\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.


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