# 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