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).
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Notice that the display above shows that these are different filtered complexes. The resulting spectral sequences take the form:
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The source of this document is in SpectralSequences.m2:1328:0.