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Complexes :: gradedModule(Complex)

gradedModule(Complex) -- a new complex in which the differential is zero

Synopsis

Description

This routine isolates the terms in the complex and forgets the differentials

i1 : R = ZZ/101[a,b,c,d,e];
i2 : I = intersect(ideal(a,b),ideal(c,d,e))

o2 = ideal (b*e, a*e, b*d, a*d, b*c, a*c)

o2 : Ideal of R
i3 : C = (dual freeResolution I)[-4]

      1      5      9      6      1
o3 = R  <-- R  <-- R  <-- R  <-- R
                                  
     0      1      2      3      4

o3 : Complex
i4 : dd^C

          1                             5
o4 = 0 : R  <------------------------- R  : 1
               {-5} | -e d -c -b a |

          5                                        9
     1 : R  <------------------------------------ R  : 2
               {-4} | d -c -b a 0  0  0  0  0 |
               {-4} | e 0  0  0 -c -b a  0  0 |
               {-4} | 0 e  0  0 -d 0  0  -b a |
               {-4} | 0 0  e  0 0  -d 0  c  0 |
               {-4} | 0 0  0  e 0  0  -d 0  c |

          9                                 6
     2 : R  <----------------------------- R  : 3
               {-3} | -b a  0  0  0  0 |
               {-3} | 0  0  -b a  0  0 |
               {-3} | -d 0  c  0  0  0 |
               {-3} | 0  -d 0  c  0  0 |
               {-3} | 0  0  0  0  -b a |
               {-3} | -e 0  0  0  c  0 |
               {-3} | 0  -e 0  0  0  c |
               {-3} | 0  0  -e 0  d  0 |
               {-3} | 0  0  0  -e 0  d |

          6                   1
     3 : R  <--------------- R  : 4
               {-2} | ac |
               {-2} | bc |
               {-2} | ad |
               {-2} | bd |
               {-2} | ae |
               {-2} | be |

o4 : ComplexMap
i5 : G = gradedModule C

      1      5      9      6      1
o5 = R  <-- R  <-- R  <-- R  <-- R
                                  
     0      1      2      3      4

o5 : Complex
i6 : dd^G

o6 = 0

o6 : ComplexMap
i7 : assert(isWellDefined G)
i8 : assert(G != C)

The homology of a complex already has zero differential.

i9 : H = HH C

o9 = cokernel {-5} | -e d -c -b a | <-- subquotient ({-4} | d c  0 b  0 0  a 0  0 0 |, {-4} | d -c -b a 0  0  0  0  0 |) <-- subquotient ({-3} | c b a  0  0  0  0 |, {-3} | -b a  0  0  0  0 |) <-- subquotient ({-2} | ac |, {-2} | ac |) <-- image 0
                                                     {-4} | e 0  c 0  b 0  0 a  0 0 |  {-4} | e 0  0  0 -c -b a  0  0 |                   {-3} | d 0 0  b  0  a  0 |  {-3} | 0  0  -b a  0  0 |                   {-2} | bc |  {-2} | bc |       
     0                                               {-4} | 0 -e d 0  0 b  0 0  a 0 |  {-4} | 0 e  0  0 -d 0  0  -b a |                   {-3} | 0 d 0  -c 0  0  0 |  {-3} | -d 0  c  0  0  0 |                   {-2} | ad |  {-2} | ad |      4
                                                     {-4} | 0 0  0 -e d -c 0 0  0 a |  {-4} | 0 0  e  0 0  -d 0  c  0 |                   {-3} | 0 0 -d 0  0  c  0 |  {-3} | 0  -d 0  c  0  0 |                   {-2} | bd |  {-2} | bd |
                                                     {-4} | 0 0  0 0  0 0  e -d c b |  {-4} | 0 0  0  e 0  0  -d 0  c |                   {-3} | e 0 0  0  b  0  a |  {-3} | 0  0  0  0  -b a |                   {-2} | ae |  {-2} | ae |
                                                                                                                                          {-3} | 0 e 0  0  -c 0  0 |  {-3} | -e 0  0  0  c  0 |                   {-2} | be |  {-2} | be |
                                        1                                                                                                 {-3} | 0 0 -e 0  0  0  c |  {-3} | 0  -e 0  0  0  c |       
                                                                                                                                          {-3} | 0 0 0  e  -d 0  0 |  {-3} | 0  0  -e 0  d  0 |      3
                                                                                                                                          {-3} | 0 0 0  0  0  -e d |  {-3} | 0  0  0  -e 0  d |
                                                                                                                              
                                                                                                                             2

o9 : Complex
i10 : prune H

o10 = cokernel {-5} | e d c b a | <-- cokernel {-3} | e d c | <-- cokernel {-2} | b a |
                                                                   
      0                               1                           2

o10 : Complex
i11 : dd^H == 0

o11 = true
i12 : assert(H == gradedModule H)

See also