Pushforward along a closed immersion preserves sheaf cohomology

By [F1] and [F4], the stalk of $i_{\ast }\mathcal{F}$ at $x\in X$ is the filtered colimit of the groups $\mathcal{F}(U\cap Z)$ over the open neighbourhoods $U$ of $x$ in $X$. Suppose first that $x\in Z$. For an open $W\subseteq Z$ with $x\in W$ the set $W\cup (X\setminus Z)$ is open in $X$, because $X\setminus Z$ is open and $W$ is a trace of an open set by [F3], it contains $x$, and $(W\cup (X\setminus Z))\cap Z=W$; conversely $U\cap Z$ is such an open $W$ for every open $U\ni x$. The assignment $U\mapsto U\cap Z$ therefore carries the neighbourhood system of $x$ in $X$ cofinally onto the neighbourhood system of $x$ in $Z$, and the two filtered colimits agree: $(i_{\ast }\mathcal{F}{)}_x\cong {\mathcal{F}}_x$. Suppose now that $x\mathrm{\notin }Z$. Then $X\setminus Z$ is open and contains $x$, and for every open $U\ni x$ the set $U\cap (X\setminus Z)\subseteq U$ is an open neighbourhood of $x$ with $(U\cap (X\setminus Z))\cap Z=\mathrm{\varnothing }$; the colimit is thus computed by the constant subdiagram with value $\mathcal{F}(\mathrm{\varnothing })$, and $\mathcal{F}(\mathrm{\varnothing })=0$ is the zero group by [F5]. Hence $(i_{\ast }\mathcal{F}{)}_x=0$ for $x\mathrm{\notin }Z$, and $i$ is continuous by [F3] so that $i_{\ast }\mathcal{F}$ is an abelian sheaf on $X$ by [F2].

Let $\mathcal{I}$ be an injective object of $\mathrm{A}\mathrm{b}(Z)$. We show that $i_{\ast }\mathcal{I}$ is injective in $\mathrm{A}\mathrm{b}(X)$. Let $m:\mathcal{A}\rightarrowtail \mathcal{B}$ be a monomorphism in $\mathrm{A}\mathrm{b}(X)$. By [F6], $m$ being a monomorphism means that every stalk map $m_x$ has zero kernel; by [F8] the stalk of $i^{-1}m:i^{-1}\mathcal{A}\to i^{-1}\mathcal{B}$ at $x$ is the map $m_{i(x)}$, which again has zero kernel, so $i^{-1}m$ is a monomorphism by [F6]. Since $\mathcal{I}$ is injective [F9], the map ${\mathrm{Hom}}_Z(i^{-1}\mathcal{B},\mathcal{I})\to {\mathrm{Hom}}_Z(i^{-1}\mathcal{A},\mathcal{I})$ given by precomposition with $i^{-1}m$ is surjective. The adjunction [F7] provides natural bijections ${\mathrm{Hom}}_X(\mathcal{A},i_{\ast }\mathcal{I})\cong {\mathrm{Hom}}_Z(i^{-1}\mathcal{A},\mathcal{I})$ and ${\mathrm{Hom}}_X(\mathcal{B},i_{\ast }\mathcal{I})\cong {\mathrm{Hom}}_Z(i^{-1}\mathcal{B},\mathcal{I})$ under which precomposition by $m$ corresponds to precomposition by $i^{-1}m$; hence ${\mathrm{Hom}}_X(\mathcal{B},i_{\ast }\mathcal{I})\to {\mathrm{Hom}}_X(\mathcal{A},i_{\ast }\mathcal{I})$ is surjective, which is exactly the extension property of [F9] for $i_{\ast }\mathcal{I}$.

We show that $i_{\ast }$ carries short exact sequences of abelian sheaves on $Z$ to short exact sequences on $X$. Let $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be exact in $\mathrm{A}\mathrm{b}(Z)$. Its stalk sequence $0\to {\mathcal{F}}_z^{\mathrm{\prime }}\to {\mathcal{F}}_z\to {\mathcal{F}}_z^{\mathrm{\prime }\mathrm{\prime }}\to 0$ is exact at every $z\in Z$, by the only-if direction of [F6] applied on $Z$; for $x\in Z$ the stalk sequence of $0\to i_{\ast }{\mathcal{F}}^{\mathrm{\prime }}\to i_{\ast }\mathcal{F}\to i_{\ast }{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ is that same sequence by [step 1.1], and for $x\mathrm{\notin }Z$ it is the sequence $0\to 0\to 0\to 0$ of zero groups, again exact. By the if direction of [F6] applied on $X$, the sequence $0\to i_{\ast }{\mathcal{F}}^{\mathrm{\prime }}\to i_{\ast }\mathcal{F}\to i_{\ast }{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ is exact.

By [F10] there is an injective resolution $0\to \mathcal{F}\to I_Z^0(\mathcal{F})\to I_Z^1(\mathcal{F})\to \cdots$ of $\mathcal{F}$ in $\mathrm{A}\mathrm{b}(Z)$, exact at every displayed term, with each $I_Z^n(\mathcal{F})$ injective. Put $J^n:=i_{\ast }{I}_Z^n(\mathcal{F})$, an abelian sheaf on $X$ by [step 1.1]. For every $x\in X$ the stalk complex $0\to (i_{\ast }\mathcal{F}{)}_x\to J_x^0\to J_x^1\to \cdots$ is exact at every term: for $x\in Z$ it is, term by term, the stalk complex of the given resolution by [step 1.1], and for $x\mathrm{\notin }Z$ all its terms are $0$ by [step 1.1]. Extending the complex by zero objects on the left and applying the if direction of [F6] on $X$, the coaugmented complex $0\to i_{\ast }\mathcal{F}\to J^0\to J^1\to \cdots$ is exact at every displayed term, and each $J^n$ is injective by [step 1.2]; hence it is an injective resolution of $i_{\ast }\mathcal{F}$ on $X$ in the sense of [F10].

Let $0\to i_{\ast }\mathcal{F}\to I^0(i_{\ast }\mathcal{F})\to I^1(i_{\ast }\mathcal{F})\to \cdots$ be the injective resolution of $i_{\ast }\mathcal{F}$ supplied by the datum $I^{\bullet }$ of [F10] on $\mathrm{A}\mathrm{b}(X)$, and let $J^{\bullet }$ be the resolution of [step 2.2]. Both complexes are exact at every displayed term and have injective terms, so [F12], whose hypothesis DC holds by [F16], applies with $u$ the identity of $i_{\ast }\mathcal{F}$ in both directions: there are coaugmentation-preserving cochain maps $\psi :I^{\bullet }(i_{\ast }\mathcal{F})\to J^{\bullet }$ and $\chi :J^{\bullet }\to I^{\bullet }(i_{\ast }\mathcal{F})$, and the composites $\psi \chi$ and $\chi \psi$ are cochain-homotopic to the respective identities. The global-sections functor is additive by [F15] and an additive functor carries the homotopy identities of [F14] to homotopy identities between the induced cochain maps of complexes of abelian groups; by [F13] homotopic cochain maps induce the same map on cohomology and cohomology respects identities and composition, so $\mathrm{\Gamma }(X,\psi )$ and $\mathrm{\Gamma }(X,\chi )$ induce mutually inverse isomorphisms $H^q(\mathrm{\Gamma }(X,I^{\bullet }(i_{\ast }\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}))\cong H^q(\mathrm{\Gamma }(X,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))$ for every $q$. By the definition of cohomology [F11] the left-hand group is $H^q(X,i_{\ast }\mathcal{F})$, so $H^q(X,i_{\ast }\mathcal{F})\cong H^q(\mathrm{\Gamma }(X,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))\mathrm{.}$

For every $n\ge 0$ the group of global sections of $J^n=i_{\ast }{I}_Z^n(\mathcal{F})$ over $X$ is $J^n(X)=I_Z^n(\mathcal{F})(i^{-1}(X))=I_Z^n(\mathcal{F})(Z)$ by [F1], and the differentials of the two complexes correspond under these identifications because the differential of $J^{\bullet }$ is the direct image of the differential of $I_Z^{\bullet }(\mathcal{F})$ and direct image is functorial in the evident way [F1]. Hence the complexes of abelian groups $\mathrm{\Gamma }(X,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet })$ and $\mathrm{\Gamma }(Z,I_Z^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}})$ have the same terms and the same differentials, so they have isomorphic cohomology: $H^q(\mathrm{\Gamma }(X,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))=H^q(\mathrm{\Gamma }(Z,I_Z^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}))\mathrm{.}$

Combining the steps, for every $q\ge 0$: $H^q(X,i_{\ast }\mathcal{F})\cong H^q(\mathrm{\Gamma }(X,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))=H^q(\mathrm{\Gamma }(Z,I_Z^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}))=H^q(Z,\mathcal{F}),$ the first isomorphism by [step 3.1], the equality by [step 3.2], and the last equality by the definition of the cohomology of $\mathcal{F}$ on $Z$ from the supplied datum $I_Z^{\bullet }$ [F11]. The Axiom of Choice is used in [F10] to supply the two injective resolution data, and through [F16] to provide the Dependent Choice required for the comparison maps of [F12]; no further selection is made, since the two resolutions are the fixed supplied data and the maps $\psi ,\chi$ of [step 3.1] are obtained from [F12] at the single pair of complexes. ∎