# Monday, April 05

## Restriction and Inflation

:::{.definition title="Restriction and Corestriction"}
Let $\rho: H\to G$ be a group morphism, this induces an exact functor $\rho^\sharp: \mods{G} \to \mods{H}$.
We define

- $T_n(A) \da H_n(G; A)$
- $T^n(A) \da H^n(G; A)$

- $S_n(A) \da H_n(\rho^\sharp G; A)$
- $S^n(A) \da H^n(\rho^\sharp G; A)$

These are all functors $\mods{G} \to \mods{\ZZ}$.
As in section 2.1, $H_n$ defines a homological $\delta\dash$functor, and since $\rho^\sharp$ is exact, $T_n, S_n$ are homological $\delta\dash$functors as well.
We have a map 
\[
A^G &\injects (\rho^\sharp A)^H \\
T^0 A&\to S^0 A
.\]
and similarly
\[
(\rho^\sharp A)_H &\to A_G\\
S_0 A &\to T_0 A
.\] 
These maps on the 0th terms extend to morphisms of $\delta\dash$functors.

There thus exist two maps
\[
\res_H^G H^*(G; A) \to H^*(G; \rho^\sharp A) && \text{restriction} \\
\cores_H^G H_*(G; \rho^\sharp A) \to H_*(G; A) && \text{corestriction}
.\]

:::

:::{.remark}
A special case is when $H\leq G$ is a subgroup and $\rho: H\embeds G$ is the inclusion.
Then we define a capital $\Res$ as 
\[
\rho^\sharp = \Res_H^G: \mods{G} \to \mods{H}
,\]
which is a restriction of the action to a subgroup and thus a type of forgetful functor.
:::

:::{.remark}
Note that $\ZZG$ is a free $\ZZ H\dash$module with basis being any set of coset representatives, thus any projective \(G\dash\)module restricts to a projective \(H\dash\)module, using the characterization of projective modules as direct summands of free modules.
:::

:::{.remark}
Recall that 
\[
H_*(G; A) &\cong \Tor_*^{\ZZG}(\ZZ, A) \\
H_*(G; A) &\cong \Ext_{\ZZG}^*(\ZZ, A)
.\]
We can compute both using a $\ZZG\dash$projective resolution $P_* \to \ZZ$.
This is also a $\ZZ H\dash$projective resolution, so we can use this to compute $H^*(H; \wait)$ and $H_*(H; \wait)$ as well.
:::

:::{.fact}
\envlist

1. There's a natural chain map induced by the forgetful functor:
\[
\beta: \Hom_G(P_*, A) \to \Hom_H(P^*, A)
.\]

2. There is an induced map
\[
H^*(\beta): \Ext_G^*(\ZZ, A) \to \Ext_H^*(\ZZ, A)
,\]
which is equal to the map
\[
\res_H^G: H^*(G; A) \to H^*(H; A)
,\]
giving a way to calculate $\res$ from something just coming from restriction of functions.

3. There is a chain map 
\[
\alpha: P_* \tensor_{\ZZH} A &\to P_* \tensor{\ZZH} P_* \tensor_{\ZZG} A \\
p\tensor a &\mapsto p\tensor a
,\]
which induces
\[
H( \alpha): \Tor_*^H(\ZZ, A) \to \Tor_*^G(\ZZ, A)
\]
which is equal to
\[
\cores_H^G: H_*(H; A) \to H_*(G; A)
.\]
So this can be computed from tensor products.
:::

:::{.definition title="Inflation and Coinflation"}
Now consider quotient groups instead: assume $H\normal G$ and let $\rho:G\to G/H$.
By precomposing with $\rho$, we get a map $\rho^\sharp: \mods{G\over H}\to\mods{G}$.
Given a \(G\dash\)module, taking $H$ invariants yields a $G/H\dash$module, so $H^*(G/H; A^H) \in \mods{G\over H}$.
We form the following composition:

\begin{tikzcd}
	{H^*\qty{{G\over H}; A^H}} && {H^*(G; A^H)} &&&& {H^*(G; A)}
	\arrow["{H^*(G; \wait)(A^H \injects A)}", from=1-3, to=1-7]
	\arrow["\res", from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJIXipcXHF0eXt7R1xcb3ZlciBIfTsgQV5IfSJdLFsyLDAsIkheKihHOyBBXkgpIl0sWzYsMCwiSF4qKEc7IEEpIl0sWzEsMiwiSF4qKEc7IFxcd2FpdCkoQV5IIFxcaW5qZWN0cyBBKSJdLFswLDEsIlxccmVzIl1d)

We'll refer to this as **inflation**.
We similarly define **coinflation** as the following composition:

\begin{tikzcd}
	{H_*(G; A)} &&& {H(G; A_H)} && {H_*\qty{{G\over H}, A_H}}
	\arrow["\cores", from=1-4, to=1-6]
	\arrow["{H_*(G; \wait)(A \surjects A_H)}", from=1-1, to=1-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJIXyooRzsgQSkiXSxbMywwLCJIKEc7IEFfSCkiXSxbNSwwLCJIXypcXHF0eXt7R1xcb3ZlciBIfSwgQV9IfSJdLFsxLDIsIlxcY29yZXMiXSxbMCwxLCJIXyooRzsgXFx3YWl0KShBIFxcc3VyamVjdHMgQV9IKSJdXQ==)

:::

:::{.remark}
When $*=0$, we can write 
\[
\infl: (A^H)^{G\over H} \to (A^H)^G \to A^G
,\]
and note that this is exactly the functor composition we needed to get the LHS spectral sequence.
Similarly there is a LHS for homology, and an isomorphism
\[
\coinfl: A_G \to (A_H)_G \to (A_H)_{G\over H}
.\]
:::

:::{.remark}
When $A \in \mods{H}^{\Triv}$, $A_H\injects A$ is the identity, so $A^H = A = A_H$.
In this case $\infl = \res$ and $\coinfl = \cores$.
:::

:::{.remark}
Back to the LHS spectral sequence, the five-term exact sequence yields
\[
0 \to E_{2}^{1, 0} \to H^1(T) \to E_2^{0, 1} \mapsvia{d_2} E_{2, 0} \to H^2(T)
,\]
which we can identify as 
\[
0\to H^1\qty{{G\over H}; A^H}
\mapsvia{\infl} 
H^1(G; A)
\mapsvia{\res} 
H^1(H; A)^{G\over H}
\mapsvia{d_2} 
H^2\qty{ {G\over H}; A^H }
\mapsvia{\infl} 
H^2(G; A)
.\]
There is a similar story in homology with coinflation and corestriction.
:::

## Shapiro's Lemma, Induced/Coinduced Modules

:::{.definition title="Induced and Coinduced Modules"}
Let $H\leq G$ and $B\in \mods{\ZZ H}$.
Define the **induced \(G\dash\)module** (or tensor-induced \(G\dash\)module)
\[
\Ind_H^G(B) \da \ZZG \tensor_{\ZZH} B \in \mods{\ZZ G}
.\]
This is a $\ZZG\dash$module with an action on the first tensor factor.
Similarly define the **coinduced** or **hom-induced \(G\dash\)module**.
\[
\Coind_H^G(B) \da \Hom_{H}(\ZZG, B) \in \mods{\ZZ G}
.\]
Here the action is $(g.f)(g') \da f(gg')$.
:::

:::{.lemma title="Shapiro's Lemma (Frobenius Reciprocity)"}
\[
H_*(G; \Ind_H^G B) &\cong H_*(H; B) &&(1) \\
H^*(G; \Coind^G B) &\cong H^*(H; B) &&(2)
.\]
:::

:::{.remark}
So this provides a way of computing homology on subgroups when the coefficients are in these induced/coinduced modules.
:::