The Stacks project

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20.22 Cohomology with support in a closed

Let $X$ be a topological space and let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be an abelian sheaf on $X$. We let

\[ \Gamma _ Z(X, \mathcal{F}) = \{ s \in \mathcal{F}(X) \mid \text{Supp}(s) \subset Z\} \]

be the sections with support in $Z$ (Modules, Definition 17.5.1). This is a left exact functor which is not exact in general. Hence we obtain a derived functor

\[ R\Gamma _ Z(X, -) : D(X) \longrightarrow D(\textit{Ab}) \]

and cohomology groups with support in $Z$ defined by $H^ q_ Z(X, \mathcal{F}) = R^ q\Gamma _ Z(X, \mathcal{F})$.

Let $\mathcal{I}$ be an injective abelian sheaf on $X$. Let $U = X \setminus Z$. Then the restriction map $\mathcal{I}(X) \to \mathcal{I}(U)$ is surjective (Lemma 20.9.1) with kernel $\Gamma _ Z(X, \mathcal{I})$. It immediately follows that for $K \in D(X)$ there is a distinguished triangle

\[ R\Gamma _ Z(X, K) \to R\Gamma (X, K) \to R\Gamma (U, K) \to R\Gamma _ Z(X, K)[1] \]

in $D(\textit{Ab})$. As a consequence we obtain a long exact cohomology sequence

\[ \ldots \to H^ i_ Z(X, K) \to H^ i(X, K) \to H^ i(U, K) \to H^{i + 1}_ Z(X, K) \to \ldots \]

for any $K$ in $D(X)$.

For an abelian sheaf $\mathcal{F}$ on $X$ we can consider the subsheaf of sections with support in $Z$, denoted $\mathcal{H}_ Z(\mathcal{F})$, defined by the rule

\[ \mathcal{H}_ Z(\mathcal{F})(U) = \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset U \cap Z\} \]

Using the equivalence of Modules, Lemma 17.6.1 we may view $\mathcal{H}_ Z(\mathcal{F})$ as an abelian sheaf on $Z$ (see also Modules, Lemmas 17.6.2 and 17.6.3). Thus we obtain a functor

\[ \textit{Ab}(X) \longrightarrow \textit{Ab}(Z),\quad \mathcal{F} \longmapsto \mathcal{H}_ Z(\mathcal{F})\text{ viewed as a sheaf on }Z \]

which is left exact, but in general not exact.

Lemma 20.22.1. Let $i : Z \to X$ be the inclusion of a closed subset. Let $\mathcal{I}$ be an injective abelian sheaf on $X$. Then $\mathcal{H}_ Z(\mathcal{I})$ is an injective abelian sheaf on $Z$.

Proof. Observe that for any abelian sheaf $\mathcal{G}$ on $Z$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _ Z(\mathcal{G}, \mathcal{H}_ Z(\mathcal{F})) = \mathop{\mathrm{Hom}}\nolimits _ X(i_*\mathcal{G}, \mathcal{F}) \]

because after all any section of $i_*\mathcal{G}$ has support in $Z$. Since $i_*$ is exact (Modules, Lemma 17.6.1) and $\mathcal{I}$ injective on $X$ we conclude that $\mathcal{H}_ Z(\mathcal{I})$ is injective on $Z$. $\square$

Denote

\[ R\mathcal{H}_ Z : D(X) \longrightarrow D(Z) \]

the derived functor. We set $\mathcal{H}^ q_ Z(\mathcal{F}) = R^ q\mathcal{H}_ Z(\mathcal{F})$ so that $\mathcal{H}^0_ Z(\mathcal{F}) = \mathcal{H}_ Z(\mathcal{F})$. By the lemma above we have a Grothendieck spectral sequence

\[ E_2^{p, q} = H^ p(Z, \mathcal{H}^ q_ Z(\mathcal{F})) \Rightarrow H^{p + q}_ Z(X, \mathcal{F}) \]

Lemma 20.22.2. Let $i : Z \to X$ be the inclusion of a closed subset. Let $\mathcal{G}$ be an injective abelian sheaf on $Z$. Then $\mathcal{H}^ p_ Z(i_*\mathcal{G}) = 0$ for $p > 0$.

Proof. This is true because the functor $i_*$ is exact and transforms injective abelian sheaves into injective abelian sheaves by Lemma 20.12.11. $\square$

Let $X$ be a topological space and let $Z \subset X$ be a closed subset. We denote $D_ Z(X)$ the strictly full saturated triangulated subcategory of $D(X)$ consisting of complexes whose cohomology sheaves are supported on $Z$.

Lemma 20.22.3. Let $i : Z \to X$ be the inclusion of a closed subset of a topological space $X$. The map $Ri_* = i_* : D(Z) \to D(X)$ induces an equivalence $D(Z) \to D_ Z(X)$ with quasi-inverse

\[ i^{-1}|_{D_ Z(X)} = R\mathcal{H}_ Z|_{D_ Z(X)} \]

Proof. Recall that $i^{-1}$ and $i_*$ is an adjoint pair of exact functors such that $i^{-1}i_*$ is isomorphic to the identify functor on abelian sheaves. See Modules, Lemmas 17.3.3 and 17.6.1. Thus $i_* : D(Z) \to D_ Z(X)$ is fully faithful and $i^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_ Z(X)$ and consider the adjunction map $K \to i_*i^{-1}K$. Using exactness of $i_*$ and $i^{-1}$ this induces the adjunction maps $H^ n(K) \to i_*i^{-1}H^ n(K)$ on cohomology sheaves. Since these cohomology sheaves are supported on $Z$ we see these adjunction maps are isomorphisms and we conclude that $D(Z) \to D_ Z(X)$ is an equivalence.

To finish the proof we have to show that $R\mathcal{H}_ Z(K) = i^{-1}K$ if $K$ is an object of $D_ Z(X)$. To do this we can use that $K = i_*i^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet $ for $i^{-1}K$. Since $i_*$ is the right adjoint to the exact functor $i^{-1}$, the complex $i_*\mathcal{I}^\bullet $ is K-injective (Derived Categories, Lemma 13.29.9). We see that $R\mathcal{H}_ Z(K)$ is computed by $\mathcal{H}_ Z(i_*\mathcal{I}^\bullet ) = \mathcal{I}^\bullet $ as desired. $\square$


Comments (1)

Comment #2123 by Arun Debray on

Minor typo: the title should probably be "Cohomology with support in a closed subset", rather than "Cohomology with support in a closed".


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