# Sheaf Cohomology (Friday, January 29)

Last time: we defined the Čech complex $C_{\mathfrak{U} }^p(X, \mathcal{F} ) \da \prod_{i_1, \cdots, i_p} \mathcal{F} (U_{i_1} \intersect \cdots \intersect U_{i_p})$ for \( \mathfrak{U}\da \ts{U_i}  \) is an open cover of $X$ and $F$ is a sheaf of abelian groups.

:::{.fact}
If \( \mathfrak{U}  \) is a sufficiently fine cover then $H^p_{\mathfrak{U}}(X, \mathcal{F})$ is independent of \( \mathfrak{U}  \), and we call this $H^p(X; \mathcal{F})$. 
:::

:::{.remark}
Recall that we computed \( H^p(S^1, \ul{\ZZ} = [\ZZ, \ZZ, 0, \cdots] \).
:::

:::{.theorem title="When sheaf cohomology is isomorphic to singular cohomology"}
Let $X$ be a paracompact and locally contractible topological space.
Then $H^p(X, \ul{\ZZ}) \cong H^p_{\sing}(X, \ul{\ZZ})$.
This will also hold more generally with $\ul{\ZZ}$ replaced by $\ul{A}$ for any $A\in \Ab$.
:::

:::{.definition title="Acyclic Sheaves"}
We say \( \mathcal{F}  \) is *acyclic* on $X$ if $H^{> 0 }(X; \mathcal{F}) = 0$.
:::

:::{.remark}
How to visualize when $H^1(X; \mathcal{F}) = 0$: 

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-01-29_14-01.pdf_tex} };
\end{tikzpicture}

On the intersections, we have $\im \bd^0 = \ts{ (s_{i} - s_{j})_{ij} \st s_i \in \mathcal{F}(U_i)}$, which are *cocycles*.
We have $C^1(X; \mathcal{F})$ are collections of sections of \( \mathcal{F}  \) on every double overlap. 
We can check that $\ker \bd^1 = \ts{ (s_{ij}) \st s_{ij} - s_{ik} + s_{jk} = 0}$, which is the cocycle condition. 
From the exercise from last class, $\bd^2 = 0$.
:::

:::{.theorem title="(Important!)"}
Let $X$ be a paracompact Hausdorff space and let 
\[
0 \to \mathcal{F}_1 \mapsvia{\varphi} \mathcal{F}_2 \to \mathcal{F}_3 \to 0   
\]
be a SES of sheaves of abelian groups, i.e. \( \mathcal{F}_3 = \coker(\varphi)  \) and \( \varphi \) is injective.
Then there is a LES in cohomology:

% https://q.uiver.app/?q=WzAsOCxbMCwwLCIwIl0sWzIsMCwiSF4wKFg7IFxcbWF0aGNhbHtGfV8xKSJdLFs0LDAsIkheMChYOyBcXG1hdGhjYWx7Rn1fMikiXSxbNiwwLCJIXjAoWDsgXFxtYXRoY2Fse0Z9XzMpIl0sWzIsMiwiSF4xKFg7IFxcbWF0aGNhbHtGfV8xKSJdLFs0LDIsIkheMShYOyBcXG1hdGhjYWx7Rn1fMikiXSxbNiwyLCJIXjEoWDsgXFxtYXRoY2Fse0Z9XzMpIl0sWzIsNCwiXFxjZG90cyJdLFszLDRdLFswLDFdLFsxLDJdLFsyLDNdLFs0LDVdLFs1LDZdLFs2LDddXQ==
\begin{tikzcd}
	0 && {H^0(X; \mathcal{F}_1)} && {H^0(X; \mathcal{F}_2)} && {H^0(X; \mathcal{F}_3)} \\
	\\
	&& {H^1(X; \mathcal{F}_1)} && {H^1(X; \mathcal{F}_2)} && {H^1(X; \mathcal{F}_3)} \\
	\\
	&& \cdots
	\arrow[from=1-7, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=1-7]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow[from=3-7, to=5-3]
\end{tikzcd}

:::

:::{.example title="?"}
For $X$ a manifold, we can define a map and its cokernel sheaf:

\[
0 \to \ul{\ZZ} \mapsvia{\cdot 2} \ul{\ZZ} \to \ul{\ZZ/2\ZZ} \to 0
.\]
Using that cohomology of constant sheaves reduces to singular cohomology, we obtain a LES in homology:

% https://q.uiver.app/?q=WzAsOCxbMCwwLCIwIl0sWzIsMCwiSF4wKFg7IFxcWlopIl0sWzQsMCwiSF4wKFg7IFxcWlopIl0sWzYsMCwiSF4wKFg7IFxcWlovMlxcWlopIl0sWzIsMiwiSF4xKFg7IFxcWlopIl0sWzQsMiwiSF4xKFg7IFxcWlopIl0sWzYsMiwiSF4xKFg7IFxcWlovMlxcWlopIl0sWzIsNCwiXFxjZG90cyJdLFszLDRdLFswLDFdLFsxLDJdLFsyLDNdLFs0LDVdLFs1LDZdLFs2LDddXQ==
\begin{tikzcd}
	0 && {H^0(X; \ZZ)} && {H^0(X; \ZZ)} && {H^0(X; \ZZ/2\ZZ)} \\
	\\
	&& {H^1(X; \ZZ)} && {H^1(X; \ZZ)} && {H^1(X; \ZZ/2\ZZ)} \\
	\\
	&& \cdots
	\arrow[from=1-7, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=1-7]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow[from=3-7, to=5-3]
\end{tikzcd}

:::

:::{.corollary title="of theorem"}
Suppose $0 \to \mathcal{F}\to I_0 \mapsvia{d_0}  I_1 \mapsvia{d_1}  I_2 \mapsvia{d_2}  \cdots$ is an exact sequence of sheaves, so on any sufficiently small set kernels equal images., and suppose $I_n$ is acyclic for all $n\geq 0$.
This is referred to as an **acyclic resolution**.
Then the homology can be computed at $H^p(X; \mathcal{F}) = \ker (I_p(X) \to I_{p+1}(X)) / \im (I_{p-1}(X) \to I_p(X) )$.

> Note that locally having kernels equal images is different than satisfying this globally!

:::

:::{.proof title="of corollary"}
This is a formal consequence of the existence of the LES.
We can split the LES into a collection of SESs of sheaves:

\[
0 \to \mathcal{F}\to I_0 \mapsvia{d_0} \im(d_0) \to 0 && \im(d_0) = \ker(d_1) \\ 
0 \to \ker(d_1) \injects I_1 \to I_1/\ker (d_1) = \im(d_1) && \im(d_1) = \ker(d_2) \\ 
.\]
Note that these are all exact sheaves, and thus only true on small sets.
So take the associated LESs.
For the SES involving $I_0$, we obtain:

% https://q.uiver.app/?q=WzAsOCxbMCwwXSxbMCwyXSxbMCw0LCJIXntwLTF9KFxcbWF0aGNhbHtGfSkiXSxbMiw0LCJIXntwLTF9KFxcbWF0aGNhbHtJXzB9KSA9IDAiXSxbNCw0LCJIXntwLTF9KFxcbWF0aGNhbHtcXGltKGRfMCl9KSJdLFswLDYsIkhecChcXG1hdGhjYWx7Rn0pIl0sWzIsNiwiXFxjZG90cyA9IDAiXSxbNCwyLCJcXGNkb3RzIl0sWzIsM10sWzMsNF0sWzQsNSwiXFxjb25nIl0sWzUsNl0sWzcsMl1d
\begin{tikzcd}
	{} \\
	\\
	{} &&&& \cdots \\
	\\
	{H^{p-1}(\mathcal{F})} && {H^{p-1}(\mathcal{I_0}) = 0} && {H^{p-1}(\mathcal{\im(d_0)})} \\
	\\
	{H^p(\mathcal{F})} && {\cdots = 0}
	\arrow[from=5-1, to=5-3]
	\arrow[from=5-3, to=5-5]
	\arrow["\cong", from=5-5, to=7-1]
	\arrow[from=7-1, to=7-3]
	\arrow[from=3-5, to=5-1]
\end{tikzcd}

The middle entries vanish since $I_*$ was assumed acyclic, and so we obtain $H^p(\mathcal{F}) \cong H^{p-1}(\im d_0) \cong H^{p-1}(\ker d_1)$. 
Now taking the LES associated to $I_1$, we get $H^{p-1}(\ker d_1) \cong H^{p-2}(\im d_1)$.
Continuing this inductively, these are all isomorphic to $H^p(\mathcal{F}) \cong H^0(\ker d_p)/ d_{p-1}(H^0(I_{p-1}))$ after the $p$th step.

:::

:::{.corollary title="of the previous corollary"}
Suppose \( \mathfrak{U}\covers X  \), then if \( \mathcal{F}  \) is acyclic on each $U_{i_1, \cdots, i_p}$, then \( \mathfrak{U}  \) is sufficiently fine to compute Čech cohomology, and $H^p_{\mathfrak{U}}(X; \mathcal{F}) \cong H^p(X; \mathcal{F})$. 
:::

:::{.proof title="?"}
See notes.
:::

:::{.corollary title="of corollary"}
Let $X \in \Mfd_\smooth$, then $H^p(X, \ul{\RR}) = H^p_{\dR}(X;\ RR)$.
:::

:::{.proof title="?"}
Idea: construct an acyclic resolution of the sheaf $\ul{\RR}$ on $M$.
The following exact sequence works:

\[
0 \to \ul{\RR} \to \OO \mapsvia{d} \Omega^1 \mapsvia{d} \Omega^2 \to \cdots
.\]
So we start with locally constant functions, then smooth functions, then smooth 1-forms, and so on.
This is an exact sequence of sheaves, but importantly, not exact on the total space.
To check this, it suffices to show that $\ker d^p = \im d^{p-1}$ on any contractible coordinate chart.
In other words, we want to show that if $d \omega=0$ for \( \omega\in \Omega^p(\RR^n) \) then \( \omega= d \alpha \) for some \( \alpha\in \Omega^{p-1}(\RR^n) \).
This is true by integration!
Using the previous corollary, $H^p(X; \ul{\RR}) = \ker(\Omega^p(X) \mapsvia{d} \Omega^{p+1}(X) ) / \im (\Omega^{p-1}(X) \mapsvia{d} \Omega^p(X))$.
:::

> Check Hartshorne to see how injective resolutions line up with derived functors!