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Tag 0DB3

102.43. Cohomology revisited

Exercise 102.43.1. 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 102.43.2. 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 102.43.3. 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 102.43.2.

Exercise 102.43.4. 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 102.43.3 and 102.43.2; don't give all the details.

Exercise 102.43.5. 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 102.43.4 and a cleverly chosen exact sequence.

Exercise 102.43.6. 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 102.43.7. 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 102.43.6.

    The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 4166–4273 (see updates for more information).

    \section{Cohomology revisited}
    \label{section-more-more-cohomology}
    
    
    \begin{exercise}
    \label{exercise-nonkillable}
    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.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-cohomology-cycle-curves}
    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.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-surface}
    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 \ref{exercise-cohomology-cycle-curves}.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-surface-better}
    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 \ref{exercise-surface} and
    \ref{exercise-cohomology-cycle-curves}; don't give all the details.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-surface-h2-nonzero}
    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 \ref{exercise-surface-better} and a cleverly chosen
    exact sequence.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-kunneth}
    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 $X \times Y = X \times_{\Spec(k)} Y$ 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.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-Oab}
    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 \ref{exercise-kunneth}.
    \end{exercise}

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