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Tag 0D8P

102.41. Cohomology

Exercise 102.41.1. 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 102.41.2. 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 102.41.3. 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 102.41.4. 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 102.41.5. 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 102.41.6. 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 102.41.7. 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 102.41.8. 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 102.41.9. 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.

    The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 3915–4030 (see updates for more information).

    \section{Cohomology}
    \label{section-cohomology}
    
    \begin{exercise}
    \label{exercise-cohomology-not-zero}
    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 $H^1(X, \mathcal{F})$.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-mayer-vietoris}
    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?
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-cohomology-two-point-space}
    Let $X$ be a topological space.
    \begin{enumerate}
    \item Show that $H^i(X, \mathcal{F})$ is zero for $i > 0$
    if $X$ has $2$ or fewer points.
    \item What if $X$ has $3$ points?
    \end{enumerate}
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-cohomology-spec-local-ring}
    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 $\mathcal{F}$.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-affine-morphism}
    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 {\v C}ech cohomology with cohomology
    for quasi-coherent sheaves and affine open coverings of separated schemes.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-affine-morphism-to-Pn}
    Let $A$ be a ring. Let $\mathbf{P}^n_A = \text{Proj}(A[T_0, \ldots, T_n])$
    be projective space over $A$. Let
    $\mathbf{A}^{n + 1}_A = \Spec(A[T_0, \ldots, T_n])$ 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
    $0 : \Spec(A) \to \mathbf{A}^{n + 1}_A$.
    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 $M(d)$.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-compute-Pn}
    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 {\v C}ech cohomology
    and the agreement of {\v C}ech cohomology with cohomology
    for quasi-coherent sheaves and affine open coverings of separated schemes.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-cohomology-hypersurface}
    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 $H^i(X, \mathcal{O}_X)$?
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-characterize-finite-product-fields}
    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.
    \end{exercise}

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