## III.2: Cohomology of Sheaves


### V.2.1.  #to-work

a. Let $X=\mathbf{A}_k^1$ be the affine line over an infinite field $k$. Let $P, Q$ be distinct closed points of $X$, and let $U=X-\{P, Q\}$. Show that $H^1\left(X, \mathbf{Z}_U\right) \neq 0$.

b. \* More generally, let $Y \subseteq X=\mathbf{A}_k^n$ be the union of $n+1$ hyperplanes in suitably general position, and let $U=X-Y$. Show that $H^n\left(X, Z_U\right) \neq 0$. Thus the result of $(2.7)$ is the best possible.

### V.2.2.  #to-work

Let $X=\mathbf{P}_k^1$ be the projective line over an algebraically closed field $k$. Show that the exact sequence
\[
0 \rightarrow \OO \rightarrow \mathcal{K} \rightarrow \mathcal{K} / \mathcal{O} \rightarrow 0
.\]
of (II, Ex. 1.21d) is a flasque resolution of $\mathcal{O}$. Conclude from (II, Ex. 1.21e) that $H^i(X, \mathcal{O})=0$ for all $i>0$.

### V.2.3. Cohomology with Supports. #to-work

Let $X$ be a topological space, let $Y$ be a closed subset, and let $\mathcal{F}$ be a sheaf of abelian groups. Let $\Gamma_Y(X, \mathcal{F})$ denote the group of sections of $\mathcal{F}$ with support in $Y$ (II, Ex. 1.20).

a. Show that $\Gamma_Y(X, \cdot)$ is a left exact functor from $\Ab(X)$ to $\Ab$.
We denote the right derived functors of $\Gamma_Y(X, \cdot)$ by $H_Y^i(X, \cdot)$. They are the cohomology groups of $X$ with supports in $Y$, and coefficients in a given sheaf.

b. If $0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \dot{\mcf}^{\prime \prime} \rightarrow 0$ is an exact sequence of sheaves, with $\mathcal{F}^{\prime}$ flasque, show that
\[
0 \rightarrow \Gamma_Y\left(X, \mathcal{F}^{\prime}\right) \rightarrow \Gamma_Y(X, \mathcal{F}) \rightarrow \Gamma_Y\left(X, \mathcal{F}^{\prime \prime}\right) \rightarrow 0
\]
is exact.

c. Show that if $\mathcal{F}$ is flasque, then $H_Y^i(X, \mathcal{F})=0$ for all $i>0$.

d. If $\mathcal{F}$ is flasque, show that the sequence
\[
0 \rightarrow \Gamma_Y(X, \mathcal{F}) \rightarrow \Gamma(X, \mathcal{F}) \rightarrow \Gamma(X-Y, \mathcal{F}) \rightarrow 0
\]
is exact.

e. Let $U=X-Y$. Show that for any $\mathcal{F}$, there is a long exact sequence of cohomology groups
\[
0 &\rightarrow H_Y^0(X, \mathcal{F}) \rightarrow H^0(X, \mathcal{F}) \rightarrow H^0\left(U,\left.\mathcal{F}\right|_U\right) \rightarrow \\
&\rightarrow H_Y^1(X, \mathcal{F}) \rightarrow H^1(X, \mathcal{F}) \rightarrow H^1\left(U,\left.\mathcal{F}\right|_U\right) \rightarrow \\
&\rightarrow H_Y^2(X, \mathcal{F}) \rightarrow \ldots
.\]

f. *Excision*. Let $V$ be an open subset of $X$ containing $Y$. Then there are natural functorial isomorphisms, for all $i$ and $\mathcal{F}$,
\[
H_Y^i(X, \mathcal{F}) \cong H_Y^i\left(V,\left.\mathcal{F}\right|_V\right) .
\]

### V.2.4. Mayer-Vietoris Sequence.  #to-work

Let $Y_1, Y_2$ be two closed subsets of $X$. Then there is a long exact sequence of cohomology with supports
\[
\begin{aligned}
\ldots & \rightarrow H_{Y_1 \cap Y_2}^i(X, \mathcal{F}) \rightarrow H_{Y_1}^i(X, \mathcal{F}) \oplus H_{Y_2}^i(X, \mathcal{F}) \rightarrow H_{Y_1 \cup Y_2}^i(X, \mathcal{F}) \rightarrow \\
& \rightarrow H_{Y_1 \cap Y_2}^{i+1}(X, \mathcal{F}) \rightarrow \cdots
\end{aligned}
\]

### V.2.5.  #to-work

Let $X$ be a Zariski space (II, Ex. 3.17). Let $P \in X$ be a closed point, and let $X_P$ be the subset of $X$ consisting of all points $Q \in X$ such that $P \in\{Q\}^{-}$. We call $X_P$ the **local space** of $X$ at $P$, and give it the induced topology. 

Let $j: X_P \rightarrow X$ be the inclusion, and for any sheaf $\mathcal{F}$ on $X$, let $\mathcal{F}_P=j^* \mathcal{F}$. Show that for all $i, \mathcal{F}$, we have
\[
H_P^i(X, \mathcal{F})=H_P^i\left(X_P, \mathcal{F}_P\right) .
\]

### V.2.6.  #to-work

Let $X$ be a noetherian topological space, and let $\left\{\mathcal{I}_\alpha\right\}_{\alpha \in A}$ be a direct system of injective sheaves of abelian groups on $X$. Then $\colim\, \mathcal{I}_\alpha$ is also injective.[^hint.3.2.6]

[^hint.3.2.6]: 
Hints: First show that a sheaf $\mathcal{I}$ is injective if and only if for every open set $U \subseteq X$, and for every subsheaf $\mathcal{R} \subseteq \mathbf{Z}_U$, and for every map $f: \mathcal{R} \rightarrow \mathcal{I}$, there exists an extension of $f$ to a map of $\mathbf{Z}_U \rightarrow \mathcal{I}$. Secondly, show that any such sheaf $\mathcal{R}$ is finitely generated, so any map $\mathcal{R} \rightarrow \colim\, \mathcal{I}_\alpha$ factors through one of the $\mathcal{I}_\alpha$.


### V.2.7.  #to-work

Let $S^1$ be the circle (with its usual topology), and let $\mathbf{Z}$ be the constant sheaf $\mathbf{Z}$.

a. Show that $H^1\left(S^1, \mathbf{Z}\right) \cong \mathbf{Z}$, using our definition of cohomology.

b. Now let $\mathcal{R}$ be the sheaf of germs of continuous real-valued functions on $S^1$. Show that $H^1\left(S^1, \mathcal{R}\right)=0$.