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III.2: Cohomology of Sheaves

V.2.1. #to_work

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 \(H^1\left(X, \mathbf{Z}_U\right) \neq 0\).
* 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.

Let \(X=\mathbf{P}_k^1\) be the projective line over an algebraically closed field \(k\). Show that the exact sequence \begin{align*} 0 \rightarrow {\mathcal{O}}\rightarrow \mathcal{K} \rightarrow \mathcal{K} / \mathcal{O} \rightarrow 0 .\end{align*} 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).

Show that \(\Gamma_Y(X, \cdot)\) is a left exact functor from \({\mathsf{Ab}}(X)\) to \({\mathsf{Ab}}\). 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.
If \(0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \dot{{\mathcal{F}}}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, with \(\mathcal{F}^{\prime}\) flasque, show that \begin{align*} 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 \end{align*} is exact.

Show that if \(\mathcal{F}\) is flasque, then \(H_Y^i(X, \mathcal{F})=0\) for all \(i>0\).
If \(\mathcal{F}\) is flasque, show that the sequence \begin{align*} 0 \rightarrow \Gamma_Y(X, \mathcal{F}) \rightarrow \Gamma(X, \mathcal{F}) \rightarrow \Gamma(X-Y, \mathcal{F}) \rightarrow 0 \end{align*} is exact.

Let \(U=X-Y\). Show that for any \(\mathcal{F}\), there is a long exact sequence of cohomology groups \begin{align*} 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 .\end{align*}
Excision. Let \(V\) be an open subset of \(X\) containing \(Y\). Then there are natural functorial isomorphisms, for all \(i\) and \(\mathcal{F}\), \begin{align*} H_Y^i(X, \mathcal{F}) \cong H_Y^i\left(V,\left.\mathcal{F}\right|_V\right) . \end{align*}

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{align*} \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} \end{align*}

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 \begin{align*} H_P^i(X, \mathcal{F})=H_P^i\left(X_P, \mathcal{F}_P\right) . \end{align*}

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. ¹

V.2.7. #to_work

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

Show that \(H^1\left(S^1, \mathbf{Z}\right) \cong \mathbf{Z}\), using our definition of cohomology.
Now let \(\mathcal{R}\) be the sheaf of germs of continuous real-valued functions on \(S^1\). Show that \(H^1\left(S^1, \mathcal{R}\right)=0\).

Footnotes

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 \(\mathcal{I}_\alpha\).