Exercise 111.42.1. Let $X = \mathbf{R}$ with the usual Euclidean topology. Using only formal properties of cohomology (functoriality and the long exact cohomology sequence) show that there exists a sheaf $\mathcal{F}$ on $X$ with nonzero $H^1(X, \mathcal{F})$.
111.42 Cohomology
Exercise 111.42.2. Let $X = U \cup V$ be a topological space written as the union of two opens. Then we have a long exact Mayer-Vietoris sequence What property of injective sheaves is essential for the construction of the Mayer-Vietoris long exact sequence? Why does it hold?
\[ 0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots \]
Exercise 111.42.3. Let $X$ be a topological space.
Show that $H^ i(X, \mathcal{F})$ is zero for $i > 0$ if $X$ has $2$ or fewer points.
What if $X$ has $3$ points?
Exercise 111.42.4. Let $X$ be the spectrum of a local ring. Show that $H^ i(X, \mathcal{F})$ is zero for $i > 0$ and any sheaf of abelian groups $\mathcal{F}$.
Exercise 111.42.5. Let $f : X \to Y$ be an affine morphism of schemes. Prove that $H^ i(X, \mathcal{F}) = H^ i(Y, f_*\mathcal{F})$ for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$. Feel free to impose some further conditions on $X$ and $Y$ and use the agreement of Čech cohomology with cohomology for quasi-coherent sheaves and affine open coverings of separated schemes.
Exercise 111.42.6. Let $A$ be a ring. Let $\mathbf{P}^ n_ A = \text{Proj}(A[T_0, \ldots , T_ n])$ be projective space over $A$. Let $\mathbf{A}^{n + 1}_ A = \mathop{\mathrm{Spec}}(A[T_0, \ldots , T_ n])$ and let be the complement of the image of the closed immersion $0 : \mathop{\mathrm{Spec}}(A) \to \mathbf{A}^{n + 1}_ A$. Construct an affine surjective morphism and prove that $f_*\mathcal{O}_ U = \bigoplus _{d \in \mathbf{Z}} \mathcal{O}_{\mathbf{P}^ n_ A}(d)$. More generally, show that for a graded $A[T_0, \ldots , T_ n]$-module $M$ one has where on the left hand side we have the quasi-coherent sheaf $\widetilde{M}$ associated to $M$ on $\mathbf{A}^{n + 1}_ A$ and on the right we have the quasi-coherent sheaves $\widetilde{M(d)}$ associated to the graded module $M(d)$.
\[ U = \bigcup \nolimits _{i = 0, \ldots , n} D(T_ i) \subset \mathbf{A}^{n + 1}_ A \]
\[ f : U \longrightarrow \mathbf{P}^ n_ A \]
\[ f_*(\widetilde{M}|_ U) = \bigoplus \nolimits _{d \in \mathbf{Z}} \widetilde{M(d)} \]
Exercise 111.42.7. Let $A$ be a ring and let $\mathbf{P}^ n_ A = \text{Proj}(A[T_0, \ldots , T_ n])$ be projective space over $A$. Carefully compute the cohomology of the Serre twists $\mathcal{O}_{\mathbf{P}^ n_ A}(d)$ of the structure sheaf on $\mathbf{P}^ n_ A$. Feel free to use Čech cohomology and the agreement of Čech cohomology with cohomology for quasi-coherent sheaves and affine open coverings of separated schemes.
Exercise 111.42.8. Let $A$ be a ring and let $\mathbf{P}^ n_ A = \text{Proj}(A[T_0, \ldots , T_ n])$ be projective space over $A$. Let $F \in A[T_0, \ldots , T_ n]$ be homogeneous of degree $d$. Let $X \subset \mathbf{P}^ n_ A$ be the closed subscheme corresponding to the graded ideal $(F)$ of $A[T_0, \ldots , T_ n]$. What can you say about $H^ i(X, \mathcal{O}_ X)$?
Exercise 111.42.9. Let $R$ be a ring such that for any left exact functor $F : \text{Mod}_ R \to \textit{Ab}$ we have $R^ iF = 0$ for $i > 0$. Show that $R$ is a finite product of fields.
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