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Tag 02UZ

Chapter 20: Cohomology of Sheaves > Section 20.21: Vanishing on Noetherian topological spaces

Proposition 20.21.7 (Grothendieck). Let $X$ be a Noetherian topological space. If $\dim(X) \leq d$, then $H^p(X, \mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\mathcal{F}$ on $X$.

Proof. We prove this lemma by induction on $d$. So fix $d$ and assume the lemma holds for all Noetherian topological spaces of dimension $< d$.

Let $\mathcal{F}$ be an abelian sheaf on $X$. Suppose $U \subset X$ is an open. Let $Z \subset X$ denote the closed complement. Denote $j : U \to X$ and $i : Z \to X$ the inclusion maps. Then there is a short exact sequence $$ 0 \to j_{!}j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \to 0 $$ see Modules, Lemma 17.7.1. Note that $j_!j^*\mathcal{F}$ is supported on the topological closure $Z'$ of $U$, i.e., it is of the form $i'_*\mathcal{F}'$ for some abelian sheaf $\mathcal{F}'$ on $Z'$, where $i' : Z' \to X$ is the inclusion.

We can use this to reduce to the case where $X$ is irreducible. Namely, according to Topology, Lemma 5.9.2 $X$ has finitely many irreducible components. If $X$ has more than one irreducible component, then let $Z \subset X$ be an irreducible component of $X$ and set $U = X \setminus Z$. By the above, and the long exact sequence of cohomology, it suffices to prove the vanishing of $H^p(X, i_*i^*\mathcal{F})$ and $H^p(X, i'_*\mathcal{F}')$ for $p > d$. By Lemma 20.21.1 it suffices to prove $H^p(Z, i^*\mathcal{F})$ and $H^p(Z', \mathcal{F}')$ vanish for $p > d$. Since $Z'$ and $Z$ have fewer irreducible components we indeed reduce to the case of an irreducible $X$.

If $d = 0$ and $X = \{*\}$, then every sheaf is constant and higher cohomology groups vanish (for example by Lemma 20.21.2).

Suppose $X$ is irreducible of dimension $d$. By Lemma 20.21.4 we reduce to the case where $\mathcal{F} = j_!\underline{\mathbf{Z}}_U$ for some open $U \subset X$. In this case we look at the short exact sequence $$ 0 \to j_!(\underline{\mathbf{Z}}_U) \to \underline{\mathbf{Z}}_X \to i_*\underline{\mathbf{Z}}_Z \to 0 $$ where $Z = X \setminus U$. By Lemma 20.21.2 we have the vanishing of $H^p(X, \underline{\mathbf{Z}}_X)$ for all $p \geq 1$. By induction we have $H^p(X, i_*\underline{\mathbf{Z}}_Z) = H^p(Z, \underline{\mathbf{Z}}_Z) = 0$ for $p \geq d$. Hence we win by the long exact cohomology sequence. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 3592–3601 (see updates for more information).

    \begin{proposition}[Grothendieck]
    \label{proposition-vanishing-Noetherian}
    \begin{reference}
    \cite[Theorem 3.6.5]{Tohoku}.
    \end{reference}
    Let $X$ be a Noetherian topological space.
    If $\dim(X) \leq d$, then $H^p(X, \mathcal{F}) = 0$
    for all $p > d$ and any abelian sheaf $\mathcal{F}$
    on $X$.
    \end{proposition}
    
    \begin{proof}
    We prove this lemma by induction on $d$.
    So fix $d$ and assume the lemma holds for all
    Noetherian topological spaces of dimension $< d$.
    
    \medskip\noindent
    Let $\mathcal{F}$ be an abelian sheaf on $X$.
    Suppose $U \subset X$ is an open. Let $Z \subset X$
    denote the closed complement.
    Denote $j : U \to X$ and $i : Z \to X$ the inclusion maps.
    Then there is a short exact sequence
    $$
    0 \to j_{!}j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \to 0
    $$
    see Modules, Lemma \ref{modules-lemma-canonical-exact-sequence}.
    Note that $j_!j^*\mathcal{F}$ is supported on
    the topological closure $Z'$ of $U$, i.e., it is of
    the form $i'_*\mathcal{F}'$ for some abelian sheaf $\mathcal{F}'$
    on $Z'$, where $i' : Z' \to X$ is the inclusion.
    
    \medskip\noindent
    We can use this to reduce to the case where $X$ is irreducible.
    Namely, according to
    Topology, Lemma \ref{topology-lemma-Noetherian}
    $X$ has finitely
    many irreducible components. If $X$ has more than one irreducible
    component, then let $Z \subset X$ be an irreducible component of $X$
    and set $U = X \setminus Z$. By the above, and the long exact sequence
    of cohomology, it suffices to prove the vanishing of
    $H^p(X, i_*i^*\mathcal{F})$ and $H^p(X, i'_*\mathcal{F}')$ for $p > d$.
    By Lemma \ref{lemma-cohomology-and-closed-immersions} it suffices to prove
    $H^p(Z, i^*\mathcal{F})$ and $H^p(Z', \mathcal{F}')$ vanish for $p > d$.
    Since $Z'$ and $Z$ have fewer irreducible components we indeed
    reduce to the case of an irreducible $X$.
    
    \medskip\noindent
    If $d = 0$ and $X = \{*\}$, then every sheaf is constant and
    higher cohomology
    groups vanish (for example by
    Lemma \ref{lemma-irreducible-constant-cohomology-zero}).
    
    \medskip\noindent
    Suppose $X$ is irreducible of dimension $d$.
    By Lemma \ref{lemma-vanishing-generated-one-section}
    we reduce to the case where
    $\mathcal{F} = j_!\underline{\mathbf{Z}}_U$ for some open $U \subset X$.
    In this case we look at the short exact sequence
    $$
    0 \to j_!(\underline{\mathbf{Z}}_U) \to
    \underline{\mathbf{Z}}_X \to i_*\underline{\mathbf{Z}}_Z \to 0
    $$
    where $Z = X \setminus U$.
    By Lemma \ref{lemma-irreducible-constant-cohomology-zero}
    we have the vanishing of $H^p(X, \underline{\mathbf{Z}}_X)$
    for all $p \geq 1$. By induction we have
    $H^p(X, i_*\underline{\mathbf{Z}}_Z) = H^p(Z, \underline{\mathbf{Z}}_Z) = 0$
    for $p \geq d$. Hence we win by the long exact cohomology sequence.
    \end{proof}

    References

    [Tohoku, Theorem 3.6.5].

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