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SpectralSequences :: I-adic filtrations of chain complexes and their spectral sequences

I-adic filtrations of chain complexes and their spectral sequences

By multiplying a chain complex by sucessive powers of an ideal we obtain a filtered complex.

i1 : B = QQ[a..d]

o1 = B

o1 : PolynomialRing
i2 : J = ideal vars B

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

o2 : Ideal of B
i3 : C = complete res monomialCurveIdeal(B,{1,3,4})

      1      4      4      1
o3 = B  <-- B  <-- B  <-- B  <-- 0
                                  
     0      1      2      3      4

o3 : ChainComplex
i4 : K = filteredComplex(J,C,4)

o4 = -5 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
                                                           
          0           1           2           3           4

     -4 : image | a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 | <-- image {2} | a4 0  0  0  a3b 0   0   0   a3c 0   0   0   a3d 0   0   0   a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0    0    0    ab3 0   0   0   ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0    0    0    ac3 0   0   0   ac2d 0    0    0    acd2 0    0    0    ad3 0   0   0   b4 0  0  0  b3c 0   0   0   b3d 0   0   0   b2c2 0    0    0    b2cd 0    0    0    b2d2 0    0    0    bc3 0   0   0   bc2d 0    0    0    bcd2 0    0    0    bd3 0   0   0   c4 0  0  0  c3d 0   0   0   c2d2 0    0    0    cd3 0   0   0   d4 0  0  0  | <-- image {4} | a4 0  0  0  a3b 0   0   0   a3c 0   0   0   a3d 0   0   0   a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0    0    0    ab3 0   0   0   ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0    0    0    ac3 0   0   0   ac2d 0    0    0    acd2 0    0    0    ad3 0   0   0   b4 0  0  0  b3c 0   0   0   b3d 0   0   0   b2c2 0    0    0    b2cd 0    0    0    b2d2 0    0    0    bc3 0   0   0   bc2d 0    0    0    bcd2 0    0    0    bd3 0   0   0   c4 0  0  0  c3d 0   0   0   c2d2 0    0    0    cd3 0   0   0   d4 0  0  0  | <-- image {5} | a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 | <-- image 0
                                                                                                                                                                                         {3} | 0  a4 0  0  0   a3b 0   0   0   a3c 0   0   0   a3d 0   0   0    a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0    0    0   ab3 0   0   0    ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0    0    0   ac3 0   0   0    ac2d 0    0    0    acd2 0    0    0   ad3 0   0   0  b4 0  0  0   b3c 0   0   0   b3d 0   0   0    b2c2 0    0    0    b2cd 0    0    0    b2d2 0    0    0   bc3 0   0   0    bc2d 0    0    0    bcd2 0    0    0   bd3 0   0   0  c4 0  0  0   c3d 0   0   0    c2d2 0    0    0   cd3 0   0   0  d4 0  0  |           {4} | 0  a4 0  0  0   a3b 0   0   0   a3c 0   0   0   a3d 0   0   0    a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0    0    0   ab3 0   0   0    ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0    0    0   ac3 0   0   0    ac2d 0    0    0    acd2 0    0    0   ad3 0   0   0  b4 0  0  0   b3c 0   0   0   b3d 0   0   0    b2c2 0    0    0    b2cd 0    0    0    b2d2 0    0    0   bc3 0   0   0    bc2d 0    0    0    bcd2 0    0    0   bd3 0   0   0  c4 0  0  0   c3d 0   0   0    c2d2 0    0    0   cd3 0   0   0  d4 0  0  |                                                                                                                                                                                   
          0                                                                                                                                                                              {3} | 0  0  a4 0  0   0   a3b 0   0   0   a3c 0   0   0   a3d 0   0    0    a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0    0   0   ab3 0   0    0    ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0    0   0   ac3 0   0    0    ac2d 0    0    0    acd2 0    0   0   ad3 0   0  0  b4 0  0   0   b3c 0   0   0   b3d 0   0    0    b2c2 0    0    0    b2cd 0    0    0    b2d2 0    0   0   bc3 0   0    0    bc2d 0    0    0    bcd2 0    0   0   bd3 0   0  0  c4 0  0   0   c3d 0   0    0    c2d2 0    0   0   cd3 0   0  0  d4 0  |           {4} | 0  0  a4 0  0   0   a3b 0   0   0   a3c 0   0   0   a3d 0   0    0    a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0    0   0   ab3 0   0    0    ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0    0   0   ac3 0   0    0    ac2d 0    0    0    acd2 0    0   0   ad3 0   0  0  b4 0  0   0   b3c 0   0   0   b3d 0   0    0    b2c2 0    0    0    b2cd 0    0    0    b2d2 0    0   0   bc3 0   0    0    bc2d 0    0    0    bcd2 0    0   0   bd3 0   0  0  c4 0  0   0   c3d 0   0    0    c2d2 0    0   0   cd3 0   0  0  d4 0  |     3                                                                                                                                                                            4
                                                                                                                                                                                         {3} | 0  0  0  a4 0   0   0   a3b 0   0   0   a3c 0   0   0   a3d 0    0    0    a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0   0   0   ab3 0    0    0    ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0   0   0   ac3 0    0    0    ac2d 0    0    0    acd2 0   0   0   ad3 0  0  0  b4 0   0   0   b3c 0   0   0   b3d 0    0    0    b2c2 0    0    0    b2cd 0    0    0    b2d2 0   0   0   bc3 0    0    0    bc2d 0    0    0    bcd2 0   0   0   bd3 0  0  0  c4 0   0   0   c3d 0    0    0    c2d2 0   0   0   cd3 0  0  0  d4 |           {4} | 0  0  0  a4 0   0   0   a3b 0   0   0   a3c 0   0   0   a3d 0    0    0    a2b2 0    0    0    a2bc 0    0    0    a2bd 0    0    0    a2c2 0    0    0    a2cd 0    0    0    a2d2 0   0   0   ab3 0    0    0    ab2c 0    0    0    ab2d 0    0    0    abc2 0    0    0    abcd 0    0    0    abd2 0   0   0   ac3 0    0    0    ac2d 0    0    0    acd2 0   0   0   ad3 0  0  0  b4 0   0   0   b3c 0   0   0   b3d 0    0    0    b2c2 0    0    0    b2cd 0    0    0    b2d2 0   0   0   bc3 0    0    0    bc2d 0    0    0    bcd2 0   0   0   bd3 0  0  0  c4 0   0   0   c3d 0    0    0    c2d2 0   0   0   cd3 0  0  0  d4 |
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                   1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             2

     -3 : image | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 | <-- image {2} | a3 0  0  0  a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0   0   0   b3 0  0  0  b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0   0   0   c3 0  0  0  c2d 0   0   0   cd2 0   0   0   d3 0  0  0  | <-- image {4} | a3 0  0  0  a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0   0   0   b3 0  0  0  b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0   0   0   c3 0  0  0  c2d 0   0   0   cd2 0   0   0   d3 0  0  0  | <-- image {5} | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 | <-- image 0
                                                                                                          {3} | 0  a3 0  0  0   a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0   0   0  b3 0  0  0   b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0   0   0  c3 0  0  0   c2d 0   0   0   cd2 0   0   0  d3 0  0  |           {4} | 0  a3 0  0  0   a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0   0   0  b3 0  0  0   b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0   0   0  c3 0  0  0   c2d 0   0   0   cd2 0   0   0  d3 0  0  |                                                                                                    
          0                                                                                               {3} | 0  0  a3 0  0   0   a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0   0  0  b3 0  0   0   b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0   0  0  c3 0  0   0   c2d 0   0   0   cd2 0   0  0  d3 0  |           {4} | 0  0  a3 0  0   0   a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0   0  0  b3 0  0   0   b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0   0  0  c3 0  0   0   c2d 0   0   0   cd2 0   0  0  d3 0  |     3                                                                                             4
                                                                                                          {3} | 0  0  0  a3 0   0   0   a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0  0  0  b3 0   0   0   b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0  0  0  c3 0   0   0   c2d 0   0   0   cd2 0  0  0  d3 |           {4} | 0  0  0  a3 0   0   0   a2b 0   0   0   a2c 0   0   0   a2d 0   0   0   ab2 0   0   0   abc 0   0   0   abd 0   0   0   ac2 0   0   0   acd 0   0   0   ad2 0  0  0  b3 0   0   0   b2c 0   0   0   b2d 0   0   0   bc2 0   0   0   bcd 0   0   0   bd2 0  0  0  c3 0   0   0   c2d 0   0   0   cd2 0  0  0  d3 |
                                                                                                                                                                                                                                                                                                                                                                                                                                       
                                                                                                    1                                                                                                                                                                                                                                                                                                                                 2

     -2 : image | a2 ab ac ad b2 bc bd c2 cd d2 | <-- image {2} | a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 0  0  0  | <-- image {4} | a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 0  0  0  | <-- image {5} | a2 ab ac ad b2 bc bd c2 cd d2 | <-- image 0
                                                            {3} | 0  a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 0  0  |           {4} | 0  a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 0  0  |                                                      
          0                                                 {3} | 0  0  a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 0  |           {4} | 0  0  a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 0  |     3                                               4
                                                            {3} | 0  0  0  a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 |           {4} | 0  0  0  a2 0  0  0  ab 0  0  0  ac 0  0  0  ad 0  0  0  b2 0  0  0  bc 0  0  0  bd 0  0  0  c2 0  0  0  cd 0  0  0  d2 |
                                                                                                                                                                                                 
                                                      1                                                                                                                                         2

     -1 : image | a b c d | <-- image {2} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 | <-- image {4} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 | <-- image {5} | a b c d | <-- image 0
                                      {3} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 |           {4} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 |                                
          0                           {3} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 |           {4} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 |     3                         4
                                      {3} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d |           {4} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d |
                                                                                   
                                1                                                 2

          1      4      4      1
     0 : B  <-- B  <-- B  <-- B  <-- 0
                                      
         0      1      2      3      4

o4 : FilteredComplex

Here are some higher pages of the associated spectral sequence:

i5 : E = prune spectralSequence K

o5 = E

o5 : SpectralSequence
i6 : E^4

     +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+
o6 = |cokernel {4} | -d -b 0  -c 0  0  -d -c 0  0  0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||0                                                                                                                       |0                                                                                       |0                                               |0                   |
     |         {4} | c  a  -d 0  -b 0  0  0  0  -d 0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  c  b  a  -d 0  0  -d 0  0  0  0  0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||{-3, 4}                                                                                                                 |{-2, 4}                                                                                 |{-1, 4}                                         |{0, 4}              |
     |         {4} | 0  0  0  0  0  c  b  a  0  0  0  -d 0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  b  a  c  0  0  0  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  -d c  b  a  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  ||                                                                                                                        |                                                                                        |                                                |                    |
     |         {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  ||                                                                                                                        |                                                                                        |                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |                                                |                    |
     |{-4, 4}                                                                                                                                                      |                                                                                                                        |                                                                                        |                                                |                    |
     +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+
     |0                                                                                                                                                            |cokernel {3} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||0                                                                                       |0                                               |0                   |
     |                                                                                                                                                             |         {3} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |{-4, 3}                                                                                                                                                      |         {3} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||{-2, 3}                                                                                 |{-1, 3}                                         |{0, 3}              |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |         {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a ||                                                                                        |                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |                                                |                    |
     |                                                                                                                                                             |{-3, 3}                                                                                                                 |                                                                                        |                                                |                    |
     +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+
     |0                                                                                                                                                            |0                                                                                                                       |cokernel {2} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||0                                               |0                   |
     |                                                                                                                                                             |                                                                                                                        |         {2} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                |                    |
     |{-4, 2}                                                                                                                                                      |{-3, 2}                                                                                                                 |         {2} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||{-1, 2}                                         |{0, 2}              |
     |                                                                                                                                                             |                                                                                                                        |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 ||                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 ||                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 ||                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |         {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a ||                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |{-2, 2}                                                                                 |                                                |                    |
     +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+
     |0                                                                                                                                                            |0                                                                                                                       |0                                                                                       |cokernel {1} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 ||0                   |
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |         {1} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 ||                    |
     |{-4, 1}                                                                                                                                                      |{-3, 1}                                                                                                                 |{-2, 1}                                                                                 |         {1} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 ||{0, 1}              |
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |         {1} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a ||                    |
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |                                                |                    |
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |{-1, 1}                                         |                    |
     +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+
     |0                                                                                                                                                            |0                                                                                                                       |0                                                                                       |0                                               |cokernel | d c b a ||
     |                                                                                                                                                             |                                                                                                                        |                                                                                        |                                                |                    |
     |{-4, 0}                                                                                                                                                      |{-3, 0}                                                                                                                 |{-2, 0}                                                                                 |{-1, 0}                                         |{0, 0}              |
     +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+

o6 : SpectralSequencePage
i7 : E^4 .dd

o7 = {-9, 8} : 0 <----- 0 : {-5, 5}
                    0

     {-9, 9} : 0 <----- 0 : {-5, 6}
                    0

     {-9, 10} : 0 <----- 0 : {-5, 7}
                     0

     {-9, 11} : 0 <----- 0 : {-5, 8}
                     0

     {-9, 12} : 0 <----- 0 : {-5, 9}
                     0

     {-4, 3} : 0 <----- cokernel | d c b a | : {0, 0}
                    0

     {-4, 4} : cokernel {4} | -d -b 0  -c 0  0  -d -c 0  0  0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  | <----- 0 : {0, 1}
                        {4} | c  a  -d 0  -b 0  0  0  0  -d 0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |    0
                        {4} | 0  0  c  b  a  -d 0  0  -d 0  0  0  0  0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  c  b  a  0  0  0  -d 0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  b  a  c  0  0  0  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  -d c  b  a  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  |
                        {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  |

     {-4, 5} : 0 <----- 0 : {0, 2}
                    0

     {-4, 6} : 0 <----- 0 : {0, 3}
                    0

     {-4, 7} : 0 <----- 0 : {0, 4}
                    0

     {-5, 4} : 0 <----- cokernel {1} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 | : {-1, 1}
                    0            {1} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 |
                                 {1} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 |
                                 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |

     {-5, 5} : 0 <----- 0 : {-1, 2}
                    0

     {-5, 6} : 0 <----- 0 : {-1, 3}
                    0

     {-5, 7} : 0 <----- 0 : {-1, 4}
                    0

     {-5, 8} : 0 <----- 0 : {-1, 5}
                    0

     {-6, 5} : 0 <----- cokernel {2} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | : {-2, 2}
                    0            {2} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {2} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 |
                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 |
                                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |

     {-6, 6} : 0 <----- 0 : {-2, 3}
                    0

     {-6, 7} : 0 <----- 0 : {-2, 4}
                    0

     {-6, 8} : 0 <----- 0 : {-2, 5}
                    0

     {-6, 9} : 0 <----- 0 : {-2, 6}
                    0

     {-7, 6} : 0 <----- cokernel {3} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | : {-3, 3}
                    0            {3} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 |
                                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |

     {-7, 7} : 0 <----- 0 : {-3, 4}
                    0

     {-7, 8} : 0 <----- 0 : {-3, 5}
                    0

     {-7, 9} : 0 <----- 0 : {-3, 6}
                    0

     {-7, 10} : 0 <----- 0 : {-3, 7}
                     0

     {-8, 7} : 0 <----- cokernel {4} | -d -b 0  -c 0  0  -d -c 0  0  0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  | : {-4, 4}
                    0            {4} | c  a  -d 0  -b 0  0  0  0  -d 0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  c  b  a  -d 0  0  -d 0  0  0  0  0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  c  b  a  0  0  0  -d 0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  b  a  c  0  0  0  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  -d c  b  a  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d 0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  0  0  -d |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  0  -d 0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  0  0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  -d 0  0  |
                                 {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c  b  a  |

     {-8, 8} : 0 <----- 0 : {-4, 5}
                    0

     {-8, 9} : 0 <----- 0 : {-4, 6}
                    0

     {-8, 10} : 0 <----- 0 : {-4, 7}
                     0

     {-8, 11} : 0 <----- 0 : {-4, 8}
                     0

o7 : SpectralSequencePageMap