The shifted factorization $D$ is defined by $D_j = C_{i+j}$ for all $j$ and the $j$th differential is rescaled by $t^i$, where $t$ is some primitive root of unity. Note that a factorization of period > 2 must have a root of unity adjoined in order to define the shift.
The shifted complex map $g$ is defined by $g_j = f_{i+j}$ for all $j$.
The shift defines a natural automorphism on the category of ZZ/d-graded factorizations. For factorizations with period greater than $2$, the shift is not necessarily the same as the suspension functor; see TO (suspension, ZZdFactorization) to see how these functors differ.
i1 : R = QQ[a..d]/(c^2-b*d+a^2);
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i2 : m = ideal vars R
o2 = ideal (a, b, c, d)
o2 : Ideal of R
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i3 : C = tailMF (m^2)
24 24 24
o3 = (QQ[a..d]) <-- (QQ[a..d]) <-- (QQ[a..d])
0 1 0
o3 : ZZdFactorization
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i4 : dd^C_3
o4 = {5} | a 0 0 d 0 d 0 0 0 0 0 c 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
{5} | 0 0 a 0 0 0 0 0 d 0 0 0 0 0 c d 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
{5} | 0 0 0 0 -a 0 b 0 d 0 0 0 0 0 0 0 c d 0 b 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
{5} | 0 0 d 0 0 0 -a 0 0 0 0 0 0 0 0 0 0 0 b 0 c d
{5} | 0 -b 0 d 0 0 0 -a 0 0 0 0 0 0 0 0 0 0 0 0 0 c
{5} | 0 0 -b 0 0 0 0 0 -a 0 0 0 0 0 0 0 0 0 0 0 0 0
{5} | 0 0 0 -b 0 0 0 0 0 -a 0 0 0 0 0 0 0 0 0 0 0 0
{5} | -d 0 0 0 0 0 -c 0 0 0 -a 0 0 0 d 0 -d 0 0 -c 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
{5} | 0 -d 0 0 0 0 0 0 0 0 0 0 -a 0 0 0 0 0 -c 0 -d 0
{5} | 0 c -d 0 0 0 0 0 0 0 0 0 0 -a 0 0 0 0 0 0 0 -d
{5} | 0 0 c -d 0 0 0 0 0 0 0 0 0 0 -a 0 0 0 0 0 0 0
{5} | 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 -a 0 0 0 0 0 0
{5} | 0 0 0 0 c -d 0 0 0 0 b 0 0 0 0 0 a 0 0 0 0 0
{5} | 0 0 0 0 0 c 0 0 0 0 0 b 0 0 0 0 0 a 0 0 0 0
{5} | 0 0 0 0 0 0 -d 0 0 0 0 0 -c -d 0 0 0 0 a 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
{5} | 0 0 0 0 0 0 c -d 0 0 0 0 b 0 -d 0 0 0 0 0 a 0
{5} | 0 0 0 0 0 0 0 c -d 0 0 0 0 b 0 -d 0 0 0 0 0 a
{5} | 0 0 0 0 0 0 0 0 c -d 0 0 0 0 b 0 0 0 0 0 0 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 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
o4 : Matrix (QQ[a..d]) <-- (QQ[a..d])
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i5 : D = C[1]
24 24 24
o5 = (QQ[a..d]) <-- (QQ[a..d]) <-- (QQ[a..d])
0 1 0
o5 : ZZdFactorization
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i6 : assert isWellDefined D
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i7 : assert(dd^D_2 == -dd^C_3)
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i8 : Q = ZZ/101[a..c];
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i9 : K = linearMF(a^3 + b^3 + c^3, t)
/ Q[t] \9 / Q[t] \9 / Q[t] \9 / Q[t] \9
o9 = |----------| <-- |----------| <-- |----------| <-- |----------|
| 2 | | 2 | | 2 | | 2 |
\t + t + 1/ \t + t + 1/ \t + t + 1/ \t + t + 1/
0 1 2 0
o9 : ZZdFactorization
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i10 : K.dd
/ Q[t] \9 / Q[t] \9
o10 = 2 : |----------| <------------------------------------------------------- |----------| : 0
| 2 | {0, 3} | c 0 0 b a 0 0 0 0 | | 2 |
\t + t + 1/ {0, 3} | 0 c 0 0 bt a 0 0 0 | \t + t + 1/
{0, 3} | 0 0 c a 0 -bt-b 0 0 0 |
{0, 3} | 0 0 0 ct 0 0 b a 0 |
{0, 3} | 0 0 0 0 ct 0 0 bt a |
{0, 3} | 0 0 0 0 0 ct a 0 -bt-b |
{0, 3} | b a 0 0 0 0 -ct-c 0 0 |
{0, 3} | 0 bt a 0 0 0 0 -ct-c 0 |
{0, 3} | a 0 -bt-b 0 0 0 0 0 -ct-c |
/ Q[t] \9 / Q[t] \9
0 : |----------| <------------------------------------------------------- |----------| : 1
| 2 | {0, 3} | c 0 0 b a 0 0 0 0 | | 2 |
\t + t + 1/ {0, 3} | 0 c 0 0 bt a 0 0 0 | \t + t + 1/
{0, 3} | 0 0 c a 0 -bt-b 0 0 0 |
{0, 3} | 0 0 0 ct 0 0 b a 0 |
{0, 3} | 0 0 0 0 ct 0 0 bt a |
{0, 3} | 0 0 0 0 0 ct a 0 -bt-b |
{0, 3} | b a 0 0 0 0 -ct-c 0 0 |
{0, 3} | 0 bt a 0 0 0 0 -ct-c 0 |
{0, 3} | a 0 -bt-b 0 0 0 0 0 -ct-c |
/ Q[t] \9 / Q[t] \9
1 : |----------| <------------------------------------------------------- |----------| : 2
| 2 | {0, 3} | c 0 0 b a 0 0 0 0 | | 2 |
\t + t + 1/ {0, 3} | 0 c 0 0 bt a 0 0 0 | \t + t + 1/
{0, 3} | 0 0 c a 0 -bt-b 0 0 0 |
{0, 3} | 0 0 0 ct 0 0 b a 0 |
{0, 3} | 0 0 0 0 ct 0 0 bt a |
{0, 3} | 0 0 0 0 0 ct a 0 -bt-b |
{0, 3} | b a 0 0 0 0 -ct-c 0 0 |
{0, 3} | 0 bt a 0 0 0 0 -ct-c 0 |
{0, 3} | a 0 -bt-b 0 0 0 0 0 -ct-c |
o10 : ZZdFactorizationMap
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i11 : (K[1]).dd
/ Q[t] \9 / Q[t] \9
o11 = 2 : |----------| <-------------------------------------------------------- |----------| : 0
| 2 | {0, 3} | ct 0 0 bt at 0 0 0 0 | | 2 |
\t + t + 1/ {0, 3} | 0 ct 0 0 -bt-b at 0 0 0 | \t + t + 1/
{0, 3} | 0 0 ct at 0 b 0 0 0 |
{0, 3} | 0 0 0 -ct-c 0 0 bt at 0 |
{0, 3} | 0 0 0 0 -ct-c 0 0 -bt-b at |
{0, 3} | 0 0 0 0 0 -ct-c at 0 b |
{0, 3} | bt at 0 0 0 0 c 0 0 |
{0, 3} | 0 -bt-b at 0 0 0 0 c 0 |
{0, 3} | at 0 b 0 0 0 0 0 c |
/ Q[t] \9 / Q[t] \9
0 : |----------| <-------------------------------------------------------- |----------| : 1
| 2 | {0, 3} | ct 0 0 bt at 0 0 0 0 | | 2 |
\t + t + 1/ {0, 3} | 0 ct 0 0 -bt-b at 0 0 0 | \t + t + 1/
{0, 3} | 0 0 ct at 0 b 0 0 0 |
{0, 3} | 0 0 0 -ct-c 0 0 bt at 0 |
{0, 3} | 0 0 0 0 -ct-c 0 0 -bt-b at |
{0, 3} | 0 0 0 0 0 -ct-c at 0 b |
{0, 3} | bt at 0 0 0 0 c 0 0 |
{0, 3} | 0 -bt-b at 0 0 0 0 c 0 |
{0, 3} | at 0 b 0 0 0 0 0 c |
/ Q[t] \9 / Q[t] \9
1 : |----------| <-------------------------------------------------------- |----------| : 2
| 2 | {0, 3} | ct 0 0 bt at 0 0 0 0 | | 2 |
\t + t + 1/ {0, 3} | 0 ct 0 0 -bt-b at 0 0 0 | \t + t + 1/
{0, 3} | 0 0 ct at 0 b 0 0 0 |
{0, 3} | 0 0 0 -ct-c 0 0 bt at 0 |
{0, 3} | 0 0 0 0 -ct-c 0 0 -bt-b at |
{0, 3} | 0 0 0 0 0 -ct-c at 0 b |
{0, 3} | bt at 0 0 0 0 c 0 0 |
{0, 3} | 0 -bt-b at 0 0 0 0 c 0 |
{0, 3} | at 0 b 0 0 0 0 0 c |
o11 : ZZdFactorizationMap
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i12 : isWellDefined (K[1])
o12 = true
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i13 : potential (K[2])
3 3 3
o13 = a + b + c
Q[t]
o13 : ----------
2
t + t + 1
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i14 : assert(K[3] == K)
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In order to shift the factorization by one step, and not change the differential, one can do the following.