Lemma 59.79.1. Let $i : Z \to X$ be a closed immersion of schemes. 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 $Z_{\acute{e}tale}$.
59.79 Cohomology with support in a closed subscheme
Let $X$ be a scheme and let $Z \subset X$ be a closed subscheme. 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 $Z$ (Definition 59.31.3). 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 $U = X \setminus Z$. 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 Definition 59.31.3. Using the equivalence of Proposition 59.46.4 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 (Section 59.46) 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 59.79.2. Let $i : Z \to X$ be a closed immersion of schemes. 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 and transforms injective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma 21.14.2). $\square$
Lemma 59.79.3. Let $i : Z \to X$ be a closed immersion of schemes. Let $j : U \to X$ be the inclusion of the complement of $Z$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. There is a distinguished triangle
\[ i_*R\mathcal{H}_ Z(\mathcal{F}) \to \mathcal{F} \to Rj_*(\mathcal{F}|_ U) \to i_*R\mathcal{H}_ Z(\mathcal{F})[1] \]
in $D(X_{\acute{e}tale})$. This produces an exact sequence
\[ 0 \to i_*\mathcal{H}_ Z(\mathcal{F}) \to \mathcal{F} \to j_*(\mathcal{F}|_ U) \to i_*\mathcal{H}^1_ Z(\mathcal{F}) \to 0 \]
and isomorphisms $R^ pj_*(\mathcal{F}|_ U) \cong i_*\mathcal{H}^{p + 1}_ Z(\mathcal{F})$ for $p \geq 1$.
Proof. To get the distinguished triangle, choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Then we obtain a short exact sequence of complexes
\[ 0 \to i_*\mathcal{H}_ Z(\mathcal{I}^\bullet ) \to \mathcal{I}^\bullet \to j_*(\mathcal{I}^\bullet |_ U) \to 0 \]
by the discussion above. Thus the distinguished triangle by Derived Categories, Section 13.12. $\square$
Let $X$ be a scheme and let $Z \subset X$ be a closed subscheme. We denote $D_ Z(X_{\acute{e}tale})$ the strictly full saturated triangulated subcategory of $D(X_{\acute{e}tale})$ consisting of complexes whose cohomology sheaves are supported on $Z$. Note that $D_ Z(X_{\acute{e}tale})$ only depends on the underlying closed subset of $X$.
Lemma 59.79.4. Let $i : Z \to X$ be a closed immersion of schemes. The map $Ri_{small, *} = i_{small, *} : D(Z_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ induces an equivalence $D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ with quasi-inverse
\[ i_{small}^{-1}|_{D_ Z(X_{\acute{e}tale})} = R\mathcal{H}_ Z|_{D_ Z(X_{\acute{e}tale})} \]
Proof. Recall that $i_{small}^{-1}$ and $i_{small, *}$ is an adjoint pair of exact functors such that $i_{small}^{-1}i_{small, *}$ is isomorphic to the identify functor on abelian sheaves. See Proposition 59.46.4 and Lemma 59.36.2. Thus $i_{small, *} : D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is fully faithful and $i_{small}^{-1}$ 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 $K \to i_{small, *}i_{small}^{-1}K$. Using exactness of $i_{small, *}$ and $i_{small}^{-1}$ this induces the adjunction maps $H^ n(K) \to i_{small, *}i_{small}^{-1}H^ n(K)$ 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 $R\mathcal{H}_ Z(K) = i_{small}^{-1}K$ if $K$ is an object of $D_ Z(X_{\acute{e}tale})$. To do this we can use that $K = i_{small, *}i_{small}^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet $ for $i_{small}^{-1}K$. Since $i_{small, *}$ is the right adjoint to the exact functor $i_{small}^{-1}$, the complex $i_{small, *}\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_{small, *}\mathcal{I}^\bullet ) = \mathcal{I}^\bullet $ as desired. $\square$
Lemma 59.79.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module and denote $\mathcal{F}^ a$ the associated quasi-coherent sheaf on the small étale site of $X$ (Proposition 59.17.1). Then
$H^ q_ Z(X, \mathcal{F})$ agrees with $H^ q_ Z(X_{\acute{e}tale}, \mathcal{F}^ a)$,
if the complement of $Z$ is retrocompact in $X$, then $i_*\mathcal{H}^ q_ Z(\mathcal{F}^ a)$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules equal to $(i_*\mathcal{H}^ q_ Z(\mathcal{F}))^ a$.
Proof. Let $j : U \to X$ be the inclusion of the complement of $Z$. The statement (1) on cohomology groups follows from the long exact sequences for cohomology with supports and the agreements $H^ q(X_{\acute{e}tale}, \mathcal{F}^ a) = H^ q(X, \mathcal{F})$ and $H^ q(U_{\acute{e}tale}, \mathcal{F}^ a) = H^ q(U, \mathcal{F})$, see Theorem 59.22.4. If $j : U \to X$ is a quasi-compact morphism, i.e., if $U \subset X$ is retrocompact, then $R^ qj_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Cohomology of Schemes, Lemma 30.4.5) and commutes with taking associated sheaf on étale sites (Descent, Lemma 35.9.5). We conclude by applying Lemma 59.79.3. $\square$
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