f') ** g == (f ** id_F) * (f' ** g)

o23 = true

Ways to use this method:

Module ** ZZdFactorizationMap
ZZdFactorization ** ZZdFactorizationMap
ZZdFactorizationMap ** Module
ZZdFactorizationMap ** ZZdFactorization
ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3868:0.

ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations

Description

See also

Ways to use this method: