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Complexes :: Complex Array

Complex Array -- shift a complex

Synopsis

Description

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

See also