# Wednesday, March 31

:::{.remark}
Last time we talked about hypercohomology and hyper derived functors, and we proved that two spectra sequences converging to $\LL_{p+q}F(A)$.
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## Grothendieck Spectral Sequences

:::{.remark}
We'll focus on the cohomological version, which gives a spectral sequence from a composition of functors.
Let $\cat{A}, \cat{B}, \cat{C}$ be abelian categories with enough injectives, and let $G: \cat{A} \to \cat{B}$, $F: \cat{B} \to \cat{C}$ be left exact functors.
By a previous result, $FG:\cat{A} \to \cat{C}$ is left exact, which follows from checking that it preserves 4-term exact sequences.
Recall that $B \in \cat{B}$ is $F\dash$acyclic if $R^i F(B) = 0$ for all $i>0$.
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:::{.theorem title="Grothendieck Spectral Sequence"}
Assume the above setup, and that $G$ sends injectives in $\cat{A}$ to $F\dash$acyclic objects in $\cat{B}$.
Then there is a convergent QI spectral sequence for each $A \in \cat{A}$:
\[
E_2^{p, q} = (R^p F)(R^q G)(A) \abuts R^{p+q}(FG)(A)
.\]
The edge maps are the natural maps 
\[
(R^p F)(GA) &\to R^p(FG)(A) \\
R^q (FG)(A) &\to F( R^qG(A))
.\]
The exact sequences of the low-degree terms are

\[
0 \to (R^jF)(GA) \to R^j(FG)(A) \to F(R^j G(A)) \to (R^j F)(GA) \to R^j(FG)(A)
.\]
:::

:::{.proof title="?"}
Choose an injective resolution $A\injects I$ in $\cat{A}$ and apply $G$ to form the cochain complex $G(I)\in \cat{B}$.
Using a first quadrant CE resolution of $G(I)$, form the hyper right-derived functors $\RR^i F(G(I))$.
We have the two spectral sequences that converge to this, since the complex is bounded below:
\[
{}^I E_1^{p, q} = H^p R^q F(GI) \abuts (\RR^{p+q} F)(GI)
.\]
By hypothesis $I^p$ is injective in $\cat{A}$, and thus $G(I^p)$ is $F\dash$acyclic in $\cat{B}$, so this spectral sequence collapses onto the horizontal axis at the 2nd page.
So $(\RR^p F)(GI) = H^p(FG(I))$, which is by definition $R^p(FG)(A)$, and this holds for all $p>0$.
This follows because only one term survives on each diagonal, and the associated graded is just to those terms, so it lifts to just being the actual homology.

The second spectral sequence converges to the same thing, and so by reindexing the previous limiting term $p\mapsto p+q$, we can write
\[
{}^{II} E_2^{p, q} = (R^p F)(H^q(GI)) \abuts R^{p+q} (FG)(A)
.\]
But this is $(R^p F)(R^q G)(A)$ by definition.

By example 5.2.6, the edge maps from the $p\dash$axis are 
\[
E_2^{p, 0} \to E_{\infty }^{p, 0} \injects H^p
,\]

and composing these yields $(R^p F)(GA) \to R^p(FG)(A)$.
We also have $H^q \surjects E_{\infty }^{p, 0} \injects E_2^{0, q}$.
:::

:::{.remark}
We're skipping the section on sheaf cohomology and 5.9, so we'll move into chapter 6.
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## 6.8: The Lyndon-Hochschild-Serre Spectral Sequence

:::{.remark}
Let $H\normal G$ and $A\in \modsleft{G}$, then $A_H, A^H \in \modsleft{G/H}$.
The canonical projection $p: G\to G/H$ induces a forgetful functor $p^*: \modsleft{G/H} \to \mods{G}$ given by pullback.
Note that $G/H\dash$modules are essentially \(G\dash\)modules where $H$ acts trivially, so this functor forgets the trivial $H$ action.
Generally, this works a bit like the Frobenius map, which yields a representation that can be pulled back.

:::

:::{.lemma title="?"}
The invariant functor $(\wait)_H$ has a left adjoint and the coinvariant functor $(\wait)^H$ has a right adjoint.
:::

:::{.proof title="?"}
A $G/H\dash$module is a \(G\dash\)module with a trivial $H$ action, so both $A_H, A^H$ are \(G/H\dash\)modules.
One needs to check that although $H$ preserves these submodules, so does $G$.
The universal property of $A^H \injects A$ as the largest trivial submodule and $A\to A_H$ as the largest trivial quotient imply that there are natural isomorphisms: for $A\in \modsleft{G}$ and $B\in \modsleft{G/H}$, 
\[
\Hom_G(p^* B, A) &\mapsvia{\sim} \Hom_{G/H}(B, A^H) \\
f &\mapsto f
\]
which is well-defined since $f(b) = f(hb) = hf(b) = f(b)$, putting $f(b) \in A^H$.
We also have
\[
\Hom_G(A, p^\sharp B) &\mapsvia{\sim} \Hom_{G/H}(A_H, B) \\
(\tilde f: A \mapsvia{\pi} A_H \mapsvia{f} B ) &\mapsfrom f
,\]
and these give the required adjunction.
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:::{.theorem title="Lyndon-Hochschild-Serre Spectral Sequence"}
Let $H\normal G$ for $A\in \modsleft{G}$, then there are two QI spectral sequences:
\[
E_{p, q}^2 &= H_p (G/H, H_q(H, A)) \\
E_2^{p, q} &= H^p(G/H, H^q(H, A))
.\]
:::

:::{.remark}
Note that we can identify the functors
\[
(\wait)^H, (\wait)_H : \modsleft{G} \to \modsleft{G/H}
,\]
whose derived functors are group homology/cohomology.
The idea will be that $G\dash$invariants can be written as a composition of other functors, and we can apply the Grothendieck spectral sequence construction.
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