The shift defines a natural automorphism on the category of complexes. Topologists often call the shifted complex C[1] the suspension of C.
i1 : S = ZZ/101[a..d] o1 = S o1 : PolynomialRing |
i2 : C = freeResolution coker vars S 1 4 6 4 1 o2 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o2 : Complex |
i3 : dd^C_3 o3 = {2} | c d 0 0 | {2} | -b 0 d 0 | {2} | a 0 0 d | {2} | 0 -b -c 0 | {2} | 0 a 0 -c | {2} | 0 0 a b | 6 4 o3 : Matrix S <--- S |
i4 : D = C[1] 1 4 6 4 1 o4 = S <-- S <-- S <-- S <-- S -1 0 1 2 3 o4 : Complex |
i5 : dd^D_2 == -dd^C_3 o5 = true |
In order to shift the complex one step, and not change the differential, one can do the following.
i6 : (lo,hi) = concentration C o6 = (0, 4) o6 : Sequence |
i7 : E = complex(for i from lo+1 to hi list dd^C_i, Base=>-1) 1 4 6 4 1 o7 = S <-- S <-- S <-- S <-- S -1 0 1 2 3 o7 : Complex |
i8 : dd^E_2 == dd^C_3 o8 = true |