RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map

Operator: **
Usage:

phi ** C

tensor(phi, C)

S ** C

C ** S
Inputs:
- phi, a ring map, whose source is a ring $R$ and whose target is a ring $S$
- C, a ZZ/d graded factorization, over the ring $R$
Outputs:
- a ZZ/d graded factorization, over the ring $S$

Description

These methods implement the base change of rings. As input, one can either give a ring map $\phi$, or the ring $S$ (when there is a canonical map from $R$ to $S$).

We illustrate the tensor product of a ZZ/d-graded factorization along 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 : Rf = R/(x^3+y^3+z^3)

o4 = Rf

o4 : QuotientRing

i5 : C = tailMF ideal vars Rf

      4      4      4
o5 = R  <-- R  <-- R
                    
     0      1      0

o5 : ZZdFactorization

i6 : D = phi**C

      4      4      4
o6 = S  <-- S  <-- S
                    
     0      1      0

o6 : ZZdFactorization

i7 : isdFactorization D

              3     2        2     3
o7 = (true, 2s  + 3s t + 3s*t  + 2t )

o7 : Sequence

i8 : phi(potential C) == potential D

o8 = true

i9 : dd^D

          4                                            4
o9 = 1 : S  <---------------------------------------- S  : 0
               {5} | -s  -t2  s2+2st+t2 0         |
               {5} | -t  s2   0         s2+2st+t2 |
               {5} | s+t 0    s2        t2        |
               {6} | 0   -s-t -t        s         |

          4                                            4
     0 : S  <---------------------------------------- S  : 1
               {3} | -s2 -t2 s2+2st+t2 0          |
               {4} | -t  s   0         -s2-2st-t2 |
               {4} | s+t 0   s         -t2        |
               {4} | 0   s+t t         s2         |

o9 : ZZdFactorizationMap

If a ring is used rather than a ring map, then the implicit map from the underlying ring of the complex to the given ring is used.

i10 : use R;

i11 : A = R/(x^2+y^2+z^2);

i12 : C ** A

       4      4      4
o12 = A  <-- A  <-- A
                     
      0      1      0

o12 : ZZdFactorization

i13 : assert(map(A,R) ** C == C ** A)

The commutativity of tensor product is witnessed as follows.

i14 : assert(D == C ** phi)

i15 : assert(C ** A == A ** C)

When the modules in the factorization are not free modules, this is different than the image of a factorization under a ring map.

i16 : use R

o16 = R

o16 : PolynomialRing

i17 : C' = C ** coker matrix{{x^2+y^2+z^2}}

o17 = cokernel {3} | x2+y2+z2 0        0        0        | <-- cokernel {5} | x2+y2+z2 0        0        0        | <-- cokernel {3} | x2+y2+z2 0        0        0        |
               {4} | 0        x2+y2+z2 0        0        |              {5} | 0        x2+y2+z2 0        0        |              {4} | 0        x2+y2+z2 0        0        |
               {4} | 0        0        x2+y2+z2 0        |              {5} | 0        0        x2+y2+z2 0        |              {4} | 0        0        x2+y2+z2 0        |
               {4} | 0        0        0        x2+y2+z2 |              {6} | 0        0        0        x2+y2+z2 |              {4} | 0        0        0        x2+y2+z2 |
                                                                                                                         
      0                                                        1                                                        0

o17 : ZZdFactorization

i18 : D1 = phi C

       4      4      4
o18 = S  <-- S  <-- S
                     
      0      1      0

o18 : ZZdFactorization

i19 : isdFactorization D1

               3     2        2     3
o19 = (true, 2s  + 3s t + 3s*t  + 2t )

o19 : Sequence

i20 : D2 = phi ** C'

o20 = cokernel {3} | 2s2+2st+2t2 0           0           0           | <-- cokernel {5} | 2s2+2st+2t2 0           0           0           | <-- cokernel {3} | 2s2+2st+2t2 0           0           0           |
               {4} | 0           2s2+2st+2t2 0           0           |              {5} | 0           2s2+2st+2t2 0           0           |              {4} | 0           2s2+2st+2t2 0           0           |
               {4} | 0           0           2s2+2st+2t2 0           |              {5} | 0           0           2s2+2st+2t2 0           |              {4} | 0           0           2s2+2st+2t2 0           |
               {4} | 0           0           0           2s2+2st+2t2 |              {6} | 0           0           0           2s2+2st+2t2 |              {4} | 0           0           0           2s2+2st+2t2 |
                                                                                                                                                 
      0                                                                    1                                                                    0

o20 : ZZdFactorization

i21 : isdFactorization D2

              3
o21 = (true, t )

o21 : Sequence

i22 : prune D1

       4      4      4
o22 = S  <-- S  <-- S
                     
      0      1      0

o22 : ZZdFactorization

i23 : prune D2

o23 = cokernel {3} | s2+st+t2 0        0        0        | <-- cokernel {5} | s2+st+t2 0        0        0        | <-- cokernel {3} | s2+st+t2 0        0        0        |
               {4} | 0        s2+st+t2 0        0        |              {5} | 0        s2+st+t2 0        0        |              {4} | 0        s2+st+t2 0        0        |
               {4} | 0        0        s2+st+t2 0        |              {5} | 0        0        s2+st+t2 0        |              {4} | 0        0        s2+st+t2 0        |
               {4} | 0        0        0        s2+st+t2 |              {6} | 0        0        0        s2+st+t2 |              {4} | 0        0        0        s2+st+t2 |
                                                                                                                         
      0                                                        1                                                        0

o23 : ZZdFactorization

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

i24 : R = ZZ/101[a..d];

i25 : S = ZZ/101[s,t];

i26 : f = map(S, R, {s^4, s^3*t, s*t^3, t^4}, DegreeMap => i -> 4*i)

                   4   3      3   4
o26 = map (S, R, {s , s t, s*t , t })

o26 : RingMap S <-- R

i27 : C = koszulMF({a,b,c,d}, a^2+b^2+c^2+d^2)

       8      8      8
o27 = R  <-- R  <-- R
                     
      0      1      0

o27 : ZZdFactorization

i28 : D = f ** C

       8      8      8
o28 = S  <-- S  <-- S
                     
      0      1      0

o28 : ZZdFactorization

i29 : isdFactorization D

              8    6 2    2 6    8
o29 = (true, s  + s t  + s t  + t )

o29 : Sequence

i30 : potential D == f(potential C)

o30 = true

i31 : D == f C

o31 = true

i32 : C1 = Hom(C, image vars R)

o32 = image | d c b a 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 0 0 0 | <-- image | d c b a 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 0 0 0 | <-- image | d c b a 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 0 0 0 |
            | 0 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 0 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 0 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 0 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 0 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 0 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 d c b a 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 0 0 0 0 0 0 0 d c b a |           | 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 0 0 0 d c b a |           | 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 0 0 0 d c b a |
                                                                                                                                                                   
      0                                                                             1                                                                             0

o32 : ZZdFactorization

i33 : D1 = f ** C1

o33 = cokernel {4} | st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    | <-- cokernel {4} | st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    | <-- cokernel {4} | st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | -t4 0   s3t  0   s4   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    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | -t4 0   s3t  0   s4   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    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | -t4 0   s3t  0   s4   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    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | 0   -t4 -st3 0   0    s4   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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   -t4 -st3 0   0    s4   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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   -t4 -st3 0   0    s4   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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | 0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    0   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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   0   0    0   0    0    |
               {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    |
               {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    |              {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    |
               {4} | 0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   0   0    0   0    0    0   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   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   0   0    0   0    0    0   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   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   0   0    0   0    0    0   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   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   0   0    0   0    0    0   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   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   0   0    0   0    0    0   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   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   0   0    0   0    0    0   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   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  0    0    0   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   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  0    0    0   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   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  0    0    0   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   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   0    0   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   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   0    0   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   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   0    0   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   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   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   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   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   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   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    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   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    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   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    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   0   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    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   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    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   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    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t 0   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    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  0    0    |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  0    0    |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    st3 s3t 0    s4  0    0    |
               {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   0    |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   0    |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    -t4 0   s3t  0   s4   0    |
               {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   -t4 -st3 0   0    s4   |
               {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t |              {4} | 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   0   0    0   0    0    0   0   0    0   0    0    0   0   0    0   0    0    0   0   0    -t4 -st3 -s3t |
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
      0                                                                                                                                                                                                                                            1                                                                                                                                                                                                                                            0

o33 : ZZdFactorization

i34 : isdFactorization C1

o34 = (false, no potential)

o34 : Sequence

i35 : isdFactorization D1

o35 = (false, no potential)

o35 : Sequence

Ways to use this method:

Ring ** ZZdFactorization
RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
ZZdFactorization ** Ring
ZZdFactorization ** RingMap

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:1990:0.

RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map

Description

See also

Ways to use this method: