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Complexes :: Hom(Complex,Complex)

Hom(Complex,Complex) -- the complex of homomorphisms between two complexes

Synopsis

Description

The complex of homomorphisms is a complex $D$ whose $i$th component is the direct sum of $Hom(C1_j, C2_{j+i})$ over all $j$. The differential on $Hom(C1_j, C2_{j+i})$ is the differential $Hom(id_{C1}, dd^{C2}) + (-1)^j Hom(dd^{C1}, id_{C2})$. $dd^{C1} \otimes id_{C2} + (-1)^j id_{C1} \otimes dd^{C2}$.

In particular, for this operation to be well-defined, both arguments must have the same underlying ring.

i1 : S = ZZ/101[a..c]

o1 = S

o1 : PolynomialRing
i2 : C = freeResolution coker vars S

      1      3      3      1
o2 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o2 : Complex
i3 : D = Hom(C,C)

      1      6      15      20      15      6      1
o3 = S  <-- S  <-- S   <-- S   <-- S   <-- S  <-- S
                                                   
     -3     -2     -1      0       1       2      3

o3 : Complex
i4 : dd^D

           1                                6
o4 = -3 : S  <---------------------------- S  : -2
                {-3} | c -b a -a -b -c |

           6                                                      15
     -2 : S  <-------------------------------------------------- S   : -1
                {-2} | -b a  0 a b c 0  0  0  0 0 0 0  0  0  |
                {-2} | -c 0  a 0 0 0 a  b  c  0 0 0 0  0  0  |
                {-2} | 0  -c b 0 0 0 0  0  0  a b c 0  0  0  |
                {-2} | 0  0  0 c 0 0 -b 0  0  a 0 0 -b -c 0  |
                {-2} | 0  0  0 0 c 0 0  -b 0  0 a 0 a  0  -c |
                {-2} | 0  0  0 0 0 c 0  0  -b 0 0 a 0  a  b  |

           15                                                                           20
     -1 : S   <----------------------------------------------------------------------- S   : 0
                 {-1} | a -a -b -c 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                 {-1} | b 0  0  0  -a -b -c 0  0  0  0  0  0  0  0  0  0  0  0  0  |
                 {-1} | c 0  0  0  0  0  0  -a -b -c 0  0  0  0  0  0  0  0  0  0  |
                 {-1} | 0 -b 0  0  a  0  0  0  0  0  b  c  0  0  0  0  0  0  0  0  |
                 {-1} | 0 0  -b 0  0  a  0  0  0  0  -a 0  c  0  0  0  0  0  0  0  |
                 {-1} | 0 0  0  -b 0  0  a  0  0  0  0  -a -b 0  0  0  0  0  0  0  |
                 {-1} | 0 -c 0  0  0  0  0  a  0  0  0  0  0  b  c  0  0  0  0  0  |
                 {-1} | 0 0  -c 0  0  0  0  0  a  0  0  0  0  -a 0  c  0  0  0  0  |
                 {-1} | 0 0  0  -c 0  0  0  0  0  a  0  0  0  0  -a -b 0  0  0  0  |
                 {-1} | 0 0  0  0  -c 0  0  b  0  0  0  0  0  0  0  0  b  c  0  0  |
                 {-1} | 0 0  0  0  0  -c 0  0  b  0  0  0  0  0  0  0  -a 0  c  0  |
                 {-1} | 0 0  0  0  0  0  -c 0  0  b  0  0  0  0  0  0  0  -a -b 0  |
                 {-1} | 0 0  0  0  0  0  0  0  0  0  c  0  0  -b 0  0  a  0  0  -c |
                 {-1} | 0 0  0  0  0  0  0  0  0  0  0  c  0  0  -b 0  0  a  0  b  |
                 {-1} | 0 0  0  0  0  0  0  0  0  0  0  0  c  0  0  -b 0  0  a  -a |

          20                                                     15
     0 : S   <------------------------------------------------- S   : 1
                | a b c 0  0  0  0  0  0  0  0  0  0  0  0  |
                | a 0 0 -b -c 0  0  0  0  0  0  0  0  0  0  |
                | 0 a 0 a  0  -c 0  0  0  0  0  0  0  0  0  |
                | 0 0 a 0  a  b  0  0  0  0  0  0  0  0  0  |
                | b 0 0 0  0  0  -b -c 0  0  0  0  0  0  0  |
                | 0 b 0 0  0  0  a  0  -c 0  0  0  0  0  0  |
                | 0 0 b 0  0  0  0  a  b  0  0  0  0  0  0  |
                | c 0 0 0  0  0  0  0  0  -b -c 0  0  0  0  |
                | 0 c 0 0  0  0  0  0  0  a  0  -c 0  0  0  |
                | 0 0 c 0  0  0  0  0  0  0  a  b  0  0  0  |
                | 0 0 0 -b 0  0  a  0  0  0  0  0  c  0  0  |
                | 0 0 0 0  -b 0  0  a  0  0  0  0  -b 0  0  |
                | 0 0 0 0  0  -b 0  0  a  0  0  0  a  0  0  |
                | 0 0 0 -c 0  0  0  0  0  a  0  0  0  c  0  |
                | 0 0 0 0  -c 0  0  0  0  0  a  0  0  -b 0  |
                | 0 0 0 0  0  -c 0  0  0  0  0  a  0  a  0  |
                | 0 0 0 0  0  0  -c 0  0  b  0  0  0  0  c  |
                | 0 0 0 0  0  0  0  -c 0  0  b  0  0  0  -b |
                | 0 0 0 0  0  0  0  0  -c 0  0  b  0  0  a  |
                | 0 0 0 0  0  0  0  0  0  0  0  0  c  -b a  |

          15                                 6
     1 : S   <----------------------------- S  : 2
                {1} | b  c  0  0  0  0  |
                {1} | -a 0  c  0  0  0  |
                {1} | 0  -a -b 0  0  0  |
                {1} | a  0  0  -c 0  0  |
                {1} | 0  a  0  b  0  0  |
                {1} | 0  0  a  -a 0  0  |
                {1} | b  0  0  0  -c 0  |
                {1} | 0  b  0  0  b  0  |
                {1} | 0  0  b  0  -a 0  |
                {1} | c  0  0  0  0  -c |
                {1} | 0  c  0  0  0  b  |
                {1} | 0  0  c  0  0  -a |
                {1} | 0  0  0  -b a  0  |
                {1} | 0  0  0  -c 0  a  |
                {1} | 0  0  0  0  -c b  |

          6                  1
     2 : S  <-------------- S  : 3
               {2} | c  |
               {2} | -b |
               {2} | a  |
               {2} | a  |
               {2} | b  |
               {2} | c  |

o4 : ComplexMap
i5 : assert isWellDefined D

The homology of this complex is $Hom(C, ZZ/101)$

i6 : prune HH D == Hom(C, coker vars S)

o6 = true

If one of the arguments is a module or a ring, it is considered as a complex concentrated in homological degree 0.

i7 : E = Hom(C, S^2)

      2      6      6      2
o7 = S  <-- S  <-- S  <-- S
                           
     -3     -2     -1     0

o7 : Complex
i8 : prune HH E

o8 = cokernel {-3} | c b a 0 0 0 |
              {-3} | 0 0 0 c b a |
      
     -3

o8 : Complex

There is a simple relationship between Hom complexes and shifts. Specifically, shifting the first argument is the same as the negative shift of the result. But shifting the second argument is only the same as the positive shift of the result up to a sign.

i9 : Hom(C[3], C) == D[-3]

o9 = true
i10 : Hom(C, C[-2]) == D[-2]

o10 = true
i11 : Hom(C, C[-3]) != D[-3]

o11 = true
i12 : Hom(C, C[-3]) == complex(- dd^(D[-3]))

o12 = true

Specific maps and morphisms between complexes can be obtained with homomorphism(ComplexMap).

Because the Hom complex can be regarded as the total complex of a double complex, each term comes with pairs of indices, labelling the summands.

i13 : indices D_-1

o13 = {{0, -1}, {1, 0}, {2, 1}, {3, 2}}

o13 : List
i14 : components D_-1

           3   9   3
o14 = {0, S , S , S }

o14 : List
i15 : indices D_-2

o15 = {{0, -2}, {1, -1}, {2, 0}, {3, 1}}

o15 : List
i16 : components D_-2

              3   3
o16 = {0, 0, S , S }

o16 : List

See also