Compact-support calculus along open and closed embeddings

Clause (1), degree zero. By [F1], $(j_{!}\mathcal{A})(X)=\{s\in \mathcal{A}(U):\mathrm{Supp}(s)\text{ closed in }X\}$. For such $s$ the support of $s$ read as a global section of $j_{!}\mathcal{A}$ is the same set $\mathrm{Supp}(s)\subseteq U$: at $x\in U$ the stalks of $j_{!}\mathcal{A}$ and $\mathcal{A}$ agree with compatible germ identifications, while for $x\mathrm{\notin }U$ the germ of $s$ vanishes, because the defining local condition supplies an open neighbourhood $W_x$ with $s{\mathrm{\mid }}_{W_x\cap U}=0$, so the restriction of $s$ to $W_x$ is already the zero element of $(j_{!}\mathcal{A})(W_x)$. Hence, using that compactness of a subset of $U$ may be read in $U$ or in $X$ and that compact subsets of $X$ are closed [F6], the subgroups ${\mathrm{\Gamma }}_c(X,j_{!}\mathcal{A})=\{s\in (j_{!}\mathcal{A})(X):\mathrm{Supp}(s)\text{ compact in }X\}$ and $\{s\in \mathcal{A}(U):\mathrm{Supp}(s)\text{ compact in }U\}={\mathrm{\Gamma }}_c(U,\mathcal{A})$ are equal inside $\mathcal{A}(U)$; the canonical map is the inclusion of these equal subgroups, hence a bijection, and it is natural in $\mathcal{A}$ because $j_{!}\phi$ acts as $\phi$ on sections of $\mathcal{A}(U)$.

Clause (2), degree zero and stalks of $i_{\ast }$. By [F2], $(i_{\ast }\mathcal{B})(X)=\mathcal{B}(X\cap Z)=\mathcal{B}(Z)$. The stalks are computed as follows: for $z\in Z$ the sets $V\cap Z$ with $V$ an open neighbourhood of $z$ in $X$ are exactly the open neighbourhoods of $z$ in $Z$ and are cofinal in the colimit defining the stalk, and $(i_{\ast }\mathcal{B})(V)=\mathcal{B}(V\cap Z)$ for each of them, so the two colimit diagrams agree and $(i_{\ast }\mathcal{B}{)}_z\cong {\mathcal{B}}_z$; for $x\mathrm{\notin }Z$ the open set $X\setminus Z=U$ is a neighbourhood of $x$ with $(i_{\ast }\mathcal{B})(W)=\mathcal{B}(W\cap Z)=\mathcal{B}(\mathrm{\varnothing })=0$ for every open $W\subseteq U$, so $(i_{\ast }\mathcal{B}{)}_x=0$. Consequently for $t\in \mathcal{B}(Z)$ the support of $t$ read as a global section of $i_{\ast }\mathcal{B}$ is the set $\mathrm{Supp}(t)\subseteq Z$, and compactness of a subset of $Z$ may be read in $Z$ or in $X$ [F6]; hence ${\mathrm{\Gamma }}_c(X,i_{\ast }\mathcal{B})=\{t\in \mathcal{B}(Z):\mathrm{Supp}(t)\text{ compact in }Z\}={\mathrm{\Gamma }}_c(Z,\mathcal{B})$, the canonical map is the identity, and it is natural in $\mathcal{B}$ because $i_{\ast }\phi$ acts as $\phi$ on sections over $Z$.

Clause (1), c-softness. Let $\mathcal{A}$ be c-soft on $U$, let $K\subseteq X$ be compact and represent a section over $K$ by $s\in (j_{!}\mathcal{A})(V)$ for $V\supseteq K$ open [F10]. Its support is closed in $V$, so $L:=K\cap \mathrm{Supp}(s)$ is compact and contained in $V\cap U$ [F1, F5, F6]. If $L=\mathrm{\varnothing }$, the zero global section has the same germs on $K$. Otherwise choose opens $W_0,W_1$ with $L\subseteq W_0\subseteq \overline{W_0}\subseteq W_1\subseteq \overline{W_1}\subseteq V\cap U$ and compact closures [F6]. The sets $T:=K\cap \overline{W_1}$ and $C:=\overline{W_1}\setminus W_0$ are compact in $U$. The class of $s$ on $T$ and the zero class on $C$ agree on $T\cap C$, since its points lie in $K\setminus L$ and have zero germ of $s$. Fuse these classes near their union in $U$ [F10], and use c-softness of $\mathcal{A}$ to extend the resulting class to $g\in \mathrm{\Gamma }(U,\mathcal{A})$, with $g=s$ near $T$ and $g=0$ near $C$ [F4, F10]. Glue $g{\mathrm{\mid }}_{W_1}$ with zero on $U\setminus \overline{W_0}$, since the overlap lies in $C$. The result $e$ has support closed in $U$ and contained in compact $\overline{W_0}\subseteq U$, hence $e\in {\mathrm{\Gamma }}_c(U,\mathcal{A})={\mathrm{\Gamma }}_c(X,j_{!}\mathcal{A})$ by step 1.1. On $K\cap W_1$ its germs are those of $s$; on $K\setminus \overline{W_0}$ both vanish, because $L\subseteq W_0$. These sets cover $K$, so $e$ restricts to the original section over $K$. Hence $j_{!}\mathcal{A}$ is c-soft.

$i_{\ast }$ is exact. Let $0\to {\mathcal{B}}^{\mathrm{\prime }}\to \mathcal{B}\to {\mathcal{B}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be an exact sequence of sheaves on $Z$ and apply $i_{\ast }$ degreewise; by [F7] it suffices to test exactness of the resulting sequence at every stalk of $X$. At a point $z\in Z$ the stalk sequence of the $i_{\ast }$-images is the given exact stalk sequence under the identifications $(i_{\ast }\mathcal{B}{)}_z\cong {\mathcal{B}}_z$ of [step 1.2], and at a point $x\mathrm{\notin }Z$ all three stalks vanish by [step 1.2], so the sequence $0\to 0\to 0\to 0$ is exact there. Hence $i_{\ast }$ sends exact sequences to exact sequences.

Clause (2), c-softness. Let $\mathcal{B}$ be c-soft on $Z$, let $K\subseteq X$ be compact and let $\sigma$ be a section of $i_{\ast }\mathcal{B}$ over $K$, represented by a section $s\in (i_{\ast }\mathcal{B})(V)=\mathcal{B}(V\cap Z)$ over an open $V\supseteq K$ [F4]. Then $L:=K\cap Z$ is compact in $Z$ [F6] and $V\cap Z$ is an open neighbourhood of $L$ in $Z$, so c-softness of $\mathcal{B}$ provides $t\in \mathrm{\Gamma }(Z,\mathcal{B})$ agreeing with $s$ on an open neighbourhood $N\subseteq V\cap Z$ of $L$ [F4]. Choose open $W\subseteq X$ with $W\cap Z=N$ [F6]; the global section $t\in \mathcal{B}(Z)=(i_{\ast }\mathcal{B})(X)$ agrees with the class $\sigma$ over $K$: at $z\in L$ the open set $W\cap V$ contains $z$ and $t{\mathrm{\mid }}_{W\cap V}=s{\mathrm{\mid }}_{W\cap V}$ as elements of $\mathcal{B}((W\cap V)\cap Z)$ because $(W\cap V)\cap Z\subseteq N$, so the germs agree, and at $x\in K\setminus Z$ the stalk $(i_{\ast }\mathcal{B}{)}_x$ vanishes by [step 1.2], so all germs there are equal. Hence $i_{\ast }\mathcal{B}$ is c-soft.

Clause (1), computing both derived functors by the same complex. Let $0\to \mathcal{A}\to J^0\to J^1\to \cdots$ be the injective resolution of $\mathcal{A}$ supplied by the fixed functorial injective resolution datum on $\mathrm{A}\mathrm{b}(U)$ [F7, F8]; each $J^q$ is injective, hence c-soft on the locally compact Hausdorff space $U$ [F4], hence $j_{!}{J}^q$ is c-soft on $X$ by [step 2.1], and c-soft sheaves on $X$ are acyclic for ${\mathrm{\Gamma }}_c(X,-)$ [F4]. Since $j_{!}$ is exact [F1], the complex $0\to j_{!}\mathcal{A}\to j_{!}{J}^0\to j_{!}{J}^1\to \cdots$ is a ${\mathrm{\Gamma }}_c(X,-)$-acyclic resolution of the sheaf $j_{!}\mathcal{A}$ on $X$, and the acyclic-resolution theorem gives canonical isomorphisms $R^q{\mathrm{\Gamma }}_c(X,j_{!}\mathcal{A})\cong H^q({\mathrm{\Gamma }}_c(X,j_{!}{J}_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))$ for all $q\ge 0$. For the functor $L:={\mathrm{\Gamma }}_c(X,j_{!}(-))$ on $\mathrm{A}\mathrm{b}(U)$, whose derived functors are formed with the same datum on $\mathrm{A}\mathrm{b}(U)$, the resolution $J^{\bullet }$ is likewise $L$-acyclic, and the derived functors $R^{q}L(\mathcal{A})$ are computed by the very same complex $L(J^{\bullet })={\mathrm{\Gamma }}_c(X,j_{!}{J}^{\bullet })$; since both isomorphisms are the canonical isomorphisms of the acyclic-resolution theorem for the same resolution complex, they are compatible with comparison extensions of morphisms $\mathcal{A}\to {\mathcal{A}}^{\mathrm{\prime }}$ [F7, F8].

Clause (2), the derived isomorphisms. Let $0\to \mathcal{B}\to J^0\to J^1\to \cdots$ be the injective resolution of $\mathcal{B}$ supplied by the fixed datum on $\mathrm{A}\mathrm{b}(Z)$ [F7, F8]; each $J^q$ is injective, hence c-soft on $Z$, hence $i_{\ast }{J}^q$ is c-soft on $X$ by [step 2.3] and acyclic for ${\mathrm{\Gamma }}_c(X,-)$ [F4]; since $i_{\ast }$ is exact by [step 2.2], the complex $0\to i_{\ast }\mathcal{B}\to i_{\ast }{J}^0\to i_{\ast }{J}^1\to \cdots$ is a ${\mathrm{\Gamma }}_c(X,-)$-acyclic resolution of $i_{\ast }\mathcal{B}$, and the acyclic-resolution theorem combined with the degreewise identification ${\mathrm{\Gamma }}_c(X,i_{\ast }{J}^q)\cong {\mathrm{\Gamma }}_c(Z,J^q)$ of [step 1.2] gives canonical isomorphisms $R^q{\mathrm{\Gamma }}_c(X,i_{\ast }\mathcal{B})\cong H^q({\mathrm{\Gamma }}_c(X,i_{\ast }{J}_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))\cong H^q({\mathrm{\Gamma }}_c(Z,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))\cong R^q{\mathrm{\Gamma }}_c(Z,\mathcal{B})$ for all $q\ge 0$, where the last isomorphism is the acyclic-resolution comparison on $Z$ applied to the same resolution, whose terms are acyclic for ${\mathrm{\Gamma }}_c(Z,-)$ because they are c-soft on $Z$ [F4]; naturality in $\mathcal{B}$ holds because the outer isomorphisms are the canonical ones for the fixed resolution data and the middle one is a degreewise natural identification.

Clause (1), the natural isomorphisms. Put $G:={\mathrm{\Gamma }}_c(U,-)$ on $\mathrm{A}\mathrm{b}(U)$. Step 1.1 gives a natural isomorphism $\zeta :L\Rightarrow G$ of additive left exact functors. Applied degreewise to the supplied resolution $J^{\bullet }(\mathcal{A})$, it gives an isomorphism of complexes ${\mathrm{\Gamma }}_c(X,j_{!}{J}^{\bullet })\cong {\mathrm{\Gamma }}_c(U,J^{\bullet })$. Taking cohomology gives natural isomorphisms $R^{q}L(\mathcal{A})\cong R^{q}G(\mathcal{A})$ by [F8]. Composing with the canonical comparison of step 3.1 yields $R^q{\mathrm{\Gamma }}_c(X,j_{!}\mathcal{A})\cong R^q{\mathrm{\Gamma }}_c(U,\mathcal{A})$ in every degree, naturally in $\mathcal{A}$.

Clause (3). Let $Z$ be compact. For every sheaf $\mathcal{B}$ on $Z$ the support of a global section is closed in $Z$ [F5], hence a closed subset of the compact space $Z$, hence compact, so every global section is compactly supported: ${\mathrm{\Gamma }}_c(Z,\mathcal{B})=\mathrm{\Gamma }(Z,\mathcal{B})$ and $H_c^q(Z,\mathcal{B})=H^q(Z,\mathcal{B})$ for all $q$ [F3]. Substituting into the isomorphisms of [step 1.2] for $q=0$ and of [step 3.2] for $q\ge 1$ gives ${\mathrm{\Gamma }}_c(X,i_{\ast }\mathcal{B})\cong \mathrm{\Gamma }(Z,\mathcal{B})$ and $R^q{\mathrm{\Gamma }}_c(X,i_{\ast }\mathcal{B})\cong H^q(Z,\mathcal{B})$ for all $q\ge 0$.

Conclusion. Clause (1) is [step 1.1], [step 2.1] and [step 4.1]; clause (2) is [step 1.2], [step 2.3] and [step 3.2]; clause (3) is [step 4.2]. The Axiom of Choice is used only through the injective resolution data of [F7] and the comparison theorems of [F8], where dependent choice is required and follows from AC [F9].