\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

103.43 Cohomology revisited

Exercise 103.43.1. Make an example of a field $k$, a curve $X$ over $k$, an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ and a cohomology class $\xi \in H^1(X, \mathcal{L})$ with the following property: for every surjective finite morphism $\pi : Y \to X$ of schemes the element $\xi $ pulls back to a nonzero element of $H^1(Y, \pi ^*\mathcal{L})$. Hint: construct $X$, $k$, $\mathcal{L}$ such that there is a short exact sequence $0 \to \mathcal{L} \to \mathcal{O}_ X \to i_*\mathcal{O}_ Z \to 0$ where $Z \subset X$ is a closed subscheme consisting of more than $1$ closed point. Then look at what happens when you pullback.

Exercise 103.43.2. Let $k$ be an algebraically closed field. Let $X$ be a projective $1$-dimensional scheme. Suppose that $X$ contains a cycle of curves, i.e., suppose there exist an $n \geq 2$ and pairwise distinct $1$-dimensional integral closed subschemes $C_1, \ldots , C_ n$ and pairwise distinct closed points $x_1, \ldots , x_ n \in X$ such that $x_ n \in C_ n \cap C_1$ and $x_ i \in C_ i \cap C_{i + 1}$ for $i = 1, \ldots , n - 1$. Prove that $H^1(X, \mathcal{O}_ X)$ is nonzero. Hint: Let $\mathcal{F}$ be the image of the map $\mathcal{O}_ X \to \bigoplus \mathcal{O}_{C_ i}$, and show $H^1(X, \mathcal{F})$ is nonzero using that $\kappa (x_ i) = k$ and $H^0(C_ i, \mathcal{O}_{C_ i}) = k$. Also use that $H^2(X, -) = 0$ by Grothendieck's theorem.

Exercise 103.43.3. Let $X$ be a projective surface over an algebraically closed field $k$. Prove there exists a proper closed subscheme $Z \subset X$ such that $H^1(Z, \mathcal{O}_ Z)$ is nonzero. Hint: Use Exercise 103.43.2.

Exercise 103.43.4. Let $X$ be a projective surface over an algebraically closed field $k$. Show that for every $n \geq 0$ there exists a proper closed subscheme $Z \subset X$ such that $\dim _ k H^1(Z, \mathcal{O}_ Z) > n$. Only explain how to do this by modifying the arguments in Exercise 103.43.3 and 103.43.2; don't give all the details.

Exercise 103.43.5. Let $X$ be a projective surface over an algebraically closed field $k$. Prove there exists a coherent $\mathcal{O}_ X$-module $\mathcal{F}$ such that $H^2(X, \mathcal{F})$ is nonzero. Hint: Use the result of Exercise 103.43.4 and a cleverly chosen exact sequence.

Exercise 103.43.6. Let $X$ and $Y$ be schemes over a field $k$ (feel free to assume $X$ and $Y$ are nice, for example qcqs or projective over $k$). Set $X \times Y = X \times _{\mathop{\mathrm{Spec}}(k)} Y$ with projections $p : X \times Y \to X$ and $q : X \times Y \to Y$. For a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and a quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{G}$ prove that

\[ H^ n(X \times Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) = \bigoplus \nolimits _{a + b = n} H^ a(X, \mathcal{F}) \otimes _ k H^ b(Y, \mathcal{G}) \]

or just show that this holds when one takes dimensions. Extra points for “clean” solutions.

Exercise 103.43.7. Let $k$ be a field. Let $X = \mathbf{P}|^1 \times \mathbf{P}^1$ be the product of the projective line over $k$ with itself with projections $p : X \to \mathbf{P}^1_ k$ and $q : X \to \mathbf{P}^1_ k$. Let

\[ \mathcal{O}(a, b) = p^*\mathcal{O}_{\mathbf{P}^1_ k}(a) \otimes _{\mathcal{O}_ X} q^*\mathcal{O}_{\mathbf{P}^1_ k}(b) \]

Compute the dimensions of $H^ i(X, \mathcal{O}(a, b))$ for all $i, a, b$. Hint: Use Exercise 103.43.6.

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 0DB3. Beware of the difference between the letter 'O' and the digit '0'.