\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

103.41 Cohomology

Exercise 103.41.1. Let $X = \mathbf{R}$ with the usual Euclidian 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 $H^1(X, \mathcal{F})$.

Exercise 103.41.2. 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

\[ 0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots \]

What property of injective sheaves is essential for the construction of the Mayer-Vietoris long exact sequence? Why does it hold?

Exercise 103.41.3. Let $X$ be a topological space.

  1. Show that $H^ i(X, \mathcal{F})$ is zero for $i > 0$ if $X$ has $2$ or fewer points.

  2. What if $X$ has $3$ points?

Exercise 103.41.4. 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 103.41.5. 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 103.41.6. 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

\[ U = \bigcup \nolimits _{i = 0, \ldots , n} D(T_ i) \subset \mathbf{A}^{n + 1}_ A \]

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

\[ f : U \longrightarrow \mathbf{P}^ n_ A \]

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

\[ f_*(\widetilde{M}|_ U) = \bigoplus \nolimits _{d \in \mathbf{Z}} \widetilde{M(d)} \]

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 $M(d)$.

Exercise 103.41.7. 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 103.41.8. 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 103.41.9. 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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0D8P. Beware of the difference between the letter 'O' and the digit '0'.