ZZdFactorizationMap _ ZZ -- access individual matrices in a ZZ/d-graded factorization map

Operator: _
Usage:

f_i
Inputs:
- f, a map of ZZ/d-graded factorizations,
- i, an integer, the homological index
Outputs:
- a matrix, the i-th map

Description

A ZZ/d-graded factorization map $f : C \to D$ of degree $d$ is a sequence of maps $f_i : C_i \to D_{i+d}$. This method allows one to access the individual $f_i$.

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

o1 = S

o1 : PolynomialRing

i2 : R = S/(a^2+b^2+c^2);

i3 : m = ideal vars R

o3 = ideal (a, b, c)

o3 : Ideal of R

i4 : C = tailMF m

      4      4      4
o4 = S  <-- S  <-- S
                    
     0      1      0

o4 : ZZdFactorization

i5 : D = tailMF (m^2)

      8      8      8
o5 = S  <-- S  <-- S
                    
     0      1      0

o5 : ZZdFactorization

i6 : f = randomFactorizationMap(C, D, Cycle=>true, InternalDegree => 1)

          4                                                                                                                                                                                                                                                                 8
o6 = 0 : S  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 0
               {3} | 18a2+ab-13b2-31ac-34bc+34c2   50a2+32ab-13b2+29ac-23bc-39c2 -30a2+15ab+36b2+26ac+8bc+33c2  6a2+33ab+38b2-22ac+38bc+4c2   -36a2+33ab+50b2-39ac+29bc-16c2 -a2-31ab-5b2-17ac-2bc+14c2   10a2+33ab-49b2+17ac+44bc+28c2 -45a2+47ab+21b2+20ac-31bc+16c2 |
               {3} | 40a2+12ab-16b2+44ac-29bc+49c2 4a2+35ab+29b2+33ac+39bc-11c2  -43a2+6ab-2b2+47ac+46bc-22c2   -30a2-46ab+44b2-38ac-15bc+8c2 -50a2-40ab-42b2-7ac+36bc+3c2   32a2+38ab-38b2-24ac-34bc+2c2 -38a2-44ab+22b2-26ac+28bc-3c2 -16a2-23ab+18b2+37ac+8bc+20c2  |
               {3} | 39a2-16ab-27b2+5ac-2bc+16c2   19a2+23ab+21b2+3ac-20bc+4c2   25a2-45ab-23b2+28ac+28bc-3c2   17a2-50ab-26b2+36ac-7c2       8a2-11ab-8b2+20ac-11bc-35c2    9a2-8ab+18b2+12ac-18c2       24a2+8ab+33b2+40ac-21bc+17c2  -28a2+13ab-47b2+19ac+20bc+2c2  |
               {3} | -26a2+41ab-43b2+48ac-50bc-6c2 -42a2-20ab-22b2+24ac+6bc+7c2  -27a2+14ab-47b2+23ac-45bc-47c2 -6a2+21ab-11b2-49ac-44bc+29c2 -12a2-43ab+45b2-ac-10bc-5c2    -12a2-20ab+48b2+6bc+35c2     -47a2-7ab-22b2+44ac-5bc+38c2  -17a2-40ab-12b2+11ac-49bc+c2   |

          4                                                                                                                                                                                                                                                                   8
     1 : S  <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 1
               {4} | 8a2+22ab+44b2+23ac+37bc+35c2  -39a2-36ab+38b2-48ac+30bc+29c2 -19a2+5ab-22b2-34ac-8bc+5c2   -25a2-26ab-18b2+14ac-16bc+33c2 17a2-12ab+42b2+3ac+4bc-6c2    9a2-ab+16b2+32ac+13bc+5c2     24a2-36ab-29b2-30ac+19bc+19c2  -28a2-47ab+2b2+38ac+16bc+22c2  |
               {4} | 36a2-39ab+11b2-13ac+40bc+2c2  18a2+28ab+48b2+15ac-46bc+8c2   50a2-37ab-22b2-41ac+19bc+32c2 -30a2-14ab-12b2-43ac+11bc      -6a2-3ab+45b2-31ac-26bc+49c2  a2+ab-43b2+21ac+11bc-3c2      -10a2-29ab-22b2-8ac-29bc-24c2  45a2-34ab-47b2-48ac+47bc+19c2  |
               {4} | -50a2-2ab+26b2-3ac+20bc+14c2  -40a2-19ab+18b2+4ac+31bc+4c2   -4a2+9ab+33b2+41ac-46bc-50c2  43a2-47ab-47b2+11ac-26bc+12c2  -30a2+22ab-8b2+17ac-34bc+34c2 32a2-14ab-27b2+28ac+3bc+16c2  -38a2-16ab+21b2+39ac+34bc+19c2 -16a2+7ab-23b2+15ac+39bc+43c2  |
               {4} | -12a2+28ab-38b2+19ac-7bc+18c2 26a2+30ab-5b2+20ac-22bc+7c2    42a2+24ab-49b2+15ac-16bc+18c2 27a2-ab+21b2+11ac+22bc+18c2    -6a2+15ab+50b2-6ac+34bc-16c2  -12a2-47ab-13b2-3ac-11bc-35c2 -47a2-39ab-13b2-18ac-43bc-15c2 -17a2-11ab+36b2+48ac+35bc+11c2 |

o6 : ZZdFactorizationMap

i7 : f_1

o7 = {4} | 8a2+22ab+44b2+23ac+37bc+35c2  -39a2-36ab+38b2-48ac+30bc+29c2
     {4} | 36a2-39ab+11b2-13ac+40bc+2c2  18a2+28ab+48b2+15ac-46bc+8c2  
     {4} | -50a2-2ab+26b2-3ac+20bc+14c2  -40a2-19ab+18b2+4ac+31bc+4c2  
     {4} | -12a2+28ab-38b2+19ac-7bc+18c2 26a2+30ab-5b2+20ac-22bc+7c2   
     ------------------------------------------------------------------------
     -19a2+5ab-22b2-34ac-8bc+5c2   -25a2-26ab-18b2+14ac-16bc+33c2
     50a2-37ab-22b2-41ac+19bc+32c2 -30a2-14ab-12b2-43ac+11bc     
     -4a2+9ab+33b2+41ac-46bc-50c2  43a2-47ab-47b2+11ac-26bc+12c2 
     42a2+24ab-49b2+15ac-16bc+18c2 27a2-ab+21b2+11ac+22bc+18c2   
     ------------------------------------------------------------------------
     17a2-12ab+42b2+3ac+4bc-6c2    9a2-ab+16b2+32ac+13bc+5c2    
     -6a2-3ab+45b2-31ac-26bc+49c2  a2+ab-43b2+21ac+11bc-3c2     
     -30a2+22ab-8b2+17ac-34bc+34c2 32a2-14ab-27b2+28ac+3bc+16c2 
     -6a2+15ab+50b2-6ac+34bc-16c2  -12a2-47ab-13b2-3ac-11bc-35c2
     ------------------------------------------------------------------------
     24a2-36ab-29b2-30ac+19bc+19c2  -28a2-47ab+2b2+38ac+16bc+22c2  |
     -10a2-29ab-22b2-8ac-29bc-24c2  45a2-34ab-47b2-48ac+47bc+19c2  |
     -38a2-16ab+21b2+39ac+34bc+19c2 -16a2+7ab-23b2+15ac+39bc+43c2  |
     -47a2-39ab-13b2-18ac-43bc-15c2 -17a2-11ab+36b2+48ac+35bc+11c2 |

             4      8
o7 : Matrix S  <-- S

i8 : f_0

o8 = {3} | 18a2+ab-13b2-31ac-34bc+34c2   50a2+32ab-13b2+29ac-23bc-39c2
     {3} | 40a2+12ab-16b2+44ac-29bc+49c2 4a2+35ab+29b2+33ac+39bc-11c2 
     {3} | 39a2-16ab-27b2+5ac-2bc+16c2   19a2+23ab+21b2+3ac-20bc+4c2  
     {3} | -26a2+41ab-43b2+48ac-50bc-6c2 -42a2-20ab-22b2+24ac+6bc+7c2 
     ------------------------------------------------------------------------
     -30a2+15ab+36b2+26ac+8bc+33c2  6a2+33ab+38b2-22ac+38bc+4c2  
     -43a2+6ab-2b2+47ac+46bc-22c2   -30a2-46ab+44b2-38ac-15bc+8c2
     25a2-45ab-23b2+28ac+28bc-3c2   17a2-50ab-26b2+36ac-7c2      
     -27a2+14ab-47b2+23ac-45bc-47c2 -6a2+21ab-11b2-49ac-44bc+29c2
     ------------------------------------------------------------------------
     -36a2+33ab+50b2-39ac+29bc-16c2 -a2-31ab-5b2-17ac-2bc+14c2  
     -50a2-40ab-42b2-7ac+36bc+3c2   32a2+38ab-38b2-24ac-34bc+2c2
     8a2-11ab-8b2+20ac-11bc-35c2    9a2-8ab+18b2+12ac-18c2      
     -12a2-43ab+45b2-ac-10bc-5c2    -12a2-20ab+48b2+6bc+35c2    
     ------------------------------------------------------------------------
     10a2+33ab-49b2+17ac+44bc+28c2 -45a2+47ab+21b2+20ac-31bc+16c2 |
     -38a2-44ab+22b2-26ac+28bc-3c2 -16a2-23ab+18b2+37ac+8bc+20c2  |
     24a2+8ab+33b2+40ac-21bc+17c2  -28a2+13ab-47b2+19ac+20bc+2c2  |
     -47a2-7ab-22b2+44ac-5bc+38c2  -17a2-40ab-12b2+11ac-49bc+c2   |

             4      8
o8 : Matrix S  <-- S

Indices that are outside of the period are taken modulo the period.

i9 : period f

o9 = 2

i10 : f_-1

o10 = {4} | 8a2+22ab+44b2+23ac+37bc+35c2  -39a2-36ab+38b2-48ac+30bc+29c2
      {4} | 36a2-39ab+11b2-13ac+40bc+2c2  18a2+28ab+48b2+15ac-46bc+8c2  
      {4} | -50a2-2ab+26b2-3ac+20bc+14c2  -40a2-19ab+18b2+4ac+31bc+4c2  
      {4} | -12a2+28ab-38b2+19ac-7bc+18c2 26a2+30ab-5b2+20ac-22bc+7c2   
      -----------------------------------------------------------------------
      -19a2+5ab-22b2-34ac-8bc+5c2   -25a2-26ab-18b2+14ac-16bc+33c2
      50a2-37ab-22b2-41ac+19bc+32c2 -30a2-14ab-12b2-43ac+11bc     
      -4a2+9ab+33b2+41ac-46bc-50c2  43a2-47ab-47b2+11ac-26bc+12c2 
      42a2+24ab-49b2+15ac-16bc+18c2 27a2-ab+21b2+11ac+22bc+18c2   
      -----------------------------------------------------------------------
      17a2-12ab+42b2+3ac+4bc-6c2    9a2-ab+16b2+32ac+13bc+5c2    
      -6a2-3ab+45b2-31ac-26bc+49c2  a2+ab-43b2+21ac+11bc-3c2     
      -30a2+22ab-8b2+17ac-34bc+34c2 32a2-14ab-27b2+28ac+3bc+16c2 
      -6a2+15ab+50b2-6ac+34bc-16c2  -12a2-47ab-13b2-3ac-11bc-35c2
      -----------------------------------------------------------------------
      24a2-36ab-29b2-30ac+19bc+19c2  -28a2-47ab+2b2+38ac+16bc+22c2  |
      -10a2-29ab-22b2-8ac-29bc-24c2  45a2-34ab-47b2-48ac+47bc+19c2  |
      -38a2-16ab+21b2+39ac+34bc+19c2 -16a2+7ab-23b2+15ac+39bc+43c2  |
      -47a2-39ab-13b2-18ac-43bc-15c2 -17a2-11ab+36b2+48ac+35bc+11c2 |

              4      8
o10 : Matrix S  <-- S

i11 : f_3

o11 = {4} | 8a2+22ab+44b2+23ac+37bc+35c2  -39a2-36ab+38b2-48ac+30bc+29c2
      {4} | 36a2-39ab+11b2-13ac+40bc+2c2  18a2+28ab+48b2+15ac-46bc+8c2  
      {4} | -50a2-2ab+26b2-3ac+20bc+14c2  -40a2-19ab+18b2+4ac+31bc+4c2  
      {4} | -12a2+28ab-38b2+19ac-7bc+18c2 26a2+30ab-5b2+20ac-22bc+7c2   
      -----------------------------------------------------------------------
      -19a2+5ab-22b2-34ac-8bc+5c2   -25a2-26ab-18b2+14ac-16bc+33c2
      50a2-37ab-22b2-41ac+19bc+32c2 -30a2-14ab-12b2-43ac+11bc     
      -4a2+9ab+33b2+41ac-46bc-50c2  43a2-47ab-47b2+11ac-26bc+12c2 
      42a2+24ab-49b2+15ac-16bc+18c2 27a2-ab+21b2+11ac+22bc+18c2   
      -----------------------------------------------------------------------
      17a2-12ab+42b2+3ac+4bc-6c2    9a2-ab+16b2+32ac+13bc+5c2    
      -6a2-3ab+45b2-31ac-26bc+49c2  a2+ab-43b2+21ac+11bc-3c2     
      -30a2+22ab-8b2+17ac-34bc+34c2 32a2-14ab-27b2+28ac+3bc+16c2 
      -6a2+15ab+50b2-6ac+34bc-16c2  -12a2-47ab-13b2-3ac-11bc-35c2
      -----------------------------------------------------------------------
      24a2-36ab-29b2-30ac+19bc+19c2  -28a2-47ab+2b2+38ac+16bc+22c2  |
      -10a2-29ab-22b2-8ac-29bc-24c2  45a2-34ab-47b2-48ac+47bc+19c2  |
      -38a2-16ab+21b2+39ac+34bc+19c2 -16a2+7ab-23b2+15ac+39bc+43c2  |
      -47a2-39ab-13b2-18ac-43bc-15c2 -17a2-11ab+36b2+48ac+35bc+11c2 |

              4      8
o11 : Matrix S  <-- S

i12 : f_4

o12 = {3} | 18a2+ab-13b2-31ac-34bc+34c2   50a2+32ab-13b2+29ac-23bc-39c2
      {3} | 40a2+12ab-16b2+44ac-29bc+49c2 4a2+35ab+29b2+33ac+39bc-11c2 
      {3} | 39a2-16ab-27b2+5ac-2bc+16c2   19a2+23ab+21b2+3ac-20bc+4c2  
      {3} | -26a2+41ab-43b2+48ac-50bc-6c2 -42a2-20ab-22b2+24ac+6bc+7c2 
      -----------------------------------------------------------------------
      -30a2+15ab+36b2+26ac+8bc+33c2  6a2+33ab+38b2-22ac+38bc+4c2  
      -43a2+6ab-2b2+47ac+46bc-22c2   -30a2-46ab+44b2-38ac-15bc+8c2
      25a2-45ab-23b2+28ac+28bc-3c2   17a2-50ab-26b2+36ac-7c2      
      -27a2+14ab-47b2+23ac-45bc-47c2 -6a2+21ab-11b2-49ac-44bc+29c2
      -----------------------------------------------------------------------
      -36a2+33ab+50b2-39ac+29bc-16c2 -a2-31ab-5b2-17ac-2bc+14c2  
      -50a2-40ab-42b2-7ac+36bc+3c2   32a2+38ab-38b2-24ac-34bc+2c2
      8a2-11ab-8b2+20ac-11bc-35c2    9a2-8ab+18b2+12ac-18c2      
      -12a2-43ab+45b2-ac-10bc-5c2    -12a2-20ab+48b2+6bc+35c2    
      -----------------------------------------------------------------------
      10a2+33ab-49b2+17ac+44bc+28c2 -45a2+47ab+21b2+20ac-31bc+16c2 |
      -38a2-44ab+22b2-26ac+28bc-3c2 -16a2-23ab+18b2+37ac+8bc+20c2  |
      24a2+8ab+33b2+40ac-21bc+17c2  -28a2+13ab-47b2+19ac+20bc+2c2  |
      -47a2-7ab-22b2+44ac-5bc+38c2  -17a2-40ab-12b2+11ac-49bc+c2   |

              4      8
o12 : Matrix S  <-- S

Ways to use this method:

ZZdFactorizationMap _ ZZ -- access individual matrices in a ZZ/d-graded factorization map

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3000:0.

ZZdFactorizationMap _ ZZ -- access individual matrices in a ZZ/d-graded factorization map

Description

See also

Ways to use this method: