differential of a ZZ/d-graded factorization -- get the maps between the terms in a ZZ/d-graded factorization

Usage:

dd^C

dd_C
Inputs:
- C, a ZZ/d graded factorization,
Outputs:
- a map of ZZ/d-graded factorizations, a map of degree -1

A Z/d-graded factorization is a sequence of modules connected by homomorphisms, called differentials, such that any d-fold composition of the maps is a scalar multiple of the identity.

One can access the differential of a factorization as follows.

i1 : R = QQ[a..d]/(c^2-b*d+a^2);

i2 : m = ideal vars R

o2 = ideal (a, b, c, d)

o2 : Ideal of R

i3 : C = tailMF (m^2)

               24               24               24
o3 = (QQ[a..d])   <-- (QQ[a..d])   <-- (QQ[a..d])
                                        
     0                1                0

o3 : ZZdFactorization

i4 : dd^C

                   24                                                                                                24
o4 = 1 : (QQ[a..d])   <----------------------------------------------------------------------------------- (QQ[a..d])   : 0
                         {6} | a  0  0  -d 0  d  0  0  0  0  0  c  0  0  0  0  0  0  0  0  0  0  0  0  |
                         {6} | 0  a  0  0  0  0  0  d  0  0  0  0  0  c  d  0  0  0  0  0  0  0  0  0  |
                         {6} | 0  0  a  0  0  0  0  0  d  0  0  0  0  0  c  d  0  0  0  0  0  0  0  0  |
                         {6} | 0  0  0  a  0  0  0  0  0  d  0  0  0  0  0  c  0  0  0  0  0  0  0  0  |
                         {6} | 0  0  0  0  -a 0  -b 0  -d 0  0  0  0  0  0  0  c  d  0  b  0  0  0  0  |
                         {6} | -b 0  0  0  0  -a 0  0  0  -d 0  0  0  0  0  0  0  c  0  0  0  0  0  0  |
                         {6} | 0  0  d  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  b  0  c  d  0  0  |
                         {6} | 0  -b 0  d  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0  c  d  0  |
                         {6} | 0  0  -b 0  0  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0  c  d  |
                         {6} | 0  0  0  -b 0  0  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0  c  |
                         {6} | -d 0  0  0  0  0  c  0  0  0  -a 0  0  0  -d 0  -d 0  0  -c 0  0  0  0  |
                         {6} | c  0  0  0  0  0  0  0  0  0  0  -a 0  0  0  -d 0  -d 0  0  0  0  0  0  |
                         {6} | 0  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  -c 0  -d 0  0  0  |
                         {6} | 0  c  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0  -d 0  0  |
                         {6} | 0  0  c  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0  -d 0  |
                         {6} | 0  0  0  c  0  0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0  -d |
                         {6} | 0  0  0  0  c  -d 0  0  0  0  b  0  0  0  0  0  a  0  0  0  0  0  -d 0  |
                         {6} | 0  0  0  0  0  c  0  0  0  0  0  b  0  0  0  0  0  a  0  0  0  0  0  -d |
                         {6} | 0  0  0  0  0  0  -d 0  0  0  0  0  -c -d 0  0  0  0  a  0  0  0  0  0  |
                         {6} | 0  0  0  0  -d 0  0  0  0  0  -c -d 0  0  0  0  0  0  -b a  -c -d 0  0  |
                         {6} | 0  0  0  0  0  0  c  -d 0  0  0  0  b  0  -d 0  0  0  0  0  a  0  0  0  |
                         {6} | 0  0  0  0  0  0  0  c  -d 0  0  0  0  b  0  -d 0  0  0  0  0  a  0  0  |
                         {6} | 0  0  0  0  0  0  0  0  c  -d 0  0  0  0  b  0  0  0  0  0  0  0  a  0  |
                         {6} | 0  0  0  0  0  0  0  0  0  c  0  0  0  0  0  b  0  0  0  0  0  0  0  a  |

                   24                                                                                                24
     0 : (QQ[a..d])   <----------------------------------------------------------------------------------- (QQ[a..d])   : 1
                         {5} | a  0  0  d  0  d  0  0  0  0  0  c  0  0  0  0  0  0  0  0  0  0  0  0  |
                         {5} | 0  a  0  0  0  0  0  d  0  0  0  0  0  c  d  0  0  0  0  0  0  0  0  0  |
                         {5} | 0  0  a  0  0  0  0  0  d  0  0  0  0  0  c  d  0  0  0  0  0  0  0  0  |
                         {5} | 0  0  0  a  0  0  0  0  0  d  0  0  0  0  0  c  0  0  0  0  0  0  0  0  |
                         {5} | 0  0  0  0  -a 0  b  0  d  0  0  0  0  0  0  0  c  d  0  b  0  0  0  0  |
                         {5} | -b 0  0  0  0  -a 0  0  0  d  0  0  0  0  0  0  0  c  0  0  0  0  0  0  |
                         {5} | 0  0  d  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  b  0  c  d  0  0  |
                         {5} | 0  -b 0  d  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0  c  d  0  |
                         {5} | 0  0  -b 0  0  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0  c  d  |
                         {5} | 0  0  0  -b 0  0  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0  c  |
                         {5} | -d 0  0  0  0  0  -c 0  0  0  -a 0  0  0  d  0  -d 0  0  -c 0  0  0  0  |
                         {5} | c  0  0  0  0  0  0  0  0  0  0  -a 0  0  0  d  0  -d 0  0  0  0  0  0  |
                         {5} | 0  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  -c 0  -d 0  0  0  |
                         {5} | 0  c  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0  -d 0  0  |
                         {5} | 0  0  c  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0  -d 0  |
                         {5} | 0  0  0  c  0  0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0  -d |
                         {5} | 0  0  0  0  c  -d 0  0  0  0  b  0  0  0  0  0  a  0  0  0  0  0  d  0  |
                         {5} | 0  0  0  0  0  c  0  0  0  0  0  b  0  0  0  0  0  a  0  0  0  0  0  d  |
                         {5} | 0  0  0  0  0  0  -d 0  0  0  0  0  -c -d 0  0  0  0  a  0  0  0  0  0  |
                         {5} | 0  0  0  0  -d 0  0  0  0  0  -c -d 0  0  0  0  0  0  b  a  c  d  0  0  |
                         {5} | 0  0  0  0  0  0  c  -d 0  0  0  0  b  0  -d 0  0  0  0  0  a  0  0  0  |
                         {5} | 0  0  0  0  0  0  0  c  -d 0  0  0  0  b  0  -d 0  0  0  0  0  a  0  0  |
                         {5} | 0  0  0  0  0  0  0  0  c  -d 0  0  0  0  b  0  0  0  0  0  0  0  a  0  |
                         {5} | 0  0  0  0  0  0  0  0  0  c  0  0  0  0  0  b  0  0  0  0  0  0  0  a  |

o4 : ZZdFactorizationMap

i5 : assert(dd^C === C.dd)

i6 : assert(source dd^C === C)

i7 : assert(target dd^C === C)

i8 : assert(degree dd^C === -1)

The composition of the differential with itself should be a scalar multiple of the identity.

i9 : (dd^C)^2 == (R.relations)_(0,0)

o9 = true

The individual maps between terms are indexed by their source.

i10 : dd^C_2

o10 = {6} | a  0  0  -d 0  d  0  0  0  0  0  c  0  0  0  0  0  0  0  0  0  0 
      {6} | 0  a  0  0  0  0  0  d  0  0  0  0  0  c  d  0  0  0  0  0  0  0 
      {6} | 0  0  a  0  0  0  0  0  d  0  0  0  0  0  c  d  0  0  0  0  0  0 
      {6} | 0  0  0  a  0  0  0  0  0  d  0  0  0  0  0  c  0  0  0  0  0  0 
      {6} | 0  0  0  0  -a 0  -b 0  -d 0  0  0  0  0  0  0  c  d  0  b  0  0 
      {6} | -b 0  0  0  0  -a 0  0  0  -d 0  0  0  0  0  0  0  c  0  0  0  0 
      {6} | 0  0  d  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  b  0  c  d 
      {6} | 0  -b 0  d  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0  c 
      {6} | 0  0  -b 0  0  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0  0 
      {6} | 0  0  0  -b 0  0  0  0  0  -a 0  0  0  0  0  0  0  0  0  0  0  0 
      {6} | -d 0  0  0  0  0  c  0  0  0  -a 0  0  0  -d 0  -d 0  0  -c 0  0 
      {6} | c  0  0  0  0  0  0  0  0  0  0  -a 0  0  0  -d 0  -d 0  0  0  0 
      {6} | 0  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  -c 0  -d 0 
      {6} | 0  c  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0  -d
      {6} | 0  0  c  -d 0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0  0 
      {6} | 0  0  0  c  0  0  0  0  0  0  0  0  0  0  0  -a 0  0  0  0  0  0 
      {6} | 0  0  0  0  c  -d 0  0  0  0  b  0  0  0  0  0  a  0  0  0  0  0 
      {6} | 0  0  0  0  0  c  0  0  0  0  0  b  0  0  0  0  0  a  0  0  0  0 
      {6} | 0  0  0  0  0  0  -d 0  0  0  0  0  -c -d 0  0  0  0  a  0  0  0 
      {6} | 0  0  0  0  -d 0  0  0  0  0  -c -d 0  0  0  0  0  0  -b a  -c -d
      {6} | 0  0  0  0  0  0  c  -d 0  0  0  0  b  0  -d 0  0  0  0  0  a  0 
      {6} | 0  0  0  0  0  0  0  c  -d 0  0  0  0  b  0  -d 0  0  0  0  0  a 
      {6} | 0  0  0  0  0  0  0  0  c  -d 0  0  0  0  b  0  0  0  0  0  0  0 
      {6} | 0  0  0  0  0  0  0  0  0  c  0  0  0  0  0  b  0  0  0  0  0  0 
      -----------------------------------------------------------------------
      0  0  |
      0  0  |
      0  0  |
      0  0  |
      0  0  |
      0  0  |
      0  0  |
      d  0  |
      c  d  |
      0  c  |
      0  0  |
      0  0  |
      0  0  |
      0  0  |
      -d 0  |
      0  -d |
      -d 0  |
      0  -d |
      0  0  |
      0  0  |
      0  0  |
      0  0  |
      a  0  |
      0  a  |

                       24               24
o10 : Matrix (QQ[a..d])   <-- (QQ[a..d])

i11 : assert(source dd^C_2 === C_2)

i12 : assert(target dd^C_2 === C_1)

differential of a ZZ/d-graded factorization -- get the maps between the terms in a ZZ/d-graded factorization

See also