# 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