Exercise 111.44.1. 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.
111.44 Cohomology revisited
Exercise 111.44.2. 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 111.44.3. 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 111.44.2.
Exercise 111.44.4. 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 111.44.3 and 111.44.2; don't give all the details.
Exercise 111.44.5. 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 111.44.4 and a cleverly chosen exact sequence.
Exercise 111.44.6. 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 111.44.7. 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 111.44.6.
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