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Tag 01E0

20.8. Locality of cohomology

The following lemma says there is no ambiguity in defining the cohomology of a sheaf $\mathcal{F}$ over an open.

Lemma 20.8.1. Let $X$ be a ringed space. Let $U \subset X$ be an open subspace.

  1. If $\mathcal{I}$ is an injective $\mathcal{O}_X$-module then $\mathcal{I}|_U$ is an injective $\mathcal{O}_U$-module.
  2. For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we have $H^p(U, \mathcal{F}) = H^p(U, \mathcal{F}|_U)$.

Proof. Denote $j : U \to X$ the open immersion. Recall that the functor $j^{-1}$ of restriction to $U$ is a right adjoint to the functor $j_!$ of extension by $0$, see Sheaves, Lemma 6.31.8. Moreover, $j_!$ is exact. Hence (1) follows from Homology, Lemma 12.25.1.

By definition $H^p(U, \mathcal{F}) = H^p(\Gamma(U, \mathcal{I}^\bullet))$ where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution in $\textit{Mod}(\mathcal{O}_X)$. By the above we see that $\mathcal{F}|_U \to \mathcal{I}^\bullet|_U$ is an injective resolution in $\textit{Mod}(\mathcal{O}_U)$. Hence $H^p(U, \mathcal{F}|_U)$ is equal to $H^p(\Gamma(U, \mathcal{I}^\bullet|_U))$. Of course $\Gamma(U, \mathcal{F}) = \Gamma(U, \mathcal{F}|_U)$ for any sheaf $\mathcal{F}$ on $X$. Hence the equality in (2). $\square$

Let $X$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $U \subset V \subset X$ be open subsets. Then there is a canonical restriction mapping \begin{equation} \tag{20.8.1.1} H^n(V, \mathcal{F}) \longrightarrow H^n(U, \mathcal{F}), \quad \xi \longmapsto \xi|_U \end{equation} functorial in $\mathcal{F}$. Namely, choose any injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. The restriction mappings of the sheaves $\mathcal{I}^p$ give a morphism of complexes $$ \Gamma(V, \mathcal{I}^\bullet) \longrightarrow \Gamma(U, \mathcal{I}^\bullet) $$ The LHS is a complex representing $R\Gamma(V, \mathcal{F})$ and the RHS is a complex representing $R\Gamma(U, \mathcal{F})$. We get the map on cohomology groups by applying the functor $H^n$. As indicated we will use the notation $\xi \mapsto \xi|_U$ to denote this map. Thus the rule $U \mapsto H^n(U, \mathcal{F})$ is a presheaf of $\mathcal{O}_X$-modules. This presheaf is customarily denoted $\underline{H}^n(\mathcal{F})$. We will give another interpretation of this presheaf in Lemma 20.12.4.

Lemma 20.8.2. Let $X$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $U \subset X$ be an open subspace. Let $n > 0$ and let $\xi \in H^n(U, \mathcal{F})$. Then there exists an open covering $U = \bigcup_{i\in I} U_i$ such that $\xi|_{U_i} = 0$ for all $i \in I$.

Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution. Then $$ H^n(U, \mathcal{F}) = \frac{\text{Ker}(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\text{Im}(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}. $$ Pick an element $\tilde \xi \in \mathcal{I}^n(U)$ representing the cohomology class in the presentation above. Since $\mathcal{I}^\bullet$ is an injective resolution of $\mathcal{F}$ and $n > 0$ we see that the complex $\mathcal{I}^\bullet$ is exact in degree $n$. Hence $\text{Im}(\mathcal{I}^{n - 1} \to \mathcal{I}^n) = \text{Ker}(\mathcal{I}^n \to \mathcal{I}^{n + 1})$ as sheaves. Since $\tilde \xi$ is a section of the kernel sheaf over $U$ we conclude there exists an open covering $U = \bigcup_{i \in I} U_i$ such that $\tilde \xi|_{U_i}$ is the image under $d$ of a section $\xi_i \in \mathcal{I}^{n - 1}(U_i)$. By our definition of the restriction $\xi|_{U_i}$ as corresponding to the class of $\tilde \xi|_{U_i}$ we conclude. $\square$

Lemma 20.8.3. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a $\mathcal{O}_X$-module. The sheaves $R^if_*\mathcal{F}$ are the sheaves associated to the presheaves $$ V \longmapsto H^i(f^{-1}(V), \mathcal{F}) $$ with restriction mappings as in Equation (20.8.1.1). There is a similar statement for $R^if_*$ applied to a bounded below complex $\mathcal{F}^\bullet$.

Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution. Then $R^if_*\mathcal{F}$ is by definition the $i$th cohomology sheaf of the complex $$ f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots $$ By definition of the abelian category structure on $\mathcal{O}_Y$-modules this cohomology sheaf is the sheaf associated to the presheaf $$ V \longmapsto \frac{\text{Ker}(f_*\mathcal{I}^i(V) \to f_*\mathcal{I}^{i + 1}(V))} {\text{Im}(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^i(V))} $$ and this is obviously equal to $$ \frac{\text{Ker}(\mathcal{I}^i(f^{-1}(V)) \to \mathcal{I}^{i + 1}(f^{-1}(V)))} {\text{Im}(\mathcal{I}^{i - 1}(f^{-1}(V)) \to \mathcal{I}^i(f^{-1}(V)))} $$ which is equal to $H^i(f^{-1}(V), \mathcal{F})$ and we win. $\square$

Lemma 20.8.4. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $V \subset Y$ be an open subspace. Denote $g : f^{-1}(V) \to V$ the restriction of $f$. Then we have $$ R^pg_*(\mathcal{F}|_{f^{-1}(V)}) = (R^pf_*\mathcal{F})|_V $$ There is a similar statement for the derived image $Rf_*\mathcal{F}^\bullet$ where $\mathcal{F}^\bullet$ is a bounded below complex of $\mathcal{O}_X$-modules.

Proof. First proof. Apply Lemmas 20.8.3 and 20.8.1 to see the displayed equality. Second proof. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ and use that $\mathcal{F}|_{f^{-1}(V)} \to \mathcal{I}^\bullet|_{f^{-1}(V)}$ is an injective resolution also. $\square$

Remark 20.8.5. Here is a different approach to the proofs of Lemmas 20.8.2 and 20.8.3 above. Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i_X : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$ be the inclusion functor and let $\#$ be the sheafification functor. Recall that $i_X$ is left exact and $\#$ is exact.

  1. First prove Lemma 20.12.4 below which says that the right derived functors of $i_X$ are given by $R^pi_X\mathcal{F} = \underline{H}^p(\mathcal{F})$. Here is another proof: The equality is clear for $p = 0$. Both $(R^pi_X)_{p \geq 0}$ and $(\underline{H}^p)_{p \geq 0}$ are delta functors vanishing on injectives, hence both are universal, hence they are isomorphic. See Homology, Section 12.11.
  2. A restatement of Lemma 20.8.2 is that $(\underline{H}^p(\mathcal{F}))^\# = 0$, $p > 0$ for any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$. To see this is true, use that ${}^\#$ is exact so $$ (\underline{H}^p(\mathcal{F}))^\# = (R^pi_X\mathcal{F})^\# = R^p(\# \circ i_X)(\mathcal{F}) = 0 $$ because $\# \circ i_X$ is the identity functor.
  3. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The presheaf $V \mapsto H^p(f^{-1}V, \mathcal{F})$ is equal to $R^p (i_Y \circ f_*)\mathcal{F}$. You can prove this by noticing that both give universal delta functors as in the argument of (1) above. Hence Lemma 20.8.3 says that $R^p f_* \mathcal{F}= (R^p (i_Y \circ f_*)\mathcal{F})^\#$. Again using that $\#$ is exact a that $\# \circ i_Y$ is the identity functor we see that $$ R^p f_* \mathcal{F} = R^p(\# \circ i_Y \circ f_*)\mathcal{F} = (R^p (i_Y \circ f_*)\mathcal{F})^\# $$ as desired.

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 490–705 (see updates for more information).

    \section{Locality of cohomology}
    \label{section-locality}
    
    \noindent
    The following lemma says there is no ambiguity in defining the cohomology
    of a sheaf $\mathcal{F}$ over an open.
    
    \begin{lemma}
    \label{lemma-cohomology-of-open}
    Let $X$ be a ringed space.
    Let $U \subset X$ be an open subspace.
    \begin{enumerate}
    \item If $\mathcal{I}$ is an injective $\mathcal{O}_X$-module
    then $\mathcal{I}|_U$ is an injective $\mathcal{O}_U$-module.
    \item For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we have
    $H^p(U, \mathcal{F}) = H^p(U, \mathcal{F}|_U)$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Denote $j : U \to X$ the open immersion.
    Recall that the functor $j^{-1}$ of restriction to $U$ is a right adjoint
    to the functor $j_!$ of extension by $0$, see
    Sheaves, Lemma \ref{sheaves-lemma-j-shriek-modules}.
    Moreover, $j_!$ is exact. Hence (1) follows from
    Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
    
    \medskip\noindent
    By definition $H^p(U, \mathcal{F}) = H^p(\Gamma(U, \mathcal{I}^\bullet))$
    where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution
    in $\textit{Mod}(\mathcal{O}_X)$.
    By the above we see that $\mathcal{F}|_U \to \mathcal{I}^\bullet|_U$
    is an injective resolution in $\textit{Mod}(\mathcal{O}_U)$.
    Hence $H^p(U, \mathcal{F}|_U)$ is equal to
    $H^p(\Gamma(U, \mathcal{I}^\bullet|_U))$.
    Of course $\Gamma(U, \mathcal{F}) = \Gamma(U, \mathcal{F}|_U)$ for
    any sheaf $\mathcal{F}$ on $X$.
    Hence the equality
    in (2).
    \end{proof}
    
    \noindent
    Let $X$ be a ringed space.
    Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
    Let $U \subset V \subset X$ be open subsets.
    Then there is a canonical {\it restriction mapping}
    \begin{equation}
    \label{equation-restriction-mapping}
    H^n(V, \mathcal{F})
    \longrightarrow
    H^n(U, \mathcal{F}), \quad
    \xi \longmapsto \xi|_U
    \end{equation}
    functorial in $\mathcal{F}$. Namely, choose any injective
    resolution $\mathcal{F} \to \mathcal{I}^\bullet$. The restriction
    mappings of the sheaves $\mathcal{I}^p$ give a morphism of complexes
    $$
    \Gamma(V, \mathcal{I}^\bullet)
    \longrightarrow
    \Gamma(U, \mathcal{I}^\bullet)
    $$
    The LHS is a complex representing $R\Gamma(V, \mathcal{F})$
    and the RHS is a complex representing $R\Gamma(U, \mathcal{F})$.
    We get the map on cohomology groups by applying the functor $H^n$.
    As indicated we will use the notation $\xi \mapsto \xi|_U$ to denote this map.
    Thus the rule $U \mapsto H^n(U, \mathcal{F})$ is a presheaf of
    $\mathcal{O}_X$-modules. This presheaf is customarily denoted
    $\underline{H}^n(\mathcal{F})$. We will give another interpretation
    of this presheaf in Lemma \ref{lemma-include}.
    
    \begin{lemma}
    \label{lemma-kill-cohomology-class-on-covering}
    Let $X$ be a ringed space.
    Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
    Let $U \subset X$ be an open subspace.
    Let $n > 0$ and let $\xi \in H^n(U, \mathcal{F})$.
    Then there exists an open covering
    $U = \bigcup_{i\in I} U_i$ such that $\xi|_{U_i} = 0$ for
    all $i \in I$.
    \end{lemma}
    
    \begin{proof}
    Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution.
    Then
    $$
    H^n(U, \mathcal{F}) =
    \frac{\Ker(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))}
    {\Im(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}.
    $$
    Pick an element $\tilde \xi \in \mathcal{I}^n(U)$ representing the
    cohomology class in the presentation above. Since $\mathcal{I}^\bullet$
    is an injective resolution of $\mathcal{F}$ and $n > 0$ we see that
    the complex $\mathcal{I}^\bullet$ is exact in degree $n$. Hence
    $\Im(\mathcal{I}^{n - 1} \to \mathcal{I}^n) =
    \Ker(\mathcal{I}^n \to \mathcal{I}^{n + 1})$ as sheaves.
    Since $\tilde \xi$ is a section of the kernel sheaf over $U$
    we conclude there exists an open covering $U = \bigcup_{i \in I} U_i$
    such that $\tilde \xi|_{U_i}$ is the image under $d$ of a section
    $\xi_i \in \mathcal{I}^{n - 1}(U_i)$. By our definition of the
    restriction $\xi|_{U_i}$ as corresponding to the class of
    $\tilde \xi|_{U_i}$ we conclude.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-describe-higher-direct-images}
    Let $f : X \to Y$ be a morphism of ringed spaces.
    Let $\mathcal{F}$ be a $\mathcal{O}_X$-module.
    The sheaves $R^if_*\mathcal{F}$ are the sheaves
    associated to the presheaves
    $$
    V \longmapsto H^i(f^{-1}(V), \mathcal{F})
    $$
    with restriction mappings as in Equation (\ref{equation-restriction-mapping}).
    There is a similar statement for $R^if_*$ applied to a
    bounded below complex $\mathcal{F}^\bullet$.
    \end{lemma}
    
    \begin{proof}
    Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution.
    Then $R^if_*\mathcal{F}$ is by definition the $i$th cohomology sheaf
    of the complex
    $$
    f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots
    $$
    By definition of the abelian category structure on $\mathcal{O}_Y$-modules
    this cohomology sheaf is the sheaf associated to the presheaf
    $$
    V
    \longmapsto
    \frac{\Ker(f_*\mathcal{I}^i(V) \to f_*\mathcal{I}^{i + 1}(V))}
    {\Im(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^i(V))}
    $$
    and this is obviously equal to
    $$
    \frac{\Ker(\mathcal{I}^i(f^{-1}(V)) \to \mathcal{I}^{i + 1}(f^{-1}(V)))}
    {\Im(\mathcal{I}^{i - 1}(f^{-1}(V)) \to \mathcal{I}^i(f^{-1}(V)))}
    $$
    which is equal to $H^i(f^{-1}(V), \mathcal{F})$
    and we win.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-localize-higher-direct-images}
    Let $f : X \to Y$ be a morphism of ringed spaces.
    Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
    Let $V \subset Y$ be an open subspace.
    Denote $g : f^{-1}(V) \to V$ the restriction of $f$.
    Then we have
    $$
    R^pg_*(\mathcal{F}|_{f^{-1}(V)}) = (R^pf_*\mathcal{F})|_V
    $$
    There is a similar statement for the
    derived image $Rf_*\mathcal{F}^\bullet$ where $\mathcal{F}^\bullet$
    is a bounded below complex of $\mathcal{O}_X$-modules.
    \end{lemma}
    
    \begin{proof}
    First proof. Apply Lemmas \ref{lemma-describe-higher-direct-images}
    and \ref{lemma-cohomology-of-open} to see the displayed equality.
    Second proof. Choose an injective resolution
    $\mathcal{F} \to \mathcal{I}^\bullet$
    and use that $\mathcal{F}|_{f^{-1}(V)} \to \mathcal{I}^\bullet|_{f^{-1}(V)}$
    is an injective resolution also.
    \end{proof}
    
    \begin{remark}
    \label{remark-daniel}
    Here is a different approach to the proofs of
    Lemmas \ref{lemma-kill-cohomology-class-on-covering} and
    \ref{lemma-describe-higher-direct-images} above.
    Let $(X, \mathcal{O}_X)$ be a ringed space.
    Let $i_X : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$
    be the inclusion functor and let $\#$ be the sheafification functor.
    Recall that $i_X$ is left exact and $\#$ is exact.
    \begin{enumerate}
    \item First prove Lemma \ref{lemma-include} below which says that the
    right derived functors of $i_X$ are given by
    $R^pi_X\mathcal{F} = \underline{H}^p(\mathcal{F})$.
    Here is another proof: The equality is clear for $p = 0$.
    Both $(R^pi_X)_{p \geq 0}$ and $(\underline{H}^p)_{p \geq 0}$
    are delta functors vanishing on injectives, hence both are universal,
    hence they are isomorphic. See Homology,
    Section \ref{homology-section-cohomological-delta-functor}.
    \item A restatement of Lemma \ref{lemma-kill-cohomology-class-on-covering}
    is that $(\underline{H}^p(\mathcal{F}))^\# = 0$, $p > 0$ for any sheaf of
    $\mathcal{O}_X$-modules $\mathcal{F}$.
    To see this is true, use that ${}^\#$ is exact so
    $$
    (\underline{H}^p(\mathcal{F}))^\# =
    (R^pi_X\mathcal{F})^\# =
    R^p(\# \circ i_X)(\mathcal{F}) = 0
    $$
    because $\# \circ i_X$ is the identity functor.
    \item Let $f : X \to Y$ be a morphism of ringed spaces.
    Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The presheaf
    $V \mapsto H^p(f^{-1}V, \mathcal{F})$ is equal to
    $R^p (i_Y \circ f_*)\mathcal{F}$. You can prove this by noticing that
    both give universal delta functors as in the argument of (1) above.
    Hence Lemma \ref{lemma-describe-higher-direct-images}
    says that $R^p f_* \mathcal{F}= (R^p (i_Y \circ f_*)\mathcal{F})^\#$.
    Again using that $\#$ is exact a that $\# \circ i_Y$ is the identity
    functor we see that
    $$
    R^p f_* \mathcal{F} =
    R^p(\# \circ i_Y \circ f_*)\mathcal{F} =
    (R^p (i_Y \circ f_*)\mathcal{F})^\#
    $$
    as desired.
    \end{enumerate}
    \end{remark}

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