The suspension of a d-fold factorization may be canonically defined by taking the cokernel of the natural embedding of a factorization into its injective cover. When $d=2$, this recovers the shift functor of a factorization, but when $d > 2$ this object is an indecomposable yet nonminimal ZZ/d-graded factorization.
i1 : Q = ZZ/101[a,b,c];
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i2 : D = koszulMF(a^3+b^3+c^3)
4 4 4
o2 = Q <-- Q <-- Q
0 1 0
o2 : ZZdFactorization
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i3 : D.dd
4 4
o3 = 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 |
o3 : ZZdFactorizationMap
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i4 : suspension D
4 4 4
o4 = Q <-- Q <-- Q
0 1 0
o4 : ZZdFactorization
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i5 : oo == D[1]
o5 = true
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i6 : suspension(2, D) == D
o6 = true
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i7 : C = linearMF(a^3+b^3,t)
/ Q[t] \3 / Q[t] \3 / Q[t] \3 / Q[t] \3
o7 = |----------| <-- |----------| <-- |----------| <-- |----------|
| 2 | | 2 | | 2 | | 2 |
\t + t + 1/ \t + t + 1/ \t + t + 1/ \t + t + 1/
0 1 2 0
o7 : ZZdFactorization
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i8 : sC = suspension(C)
/ Q[t] \6 / Q[t] \6 / Q[t] \6 / Q[t] \6
o8 = |----------| <-- |----------| <-- |----------| <-- |----------|
| 2 | | 2 | | 2 | | 2 |
\t + t + 1/ \t + t + 1/ \t + t + 1/ \t + t + 1/
0 1 2 0
o8 : ZZdFactorization
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i9 : sC.dd
/ Q[t] \6 / Q[t] \6
o9 = 2 : |----------| <------------------------------ |----------| : 0
| 2 | | 0 0 0 -b2 -abt-ab -a2 | | 2 |
\t + t + 1/ | 0 0 0 -a2 b2t+b2 ab | \t + t + 1/
| 0 0 0 abt -a2 -b2t |
| 1 0 0 -b -a 0 |
| 0 1 0 0 -bt -a |
| 0 0 1 -a 0 bt+b |
/ Q[t] \6 / Q[t] \6
0 : |----------| <------------------------------ |----------| : 1
| 2 | | -b2 -abt-ab -a2 0 0 0 | | 2 |
\t + t + 1/ | -a2 b2t+b2 ab 0 0 0 | \t + t + 1/
| abt -a2 -b2t 0 0 0 |
| -b -a 0 1 0 0 |
| 0 -bt -a 0 1 0 |
| -a 0 bt+b 0 0 1 |
/ Q[t] \6 / Q[t] \6
1 : |----------| <------------------------------ |----------| : 2
| 2 | | 1 0 0 -b -a 0 | | 2 |
\t + t + 1/ | 0 1 0 0 -bt -a | \t + t + 1/
| 0 0 1 -a 0 bt+b |
| 0 0 0 -b2 -abt-ab -a2 |
| 0 0 0 -a2 b2t+b2 ab |
| 0 0 0 abt -a2 -b2t |
o9 : ZZdFactorizationMap
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i10 : sC.dd^3
/ Q[t] \6 / Q[t] \6
o10 = 0 : |----------| <------------------------------------------- |----------| : 0
| 2 | | a3+b3 0 0 0 0 0 | | 2 |
\t + t + 1/ | 0 a3+b3 0 0 0 0 | \t + t + 1/
| 0 0 a3+b3 0 0 0 |
| 0 0 0 a3+b3 0 0 |
| 0 0 0 0 a3+b3 0 |
| 0 0 0 0 0 a3+b3 |
/ Q[t] \6 / Q[t] \6
1 : |----------| <------------------------------------------- |----------| : 1
| 2 | | a3+b3 0 0 0 0 0 | | 2 |
\t + t + 1/ | 0 a3+b3 0 0 0 0 | \t + t + 1/
| 0 0 a3+b3 0 0 0 |
| 0 0 0 a3+b3 0 0 |
| 0 0 0 0 a3+b3 0 |
| 0 0 0 0 0 a3+b3 |
/ Q[t] \6 / Q[t] \6
2 : |----------| <------------------------------------------- |----------| : 2
| 2 | | a3+b3 0 0 0 0 0 | | 2 |
\t + t + 1/ | 0 a3+b3 0 0 0 0 | \t + t + 1/
| 0 0 a3+b3 0 0 0 |
| 0 0 0 a3+b3 0 0 |
| 0 0 0 0 a3+b3 0 |
| 0 0 0 0 0 a3+b3 |
o10 : ZZdFactorizationMap
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i11 : suspension(3,C)
/ Q[t] \24 / Q[t] \24 / Q[t] \24 / Q[t] \24
o11 = |----------| <-- |----------| <-- |----------| <-- |----------|
| 2 | | 2 | | 2 | | 2 |
\t + t + 1/ \t + t + 1/ \t + t + 1/ \t + t + 1/
0 1 2 0
o11 : ZZdFactorization
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i12 : isdFactorization oo
3 3
o12 = (true, a + b )
o12 : Sequence
|