Lemma 69.9.1. 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 $Z_{\acute{e}tale}$.
69.9 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 66.20.3). 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 66.20.3. Using the equivalence of Morphisms of Spaces, Lemma 67.13.5 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.
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 69.4.1) 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 69.9.2. 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 69.4.1) and transforms injective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma 21.14.2). $\square$
Lemma 69.9.3. 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 as abelian sheaves on $Z = f^{-1}(Z)$ and we have $H^ q_ Z(Y, \mathcal{F}) = H^ q_{f^{-1}(Z)}(X, f^{-1}\mathcal{F})$.
\[ \mathcal{H}^ q_ Z(\mathcal{F}) = \mathcal{H}^ q_{f^{-1}(Z)}(f^{-1}\mathcal{F}) \]
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 69.9.4. 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 66.19.9 and Morphisms of Spaces, Lemma 67.13.5. 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. $\square$
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