Filtrations and homomorphism complexes

Let $S$ be a commutative ring and let $B : \dots \rightarrow B_{i} \rightarrow B_{i - 1} \rightarrow \cdots $ and $C : \dots \rightarrow C_{i} \rightarrow C_{i - 1} \rightarrow \cdots $ be chain complexes.

For all integers $p$ and $q$ let $K_{p,q} := Hom_S(B_{-p}, C_q)$, let $d'_{p,q} : K_{p,q} \rightarrow K_{p - 1, q}$ denote the homorphism $ \phi \mapsto \partial^B_{-p + 1} \phi$, and let $d^{''}_{p,q} : K_{p,q} \rightarrow K_{p, q - 1} $ denote the homorphism $\phi \mapsto (-1)^p \partial^C_q \phi$.

The chain complex $Hom(B, C)$ is given by $ Hom(B, C)_k := \prod_{p + q = k} Hom_S(B_{-p}, C_q) $ and the differentials by $ \partial := d^{'} + d^{''} $; it carries two natural ascending filtrations $F' ( Hom(B, C) )$ and $F''( Hom(B, C))$.

The first is obtained by letting $F'_n (Hom(B, C))$ be the chain complex determined by setting $F'_n (Hom(B, C))_k := \prod_{p + q = k , p \leq n} Hom_S(B_{-p}, C_q)$ and the differentials $\partial := d' + d''$.

The second is obtained by letting $F''_n (Hom(B, C)) := \prod_{p + q = k , q \leq n} Hom_S(B_{-p}, C_q)$ and the differentials $\partial := d' + d''$.

In Macaulay2, using this package, $F'$ and $F''$ as defined above are computed as illustrated in the following example, by using Hom(filteredComplex B, C) or Hom(B,filteredComplex C).

i1 : A = QQ[x,y,z,w];

i2 : B = res monomialCurveIdeal(A, {1,2,3});

i3 : C = res monomialCurveIdeal(A, {1,3,4});

i4 : F' = Hom(filteredComplex B, C)

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

     -2 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0
                            {-3} | 0 1 |           {-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 0 |           {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                              
          -3                                       {-2} | 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 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 |     2           3           4
                      -2                           {-1} | 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 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 1 0 0 0 0 0 0 |           {1} | 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 0 0 0 0 0 0 |
                                                   {0}  | 0 0 0 0 0 1 0 0 0 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 0 0 0 0 0 0 |
                                                   {0}  | 0 0 0 0 0 0 1 0 0 0 0 |           {1} | 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 0 0 0 0 0 0 |
                                                   {-1} | 0 0 0 0 0 0 0 1 0 0 0 |           {1} | 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 0 0 0 0 0 0 |
                                                   {0}  | 0 0 0 0 0 0 0 0 1 0 0 |           {1} | 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 0 0 0 0 0 0 |
                                                   {0}  | 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 |
                                                   {0}  | 0 0 0 0 0 0 0 0 0 0 1 |           {1} | 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 0 0 0 0 0 0 |
                                                                                            {1} | 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 0 0 0 0 0 0 |
                                             -1                                             {1} | 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 0 0 0 0 0 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |      
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |     1
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                                       
                                                                                      0

     -1 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {4} | 0 0 0 0 0 0 0 | <-- image 0 <-- image 0
                            {-3} | 0 1 |           {-2} | 0 1 0 0 0 0 0 0 0 0 0 |           {0} | 0 1 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 |           {4} | 0 0 0 0 0 0 0 |                  
          -3                                       {-2} | 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 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 |           {4} | 0 0 0 0 0 0 0 |     3           4
                      -2                           {-1} | 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 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 |           {4} | 0 0 0 0 0 0 0 |
                                                   {0}  | 0 0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 0 0 1 0 0 |
                                                   {0}  | 0 0 0 0 0 1 0 0 0 0 0 |           {0} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 0 0 0 1 0 |
                                                   {0}  | 0 0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 0 0 0 0 1 |
                                                   {-1} | 0 0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |      
                                                   {0}  | 0 0 0 0 0 0 0 0 1 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |     2
                                                   {0}  | 0 0 0 0 0 0 0 0 0 1 0 |           {0} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                                                   {0}  | 0 0 0 0 0 0 0 0 0 0 1 |           {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                                             -1                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |      
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |     1
                                                                                            {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                                       
                                                                                      0

                 2      11      21      18      7      1
     0 : 0  <-- A  <-- A   <-- A   <-- A   <-- A  <-- A  <-- 0
                                                              
         -3     -2     -1      0       1       2      3      4

o4 : FilteredComplex

i5 : F'' = Hom(B,filteredComplex C)

o5 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
                                                                                               
          -3          -2          -1          0           1           2           3           4

     0 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0 <-- image 0
                           {-3} | 0 1 |           {-2} | 0 1 0 0 0 0 0 0 0 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                                       {-2} | 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     1           2           3           4
                     -2                           {-1} | 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 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} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                  {0}  | 0 0 0 0 0 0 0 0 0 0 0 |           {0} | 0 0 0 0 0 0 0 0 0 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} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                  {-1} | 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 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} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                  {0}  | 0 0 0 0 0 0 0 0 0 0 0 |           {0} | 0 0 0 0 0 0 0 0 0 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} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                            -1                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                      
                                                                                     0

     1 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0
                           {-3} | 0 1 |           {-2} | 0 1 0 0 0 0 0 0 0 0 0 |           {0} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                              
         -3                                       {-2} | 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     2           3           4
                     -2                           {-1} | 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                  {0}  | 0 0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 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 0 0 0 0 1 0 0 0 0 0 |           {0} | 0 0 0 0 0 1 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 0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 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 |
                                                  {-1} | 0 0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 1 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 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 0 0 0 0 1 0 |           {0} | 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 1 |           {1} | 0 0 0 0 0 0 0 0 0 0 1 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 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 1 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 |
                                            -1                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 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 |
                                                                                           {1} | 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 0 0 0 0 0 0 |
                                                                                           {1} | 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 0 0 0 0 0 0 |
                                                                                           {1} | 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 0 0 0 0 0 0 |
                                                                                           {1} | 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 0 0 0 0 0 0 |
                                                                                           {1} | 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 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     1
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                      
                                                                                     0

     2 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 | <-- image 0 <-- image 0
                           {-3} | 0 1 |           {-2} | 0 1 0 0 0 0 0 0 0 0 0 |           {0} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {4} | 0 1 0 0 0 0 0 |                  
         -3                                       {-2} | 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {4} | 0 0 1 0 0 0 0 |     3           4
                     -2                           {-1} | 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {4} | 0 0 0 1 0 0 0 |
                                                  {0}  | 0 0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 1 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 1 0 0 0 0 0 |           {0} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 1 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 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {3} | 0 0 0 0 0 0 0 |
                                                  {-1} | 0 0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |      
                                                  {0}  | 0 0 0 0 0 0 0 0 1 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |     2
                                                  {0}  | 0 0 0 0 0 0 0 0 0 1 0 |           {0} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                                                  {0}  | 0 0 0 0 0 0 0 0 0 0 1 |           {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                                            -1                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |      
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |     1
                                                                                           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                                      
                                                                                     0

                 2      11      21      18      7      1
     3 : 0  <-- A  <-- A   <-- A   <-- A   <-- A  <-- A  <-- 0
                                                              
         -3     -2     -1      0       1       2      3      4

o5 : FilteredComplex

Notice that the display above shows that these are different filtered complexes. The resulting spectral sequences take the form:

i6 : E' = prune spectralSequence F';

i7 : E'' = prune spectralSequence F'' ;

i8 : E' ^0

     +-------+-------+------+
     | 2     | 3     | 1    |
o8 = |A      |A      |A     |
     |       |       |      |
     |{-2, 3}|{-1, 3}|{0, 3}|
     +-------+-------+------+
     | 8     | 12    | 4    |
     |A      |A      |A     |
     |       |       |      |
     |{-2, 2}|{-1, 2}|{0, 2}|
     +-------+-------+------+
     | 8     | 12    | 4    |
     |A      |A      |A     |
     |       |       |      |
     |{-2, 1}|{-1, 1}|{0, 1}|
     +-------+-------+------+
     | 2     | 3     | 1    |
     |A      |A      |A     |
     |       |       |      |
     |{-2, 0}|{-1, 0}|{0, 0}|
     +-------+-------+------+

o8 : SpectralSequencePage

i9 : E' ^ 0 .dd

o9 = {0, -4} : 0 <----- 0 : {0, -3}
                    0

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

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

                         1
     {0, -1} : 0 <----- A  : {0, 0}
                    0

               1                                       4
     {0, 0} : A  <----------------------------------- A  : {0, 1}
                    | yz-xw y3-x2z xz2-y2w z3-yw2 |

               4                               4
     {0, 1} : A  <--------------------------- A  : {0, 2}
                    {2} | -y2 -xz -yw -z2 |
                    {3} | z   w   0   0   |
                    {3} | x   y   -z  -w  |
                    {3} | 0   0   x   y   |

               4                  1
     {0, 2} : A  <-------------- A  : {0, 3}
                    {4} | w  |
                    {4} | -z |
                    {4} | -y |
                    {4} | x  |

               1
     {0, 3} : A  <----- 0 : {0, 4}
                    0

     {-1, -3} : 0 <----- 0 : {-1, -2}
                     0

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

                          3
     {-1, -1} : 0 <----- A  : {-1, 0}
                     0

                3                                                                                                                12
     {-1, 0} : A  <------------------------------------------------------------------------------------------------------------ A   : {-1, 1}
                     {-2} | -yz+xw -y3+x2z -xz2+y2w -z3+yw2 0      0       0        0       0      0       0        0       |
                     {-2} | 0      0       0        0       -yz+xw -y3+x2z -xz2+y2w -z3+yw2 0      0       0        0       |
                     {-2} | 0      0       0        0       0      0       0        0       -yz+xw -y3+x2z -xz2+y2w -z3+yw2 |

                12                                                   12
     {-1, 1} : A   <----------------------------------------------- A   : {-1, 2}
                      {0} | y2 xz yw z2 0  0  0  0  0  0  0  0  |
                      {1} | -z -w 0  0  0  0  0  0  0  0  0  0  |
                      {1} | -x -y z  w  0  0  0  0  0  0  0  0  |
                      {1} | 0  0  -x -y 0  0  0  0  0  0  0  0  |
                      {0} | 0  0  0  0  y2 xz yw z2 0  0  0  0  |
                      {1} | 0  0  0  0  -z -w 0  0  0  0  0  0  |
                      {1} | 0  0  0  0  -x -y z  w  0  0  0  0  |
                      {1} | 0  0  0  0  0  0  -x -y 0  0  0  0  |
                      {0} | 0  0  0  0  0  0  0  0  y2 xz yw z2 |
                      {1} | 0  0  0  0  0  0  0  0  -z -w 0  0  |
                      {1} | 0  0  0  0  0  0  0  0  -x -y z  w  |
                      {1} | 0  0  0  0  0  0  0  0  0  0  -x -y |

                12                        3
     {-1, 2} : A   <-------------------- A  : {-1, 3}
                      {2} | -w 0  0  |
                      {2} | z  0  0  |
                      {2} | y  0  0  |
                      {2} | -x 0  0  |
                      {2} | 0  -w 0  |
                      {2} | 0  z  0  |
                      {2} | 0  y  0  |
                      {2} | 0  -x 0  |
                      {2} | 0  0  -w |
                      {2} | 0  0  z  |
                      {2} | 0  0  y  |
                      {2} | 0  0  -x |

                3
     {-1, 3} : A  <----- 0 : {-1, 4}
                     0

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

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

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

                2                                                                        8
     {-2, 0} : A  <-------------------------------------------------------------------- A  : {-2, 1}
                     {-3} | yz-xw y3-x2z xz2-y2w z3-yw2 0     0      0       0      |
                     {-3} | 0     0      0       0      yz-xw y3-x2z xz2-y2w z3-yw2 |

                8                                                8
     {-2, 1} : A  <-------------------------------------------- A  : {-2, 2}
                     {-1} | -y2 -xz -yw -z2 0   0   0   0   |
                     {0}  | z   w   0   0   0   0   0   0   |
                     {0}  | x   y   -z  -w  0   0   0   0   |
                     {0}  | 0   0   x   y   0   0   0   0   |
                     {-1} | 0   0   0   0   -y2 -xz -yw -z2 |
                     {0}  | 0   0   0   0   z   w   0   0   |
                     {0}  | 0   0   0   0   x   y   -z  -w  |
                     {0}  | 0   0   0   0   0   0   x   y   |

                8                     2
     {-2, 2} : A  <----------------- A  : {-2, 3}
                     {1} | w  0  |
                     {1} | -z 0  |
                     {1} | -y 0  |
                     {1} | x  0  |
                     {1} | 0  w  |
                     {1} | 0  -z |
                     {1} | 0  -y |
                     {1} | 0  x  |

                2
     {-2, 3} : A  <----- 0 : {-2, 4}
                     0

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

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

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

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

     {-3, 1} : 0 <----- 0 : {-3, 2}
                    0

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

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

     {-3, 4} : 0 <----- 0 : {-3, 5}
                    0

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

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

o9 : SpectralSequencePageMap

i10 : E'' ^0

      +-------+-------+-------+-------+
      | 1     | 4     | 4     | 1     |
o10 = |A      |A      |A      |A      |
      |       |       |       |       |
      |{0, 0} |{1, 0} |{2, 0} |{3, 0} |
      +-------+-------+-------+-------+
      | 3     | 12    | 12    | 3     |
      |A      |A      |A      |A      |
      |       |       |       |       |
      |{0, -1}|{1, -1}|{2, -1}|{3, -1}|
      +-------+-------+-------+-------+
      | 2     | 8     | 8     | 2     |
      |A      |A      |A      |A      |
      |       |       |       |       |
      |{0, -2}|{1, -2}|{2, -2}|{3, -2}|
      +-------+-------+-------+-------+

o10 : SpectralSequencePage

i11 : E'' ^1

      +--------------------------+-----------------------------------------------------+----------------------------------------------------+-------------------------+
o11 = |cokernel {-3} | z  y  x  ||cokernel {-1} | z  y  x  0  0  0  0  0  0  0  0  0  ||cokernel {1} | z  y  x  0  0  0  0  0  0  0  0  0  ||cokernel {2} | z  y  x  ||
      |         {-3} | -w -z -y ||         {0}  | 0  0  0  z  y  x  0  0  0  0  0  0  ||         {1} | 0  0  0  z  y  x  0  0  0  0  0  0  ||         {2} | -w -z -y ||
      |                          |         {0}  | 0  0  0  0  0  0  z  y  x  0  0  0  ||         {1} | 0  0  0  0  0  0  z  y  x  0  0  0  ||                         |
      |{0, -2}                   |         {0}  | 0  0  0  0  0  0  0  0  0  z  y  x  ||         {1} | 0  0  0  0  0  0  0  0  0  z  y  x  ||{3, -2}                  |
      |                          |         {-1} | -w -z -y 0  0  0  0  0  0  0  0  0  ||         {1} | -w -z -y 0  0  0  0  0  0  0  0  0  ||                         |
      |                          |         {0}  | 0  0  0  -w -z -y 0  0  0  0  0  0  ||         {1} | 0  0  0  -w -z -y 0  0  0  0  0  0  ||                         |
      |                          |         {0}  | 0  0  0  0  0  0  -w -z -y 0  0  0  ||         {1} | 0  0  0  0  0  0  -w -z -y 0  0  0  ||                         |
      |                          |         {0}  | 0  0  0  0  0  0  0  0  0  -w -z -y ||         {1} | 0  0  0  0  0  0  0  0  0  -w -z -y ||                         |
      |                          |                                                     |                                                    |                         |
      |                          |{1, -2}                                              |{2, -2}                                             |                         |
      +--------------------------+-----------------------------------------------------+----------------------------------------------------+-------------------------+

o11 : SpectralSequencePage