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I'm attempting to wrap my head around spectral sequences, so constructed a really basic example to apply the definitions and go through the motions. My filtered chain complexes are:

$F_2C_*: 0 \rightarrow \mathbb{Z} \xrightarrow{i_1} \mathbb{Z_1}\oplus \mathbb{Z_2} \xrightarrow{p_2} \mathbb{Z} \rightarrow 0 $

$F_1C_*: 0 \rightarrow 0 \rightarrow \mathbb{Z}_2 \xrightarrow{p_2} \mathbb{Z} \rightarrow 0 $

$F_0C_*:$ The zero complex.

Here $i_1$ and $p_2$ are the inclusion and projection maps, and the sub-indices on the $\mathbb{Z}s$ are just for labeling purposes.

With $Z^r_{p,q}$ defined as $\{x \in F_pC_q : \partial x \in F_{p-r}C_{q-1}\}/F_{p-1}C_q$, the differential is meant to restrict to well-defined maps $d^r:Z^r_{p,q} \rightarrow Z^r_{p-r,q-1}$.

But in my example the map $Z^1_{2,2} \rightarrow Z^1_{1,1}$ does not appear to be well-defined:

$Z^1_{2,2} := \{x \in \mathbb{Z}\oplus \mathbb{Z} : p_2 x \in \mathbb{Z} \}/\mathbb{Z}_2 \cong \mathbb{Z}_1$,

$Z^1_{1,1} := \mathbb{Z}$.

However $[(a,b)]$ and $[(a,b')]$ in $Z^1_{2,2}$ get sent to $b$ and $b'$ in $\mathbb{Z}$ respectively, even though $[(a,b)] = [(a,b')]$.

I cannot see where I've made a mistake, but I'm pretty sure something's not right. Where have I gone wrong?

Given a map $A \xrightarrow{f} C$ with $B \subset A$ and $D \subset C$, in order for the quotient map $A/B \xrightarrow{\bar f} C/D$ to be defined we must have $f(B) \subset D$.

But it this case we have a map $(Z_1 \oplus Z_2) / Z_2 \xrightarrow{\bar p_2} Z/\{0\}$, but $p_2(Z_2) = Z_2$ which is not contained in $\{0\}!$

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$Z_{p,q}^r$ isn’t a subquotient just a subobject. $$Z_{p,q}^r := \{x \in F^pC_q: dx \in F^{p-r}C_{q-1}\}$$ check eg wikipedia.

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