68.9 Cohomology with support in a closed subspace

This section is the analogue of Cohomology, Sections 20.21 and 20.34 and Étale Cohomology, Section 59.79 for abelian sheaves on algebraic spaces.

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $Z \subset X$ be a closed subspace. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. We let

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

be the sections with support in $Z$ (Properties of Spaces, Definition 65.20.3). This is a left exact functor which is not exact in general. Hence we obtain a derived functor

$R\Gamma _ Z(X, -) : D(X_{\acute{e}tale}) \longrightarrow D(\textit{Ab})$

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 $X_{\acute{e}tale}$. Let $U \subset X$ be the open subspace which is the complement of $Z$. Then the restriction map $\mathcal{I}(X) \to \mathcal{I}(U)$ is surjective (Cohomology on Sites, Lemma 21.12.6) with kernel $\Gamma _ Z(X, \mathcal{I})$. It immediately follows that for $K \in D(X_{\acute{e}tale})$ 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 $D(X_{\acute{e}tale})$.

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

$\mathcal{H}_ Z(\mathcal{F})(U) = \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset U \times _ X Z\}$

Here we use the support of a section from Properties of Spaces, Definition 65.20.3. Using the equivalence of Morphisms of Spaces, Lemma 66.13.5 we may view $\mathcal{H}_ Z(\mathcal{F})$ as an abelian sheaf on $Z_{\acute{e}tale}$. Thus we obtain a functor

$\textit{Ab}(X_{\acute{e}tale}) \longrightarrow \textit{Ab}(Z_{\acute{e}tale}),\quad \mathcal{F} \longmapsto \mathcal{H}_ Z(\mathcal{F})$

which is left exact, but in general not exact.

Lemma 68.9.1. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{I}$ be an injective abelian sheaf on $X_{\acute{e}tale}$. Then $\mathcal{H}_ Z(\mathcal{I})$ is an injective abelian sheaf on $Z_{\acute{e}tale}$.

Proof. Observe that for any abelian sheaf $\mathcal{G}$ on $Z_{\acute{e}tale}$ 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 (Lemma 68.4.1) and as $\mathcal{I}$ is injective on $X_{\acute{e}tale}$ we conclude that $\mathcal{H}_ Z(\mathcal{I})$ is injective on $Z_{\acute{e}tale}$. $\square$

Denote

$R\mathcal{H}_ Z : D(X_{\acute{e}tale}) \longrightarrow D(Z_{\acute{e}tale})$

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 68.9.2. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{G}$ be an injective abelian sheaf on $Z_{\acute{e}tale}$. Then $\mathcal{H}^ p_ Z(i_*\mathcal{G}) = 0$ for $p > 0$.

Proof. This is true because the functor $i_*$ is exact (Lemma 68.4.1) and transforms injective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma 21.14.2). $\square$

Lemma 68.9.3. Let $S$ be a scheme. Let $f : X \to Y$ be an étale morphism of algebraic spaces over $S$. Let $Z \subset Y$ be a closed subspace such that $f^{-1}(Z) \to Z$ is an isomorphism of algebraic spaces. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then

$\mathcal{H}^ q_ Z(\mathcal{F}) = \mathcal{H}^ q_{f^{-1}(Z)}(f^{-1}\mathcal{F})$

as abelian sheaves on $Z = f^{-1}(Z)$ and we have $H^ q_ Z(Y, \mathcal{F}) = H^ q_{f^{-1}(Z)}(X, f^{-1}\mathcal{F})$.

Proof. Because $f$ is étale an injective resolution of $\mathcal{F}$ pulls back to an injective resolution of $f^{-1}\mathcal{F}$. Hence it suffices to check the equality for $\mathcal{H}_ Z(-)$ which follows from the definitions. The proof for cohomology with supports is the same. Some details omitted. $\square$

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. We denote $D_ T(X_{\acute{e}tale})$ the strictly full saturated triangulated subcategory of $D(X_{\acute{e}tale})$ consisting of objects whose cohomology sheaves are supported on $T$.

Lemma 68.9.4. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. The map $Ri_* = i_* : D(Z_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ induces an equivalence $D(Z_{\acute{e}tale}) \to D_{|Z|}(X_{\acute{e}tale})$ with quasi-inverse

$i^{-1}|_{D_ Z(X_{\acute{e}tale})} = R\mathcal{H}_ Z|_{D_{|Z|}(X_{\acute{e}tale})}$

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 Properties of Spaces, Lemma 65.19.9 and Morphisms of Spaces, Lemma 66.13.5. Thus $i_* : D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is fully faithful and $i^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_ Z(X_{\acute{e}tale})$ 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 $D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ 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 $D_ Z(X_{\acute{e}tale})$. 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 13.31.9). We see that $R\mathcal{H}_ Z(K)$ is computed by $\mathcal{H}_ Z(i_*\mathcal{I}^\bullet ) = \mathcal{I}^\bullet$ as desired. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A4L. Beware of the difference between the letter 'O' and the digit '0'.