# Wednesday, March 23

:::{.remark}
Induced and coinduced modules:

\begin{tikzcd}
	S \\
	&& {\mods{R}} && {\mods{S}} \\
	R && {R\tensor_S N} && N \\
	&& {\Hom_S(R, N)} && N
	\arrow["\Forget", from=2-3, to=2-5]
	\arrow[from=1-1, to=3-1]
	\arrow["\ind", maps to, from=3-5, to=3-3]
	\arrow["\coind", maps to, from=4-5, to=4-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbNCwxLCJcXG1vZHN7U30iXSxbMiwxLCJcXG1vZHN7Un0iXSxbMCwwLCJTIl0sWzAsMiwiUiJdLFs0LDIsIk4iXSxbMiwyLCJSXFx0ZW5zb3JfUyBOIl0sWzQsMywiTiJdLFsyLDMsIlxcSG9tX1MoUiwgTikiXSxbMSwwLCJcXEZvcmdldCJdLFsyLDNdLFs0LDUsIlxcaW5kIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFs2LDcsIlxcY29pbmQiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV1d)

Note that coinduction sends injective to injectives, and induction sends projectives to projectives.
Recall that $\Sh(X; \Ab\Grp)$ and $\mods{\OO_X}$ have enough injectives, so left exact covariant functors $F$ admit right-derived functors $\RR F$, and similarly right exact contravariant functors $F$ admit left-derived functors $\LL F$.
:::

:::{.example title="?"}
Important functors:

- Global sections $\globsec{\wait}: \Sh(X; \cat C) \to \cat{C}$ where $\mcf \mapsto \mcf(X)$, e.g. for $\cat C = \Ab\Grp$.
  $\RR \globsec{\mcf} = H^i(X; \mcf)$ is sheaf cohomology.
- For $f\in \Top(X, Y)$, the pushforwards $f_*: \Sh(X; \cat C) \to \Sh(Y; \cat C)$ where $\mcf \mapsto (U\mapsto \mcf(f\inv U))$.
  $\RR f_* \mcf$ are *derived pushforwards*.
- Inverse image, which is exact.
- $(\wait)\tensor_{\OO_X} \mcf$
- $\Hom_{\OO_X}(\wait, \mcf)$.
:::

:::{.theorem title="?"}
If $F \in [\cat A, \cat B]$ is left exact covariant and $\cat A$ has enough injectives, then for every $A\in \cat A$ there exists an acyclic resolution $0\to A \cocovers \cocomplex{J}$ whose homology computes $\RR R$.
:::

:::{.proof title="Sketch"}
Why this homology computes the derived functors: let $A = A^0$ and take an injective resolution $A \cocovers \cocomplex{J}$.
Break this into SESs, letting $Z_i$ denote images:

- $0 \to Z^0 \to J^0\to Z^1\to 0$
- $0 \to Z^1\to J^1 \to Z^2\to 0$
- $\cdots$

Note that $Z^n \cocovers \Sigma^n \cocomplex{J} = (J^n\to J^{n+1}\to\cdots)$ is an injective resolution.
Splice to obtain
\[
0 \to FA \to FJ^0 \to FZ^1\to \RR^1 FA \to 0, \qquad \RR^n F Z^1 \iso \RR^{n+1} FA \\
0 \to \ker(FJ^0\to FJ^1) \to FJ^0 \to \ker(FJ^1\to FJ^2) \to \RR^1 FA \to 0
.\]
Proceed by induction.
:::

:::{.remark}
Consider $F = \cat{A}(A, \wait)$ (covariant) or $\cat{A}(\wait, A)$ (contravariant), so $F\in \Cat(\cat A, \Ab\Grp)$.
Note that acyclic objects for $F$ are exactly injectives: take $0\to A\to B\to C\to 0$ to obtain $0\to [C, I] \to [B, I]\to [A, I] \to \Ext^1(C, I) = 0$ by acyclicity of $I$, meaning that $[B,I] \surjects [A, I]$ and thus there exist lifts:

\begin{tikzcd}
	0 & A & B & C & 0 \\
	\\
	& I
	\arrow[from=1-1, to=1-2]
	\arrow[from=1-2, to=1-3]
	\arrow[from=1-3, to=1-4]
	\arrow[from=1-4, to=1-5]
	\arrow[from=1-2, to=3-2]
	\arrow["\exists", dashed, from=1-3, to=3-2]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzEsMCwiQSJdLFsyLDAsIkIiXSxbMywwLCJDIl0sWzQsMCwiMCJdLFsxLDIsIkkiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbMSw1XSxbMiw1LCJcXGV4aXN0cyIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

:::

:::{.definition title="Flasque and soft sheaves"}
A sheaf $\mcf \in \Sh(X; \zmod)$ is **flasque** iff for all $U \subseteq X$ open, $F(X) \surjects F(U)$.
It is **soft** iff the same holds for all *closed* sets instead, and **fine** if $\mcf$ has a partition of unity property.
:::

:::{.remark}
Note that fine $\implies$ soft and flasque $\implies$ soft.
Fine sheaves are best for paracompact Hausdorff spaces, and flasque are better for e.g. the order topology.
:::