# Wednesday, March 24

:::{.remark}
Last time: we're trying to prove the classical convergence theorem in the bounded case.
We have 
\[
E_{pq}^1 = H_{p+q}( F_p C/ F_{p-1} C ) \abuts H_{p+q}C
.\]
We'd like this converge, i.e. the $E^\infty$ page will be the sections of $H_{p+q}C$.
Writing $C_n'\da F_p C_n$ for the filtered pieces, we have

\begin{tikzcd}
	{C_n'} && {C_n} \\
	{Z_n'} && {Z_n} \\
	{B_n'} && {B_n}
	\arrow[hook, from=3-1, to=3-3]
	\arrow[hook, from=2-1, to=2-3]
	\arrow[hook, from=1-1, to=1-3]
	\arrow[hook, from=1-3, to=2-3]
	\arrow[hook, from=2-3, to=3-3]
	\arrow[hook, from=1-1, to=2-1]
	\arrow[hook, from=2-1, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

Then the induced filtration on homology is 
\[
H_n' \da {Z_n' \over B_n'} &\injects H_n \da {Z_n\over B_n} \\
z' + B_n' &\mapsto z' + B_n
.\]
:::

:::{.proof title="of classical convergence theorem"}
As discussed, we have a natural bounded filtration on each $H_n C$.
Fixing $p, n$
and writing $q = n-p$, we have
\[
A_p^r = \ts{ c \in F_p C_n \st d(c) \in F_{p-r} C_{n-1} } 
.\]
This stabilizes for large $r$, namely whenever $F_{p-r} C_{n-1} = 0$ (which happens since the complex is bounded).
Call the stabilized object $A_p^{\infty } \da \ts{ c\in F_p C_n \st d(c) = 0 }$, which is $\ker d$ in the $p$th filtered piece.
Some facts:

0. $Z_p^r = \eta_p(A_p^r)$ where
\[
\eta_p: F_p C_n \to {F_p C_n \over F_{p-1} C_n }
\]
where $Z_p^{\infty } = \eta_p(A_p^{\infty })$.

1. $A_p^{\infty } \da \ker (F_p C_n \mapsvia{d} F_p C_{n-1} )$, which is the "numerator" of $F_p H_n C$.

2. $d(C_{n+1}) \intersect F_p C_n = \Union_{r\in \ZZ} d(A_{p+r}^r )$:
  \begin{tikzcd}
    &&&& {A_{p+r}} \\
    &&&& {F_{p+r}C_{n+1}} & {C_{n+1}} \\
    {F_p C_n}
    \arrow["d"', from=1-5, to=3-1]
    \arrow[hook, from=1-5, to=2-5]
    \arrow[hook, from=2-5, to=2-6]
  \end{tikzcd}

  > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbNCwwLCJBX3twK3J9Il0sWzQsMSwiRl97cCtyfUNfe24rMX0iXSxbNSwxLCJDX3tuKzF9Il0sWzAsMiwiRl9wIENfbiJdLFswLDMsImQiLDJdLFswLDEsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEsMiwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=)

3. Recall that we defined $B_p^r \da \eta_p( d A_{p+r-1}^{r-1} )$.
  We can write \( B_p^{\infty } = \eta_p (\union_r dA_{p+r}^r ) \), where the left-hand side and the inner term on the right-hand side are equal to $\Union_{r\geq 1} B_p^r$.

4. $A_{p-1}^{\infty } = A_p^{\infty } \intersect F_{p-1} C_n = \ker (A_{p}^{\infty } \mapsvia{\eta_p} E_p^0 )$.

Now to assemble this, note that
\[
{F_p H_n C \over F_{p-1} H_n C} 
&\cong 
{A_{p}^{\infty } \over A_{p-1}^{\infty } + \Union_r dA_{p+r}^r } 
&& \text{ by 1 and 2} \\
&\cong
{ \eta_p (A_p^{\infty } ) \over \eta_p\qty{\Union_{r\geq 0} dA_{p+r}^r } } 
&& \text{by 4} \\
&= {Z_p^{\infty } \over B_p^{\infty } } 
&& \text{by 0, 3} \\
&= E_p^{\infty }
.\]
where we've used that $A_{p-1}^{\infty } + \Union_{r>0} dA_{p+r}^r \subseteq \ker \eta_p = F_{p-1} C$.
:::

## Applications: Two Spectral Sequences of a Double Complex

:::{.remark}
Consider two different filtrations of the total complex $\Tot(C)$ (either sum or product) of a double complex $C_{*, *}$.
We know there is an spectral sequence associated to each and play them off of each other to get extra information about cohomology.
:::

:::{.definition title="Filtration I: by columns (of a double complex)"}
Let ${}^IF_n \Tot(C)$ be the total subcomplex obtain by applying truncation functors:
\[
\qty{{}^I \tau_{\leq n} C}_{p, q} \da 
\begin{cases}
C_{p, q} & p \leq n 
\\
0 & p > n.
\end{cases}
.\]

\begin{tikzcd}
	\ddots & \vdots & \vdots & {} & \vdots & \vdots \\
	\cdots & \bullet & \bullet && 0 & 0 & \cdots \\
	\cdots & \bullet & \bullet && 0 & 0 & \cdots \\
	\cdots & \bullet & \bullet && 0 & 0 & \cdots \\
	\bullet & \vdots & \vdots & {} & \vdots & \vdots & \bullet \\
	&& n & \bullet & {n+1}
	\arrow[no head, from=5-1, to=5-7]
	\arrow[no head, from=1-4, to=6-4]
	\arrow[from=2-3, to=2-2]
	\arrow[from=2-3, to=3-3]
	\arrow[from=3-3, to=3-2]
	\arrow[from=3-3, to=4-3]
	\arrow[from=4-3, to=4-2]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, dotted, no head, from=1-1, to=5-5]
\end{tikzcd}


> [Link to Diagram](https://q.uiver.app/?q=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)

We still have $d = d^v + d^h: {}^IF_n \to {}^I F_n$.
By the construction theorem, there is a spectral sequence $\theset{ {}^I E_{p,q}^r}$ starting with ${}^I E_{p, q}^0 = C_{p, q}$ and 
\[
{}^I E_{p, q}^0 = 
{ F_p \Tot(C)_{p+q} \over F_{p-1} \Tot(C)_{p+q}}
.\]

\begin{tikzcd}
	& \bullet &&&& 0 \\
	&& \bullet &&& 0 \\
	&&& {\bullet F_{p-1} } && 0 \\
	q &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{\bullet (F_p) = C_{p, q}} & 0 \\
	&&&&& 0 \\
	&&& {} & {p-1} & p
	\arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, no head, from=1-2, to=5-6]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMTMsWzUsMCwiMCJdLFs1LDEsIjAiXSxbNSwyLCIwIl0sWzUsMywiMCJdLFs1LDQsIjAiXSxbNCwzLCJcXGJ1bGxldCAoRl9wKSA9IENfe3AsIHF9IixbMjQwLDYwLDYwLDFdXSxbMywyLCJcXGJ1bGxldCBGX3twLTF9ICJdLFsyLDEsIlxcYnVsbGV0Il0sWzMsNV0sWzQsNSwicC0xIl0sWzUsNSwicCJdLFsxLDAsIlxcYnVsbGV0Il0sWzAsMywicSJdLFsxMSw0LCIiLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==)

Recall that $d_p^r: E_p^r \to E_{p-r}^{r}$ (going $r$ columns to the left, where we've suppressed $q$) is the map induced from $d: \Tot(C)_n \to \Tot(C)_{n-1}$.
So for $r=0$, we have $d_{p, q}^0: E_{p, q}^0 \to E_{p, q-1}^0$.
But the left-hand side is $C_{p, q}$ and the right-hand side is $C_{p, q-1}$, so it's perhaps not surprising that this coincides with the original $d^v$ from $C_{*, *}$.

Thus ${}^IE_{pq}^1= H_q^v(C_{p, *})$ by taking homology in the vertical direction.
For the differential, we want $d_{pq}^1: E_{pq}^1\to E_{p-1, q}^1$, and these will just be the maps induced on the vertical homology by $d^h$.
So we write ${}^I E_{p, q}^2 = H_p^h H_q^v (C_{**})$.

If $C$ is a first quadrant complex, the filtration is canonically bounded since $F_{-1} \Tot(C) = 0$ and $F_n \Tot(C)_n = \Tot(C)_n$.
So we get the spectral sequence that we started constructing in section 5.1, and we now know it converges to $H_* \Tot(C)$ by the classical convergence theorem.
So 
\[
{}^I E_{p, q}^2 = H_p^h H_q^v(C) \abuts H_{p+q} \Tot(C)
.\]

:::

:::{.remark}
We can say something about the unbounded case.
Suppose $C$ is 4th quadrant, then $F_{-1} \Tot(C) = 0$, so the first filtration ${}^I F$ is bounded below.
The diagonals are infinite, so we take $\Tot(C) \da \Totsum(C)$.
Every element of $(\Tot(C))_n$ lives in \( \bigoplus _{p=0}^N C_{p, n-p} \) for some finite $N$ and the filtration is exhaustive, i.e. $\Totsum C = \Union_{p\geq 0} F_p \Totsum C$.
A version of the classical convergence theorem will yield
\[
{}^I E_{pq}^r \abuts H_{p+q} \Totsum C
.\]
However, this will not hold for $\Totprod$.
:::

:::{.remark}
Next time: a second filtration and its spectral sequence, and how to play them off of each other.
:::