i1 : Q = ZZ/101[a,b,c];
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i2 : C = linearMF(a^3+b^3,t)
/ Q[t] \3 / Q[t] \3 / Q[t] \3 / Q[t] \3
o2 = |----------| <-- |----------| <-- |----------| <-- |----------|
| 2 | | 2 | | 2 | | 2 |
\t + t + 1/ \t + t + 1/ \t + t + 1/ \t + t + 1/
0 1 2 0
o2 : ZZdFactorization
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i3 : f = projectiveCover C
/ Q[t] \3 / Q[t] \9
o3 = 0 : |----------| <------------------------------------------------ |----------| : 0
| 2 | {0, 2} | 1 0 0 b a 0 b2 abt+ab a2 | | 2 |
\t + t + 1/ {0, 2} | 0 1 0 0 bt a a2 -b2t-b2 -ab | \t + t + 1/
{0, 2} | 0 0 1 a 0 -bt-b -abt a2 b2t |
/ Q[t] \3 / Q[t] \9
1 : |----------| <------------------------------------------------ |----------| : 1
| 2 | {0, 2} | b2 abt+ab a2 1 0 0 b a 0 | | 2 |
\t + t + 1/ {0, 2} | a2 -b2t-b2 -ab 0 1 0 0 bt a | \t + t + 1/
{0, 2} | -abt a2 b2t 0 0 1 a 0 -bt-b |
/ Q[t] \3 / Q[t] \9
2 : |----------| <------------------------------------------------ |----------| : 2
| 2 | {0, 2} | b a 0 b2 abt+ab a2 1 0 0 | | 2 |
\t + t + 1/ {0, 2} | 0 bt a a2 -b2t-b2 -ab 0 1 0 | \t + t + 1/
{0, 2} | a 0 -bt-b -abt a2 b2t 0 0 1 |
o3 : ZZdFactorizationMap
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i4 : isCommutative f
o4 = true
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i5 : prune coker f
o5 = 0 <-- 0 <-- 0 <-- 0
0 1 2 0
o5 : ZZdFactorization
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i6 : prune ker f
/ Q[t] \6 / Q[t] \6 / Q[t] \6 / Q[t] \6
o6 = |----------| <-- |----------| <-- |----------| <-- |----------|
| 2 | | 2 | | 2 | | 2 |
\t + t + 1/ \t + t + 1/ \t + t + 1/ \t + t + 1/
0 1 2 0
o6 : ZZdFactorization
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i7 : oo.dd
/ Q[t] \6 / Q[t] \6
o7 = 2 : |----------| <------------------------------------ |----------| : 0
| 2 | | 0 -bt -a -a2 b2t+b2 ab | | 2 |
\t + t + 1/ | -b -a 0 -b2 -abt-ab -a2 | \t + t + 1/
| 0 1 0 0 0 0 |
| -a 0 bt+b abt -a2 -b2t |
| 0 0 1 0 0 0 |
| 1 0 0 0 0 0 |
/ Q[t] \6 / Q[t] \6
0 : |----------| <------------------------------------ |----------| : 1
| 2 | | -abt-ab -a2 -b2 -b -a 0 | | 2 |
\t + t + 1/ | b2t+b2 ab -a2 0 -bt -a | \t + t + 1/
| -a2 -b2t abt -a 0 bt+b |
| 0 0 0 1 0 0 |
| 0 0 0 0 1 0 |
| 0 0 0 0 0 1 |
/ Q[t] \6 / Q[t] \6
1 : |----------| <------------------------------------ |----------| : 2
| 2 | | 1 0 0 0 0 0 | | 2 |
\t + t + 1/ | 0 0 0 1 0 0 | \t + t + 1/
| 0 1 0 0 0 0 |
| -a -b -abt-ab 0 -a2 -b2 |
| -bt 0 b2t+b2 -a ab -a2 |
| 0 -a -a2 bt+b -b2t abt |
o7 : ZZdFactorizationMap
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i8 : D = koszulMF(a^3+b^3+c^3)
4 4 4
o8 = Q <-- Q <-- Q
0 1 0
o8 : ZZdFactorization
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i9 : D.dd
4 4
o9 = 1 : Q <-------------------- Q : 0
| c2 0 b a |
| 0 c2 a2 -b2 |
| b2 a -c 0 |
| a2 -b 0 -c |
4 4
0 : Q <--------------------- Q : 1
| c 0 b a |
| 0 c a2 -b2 |
| b2 a -c2 0 |
| a2 -b 0 -c2 |
o9 : ZZdFactorizationMap
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i10 : g = projectiveCover D
4 8
o10 = 0 : Q <----------------------------- Q : 0
| 1 0 0 0 c 0 b a |
| 0 1 0 0 0 c a2 -b2 |
| 0 0 1 0 b2 a -c2 0 |
| 0 0 0 1 a2 -b 0 -c2 |
4 8
1 : Q <---------------------------- Q : 1
| c2 0 b a 1 0 0 0 |
| 0 c2 a2 -b2 0 1 0 0 |
| b2 a -c 0 0 0 1 0 |
| a2 -b 0 -c 0 0 0 1 |
o10 : ZZdFactorizationMap
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i11 : isCommutative g
o11 = true
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i12 : prune ker g
4 4 4
o12 = Q <-- Q <-- Q
0 1 0
o12 : ZZdFactorization
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i13 : oo.dd
4 4
o13 = 1 : Q <--------------------- Q : 0
| -a -b2 c2 0 |
| b -a2 0 c2 |
| -c 0 -a2 b2 |
| 0 -c -b -a |
4 4
0 : Q <---------------------- Q : 1
| -a2 b2 -c2 0 |
| -b -a 0 -c2 |
| c 0 -a -b2 |
| 0 c b -a2 |
o13 : ZZdFactorizationMap
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It turns out that the kernel of this projective when the period is 2 is isomorphic to the shift of the original factorization. In general, the kernel of the projective covering map gives a canonical method of defining the suspension of a d-fold factorization.