Filtrations and tensor product complexes

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

For all integers $p$ and $q$ let $K_{p,q} := B_p \otimes_S C_q$, let $d'_{p,q} : K_{p,q} \rightarrow K_{p - 1, q}$ denote the homorphism $\partial^B_{p} \otimes 1$, and let $d''_{p,q} : K_{p,q} \rightarrow K_{p, q - 1} $ denote the homorphism $(-1)^p \otimes \partial_q^C $.

The chain complex $B \otimes_S C$ is given by $ (B \otimes_S C)_k := \oplus_{p + q = k} B_p \otimes_S C_q$ and the differentials by $\partial := d' + d''$. It carries two natural ascending filtrations $F'B \otimes_S C$ and $F'' B \otimes_S C$.

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

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

In Macaulay2 we can compute these filtered complexes as follows.

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' = (filteredComplex B) ** C

o4 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
                                                                                               
          0           1           2           3           4           5           6           7

     0 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 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 0 0 0 0 0 |           {4} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 1 0 0 0 0 |           {4} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     4           5           6           7
                               {3} | 0 0 0 1 0 0 0 |           {4} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 |           {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                         1                                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 |           {6} | 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 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                         2                                                           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                                
                                                                                                               3

     1 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {7} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0
                               {3} | 0 1 0 0 0 0 0 |           {4} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 1 0 0 0 0 0 0 0 0 0 |                              
         0                     {3} | 0 0 1 0 0 0 0 |           {4} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 1 0 0 0 0 0 0 0 0 |     5           6           7
                               {3} | 0 0 0 1 0 0 0 |           {4} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                               {2} | 0 0 0 0 1 0 0 |           {4} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                               {2} | 0 0 0 0 0 1 0 |           {5} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                               {2} | 0 0 0 0 0 0 1 |           {5} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                         1                                     {4} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 1 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 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |     4
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           {6} | 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 |           {6} | 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 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                         2                                                           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                                                                
                                                                                                               3

          1      7      18      21      11      2
     2 : A  <-- A  <-- A   <-- A   <-- A   <-- A  <-- 0 <-- 0
                                                             
         0      1      2       3       4       5      6     7

o4 : FilteredComplex

i5 : F'' = B ** (filteredComplex C)

o5 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
                                                                                               
          0           1           2           3           4           5           6           7

     0 : image | 1 | <-- image {2} | 0 0 0 0 0 0 0 | <-- image {4} | 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 <-- image 0
                               {3} | 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                     {3} | 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 |     3           4           5           6           7
                               {3} | 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 |
                               {2} | 0 0 0 0 1 0 0 |           {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                               {2} | 0 0 0 0 0 1 0 |           {5} | 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 1 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                         1                                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 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 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 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 1 0 |
                                                               {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                          
                                                         2

     1 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 0 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 0 0 0 0 0 |           {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 1 0 0 0 0 |           {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     4           5           6           7
                               {3} | 0 0 0 1 0 0 0 |           {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 1 0 0 |           {4} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 1 0 |           {5} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 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 1 |           {5} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                         1                                     {4} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {6} | 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 1 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                                                               {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                                                               {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                                                         2                                                           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                                                                
                                                                                                               3

     2 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {7} | 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0
                               {3} | 0 1 0 0 0 0 0 |           {4} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |                              
         0                     {3} | 0 0 1 0 0 0 0 |           {4} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 0 |     5           6           7
                               {3} | 0 0 0 1 0 0 0 |           {4} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 1 0 0 0 0 0 0 0 |
                               {2} | 0 0 0 0 1 0 0 |           {4} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 1 0 0 0 0 0 0 |
                               {2} | 0 0 0 0 0 1 0 |           {5} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 1 0 0 0 0 0 |
                               {2} | 0 0 0 0 0 0 1 |           {5} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 1 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 1 0 0 0 |
                         1                                     {4} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 1 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 1 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {7} | 0 0 0 0 0 0 0 0 0 0 1 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 1 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 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |     4
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                                                               {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                                                               {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                                                               {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |           {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                                                         2                                                           {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                                                                                                                     {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                                                                                
                                                                                                               3

          1      7      18      21      11      2
     3 : A  <-- A  <-- A   <-- A   <-- A   <-- A  <-- 0 <-- 0
                                                             
         0      1      2       3       4       5      6     7

o5 : FilteredComplex

The pages of the resulting spectral sequences take the form:

i6 : E' = prune spectralSequence F';

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

i8 : E' ^0

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

o8 : SpectralSequencePage

i9 : E' ^ 1

     +----------------------------------------+----------------------------------------------------------------------------------------------------+------------------------------------------------------------------------+
o9 = |cokernel | yz-xw z3-yw2 xz2-y2w y3-x2z ||cokernel {2} | yz-xw 0     0     z3-yw2 xz2-y2w y3-x2z 0      0       0      0      0       0      ||cokernel {3} | yz-xw 0     z3-yw2 xz2-y2w y3-x2z 0      0       0      ||
     |                                        |         {2} | 0     yz-xw 0     0      0       0      z3-yw2 xz2-y2w y3-x2z 0      0       0      ||         {3} | 0     yz-xw 0      0       0      z3-yw2 xz2-y2w y3-x2z ||
     |{0, 0}                                  |         {2} | 0     0     yz-xw 0      0       0      0      0       0      z3-yw2 xz2-y2w y3-x2z ||                                                                        |
     |                                        |                                                                                                    |{2, 0}                                                                  |
     |                                        |{1, 0}                                                                                              |                                                                        |
     +----------------------------------------+----------------------------------------------------------------------------------------------------+------------------------------------------------------------------------+

o9 : SpectralSequencePage

i10 : E'' ^0

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

o10 : SpectralSequencePage

i11 : E'' ^1

      +------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------+
o11 = |cokernel | z2-yw yz-xw y2-xz ||cokernel {2} | z2-yw yz-xw y2-xz 0     0     0     0     0     0     0     0     0     ||cokernel {4} | z2-yw yz-xw y2-xz 0     0     0     0     0     0     0     0     0     ||cokernel {5} | z2-yw yz-xw y2-xz ||
      |                              |         {3} | 0     0     0     z2-yw yz-xw y2-xz 0     0     0     0     0     0     ||         {4} | 0     0     0     z2-yw yz-xw y2-xz 0     0     0     0     0     0     ||                                  |
      |{0, 0}                        |         {3} | 0     0     0     0     0     0     z2-yw yz-xw y2-xz 0     0     0     ||         {4} | 0     0     0     0     0     0     z2-yw yz-xw y2-xz 0     0     0     ||{3, 0}                            |
      |                              |         {3} | 0     0     0     0     0     0     0     0     0     z2-yw yz-xw y2-xz ||         {4} | 0     0     0     0     0     0     0     0     0     z2-yw yz-xw y2-xz ||                                  |
      |                              |                                                                                        |                                                                                        |                                  |
      |                              |{1, 0}                                                                                  |{2, 0}                                                                                  |                                  |
      +------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------+

o11 : SpectralSequencePage

Filtrations and tensor product complexes

See also