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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 ith component is the direct sum of Hom(C1j, C2(j+i)) over all j. The differential on Hom(C1j, C2(j+i)) is the differential Hom(idC1, ddC2) + (-1)j Hom(ddC1, idC2). ddC1 ⊗idC2 + (-1)j idC1 ⊗ddC2.

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

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

i5 : prune HH D == Hom(C, coker vars S)

o5 = true

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

i6 : E = Hom(C, S^1)

      1      3      3      1
o6 = S  <-- S  <-- S  <-- S
                           
     -3     -2     -1     0

o6 : Complex
i7 : prune HH E

o7 = cokernel {-3} | c b a |
      
     -3

o7 : Complex