Closed immersion preserves cohomology and coherent pushforward

The closed immersion $i$ is an affine morphism: by [F1] the inverse image of every affine open subscheme of $X$ is affine.

Apply [F2] to $f=i:Z\to X$ and $\mathcal{G}=\mathcal{F}$: the edge map $H^q(X,i_{\ast }\mathcal{F})\to H^q(Z,\mathcal{F})$ is an isomorphism for every $q\ge 0$, which is the asserted canonical isomorphism (in degree $0$ it is the identity of $\mathrm{\Gamma }(Z,\mathcal{F})$ under the canonical identification of $\mathrm{\Gamma }(X,i_{\ast }\mathcal{F})$ with $\mathrm{\Gamma }(Z,\mathcal{F})$).

Now assume $X$ locally Noetherian and $\mathcal{F}$ coherent. Coherence of $i_{\ast }\mathcal{F}$ is local on $X$, so fix an affine open $U=\mathrm{Spec}A\subseteq X$; then $A$ is Noetherian, and by [F1] there is an ideal $I\subseteq A$ with $i^{-1}(U)=\mathrm{Spec}B$, $B=A\mathrm{/}I$. By [F3] the coherent module $\mathcal{F}{\mathrm{\mid }}_{i^{-1}(U)}$ is the associated sheaf $\tilde{M}$ of a finitely generated $B$-module $M=\mathrm{\Gamma }(i^{-1}(U),\mathcal{F})$, and $B$, being a quotient of the Noetherian ring $A$, is Noetherian.

The restriction $(i_{\ast }\mathcal{F}){\mathrm{\mid }}_U$ is the associated sheaf $\tilde{M}$ of $M$ viewed as an $A$-module through $A\twoheadrightarrow B$: indeed for $f\in A$ with image $\bar{f}\in B$ one has $\mathrm{\Gamma }(D(f),(i_{\ast }\mathcal{F}){\mathrm{\mid }}_U)=\mathcal{F}(i^{-1}D(f))=\mathrm{\Gamma }(D(\bar{f}),\tilde{M})=M_{\bar{f}}$ by [F1] and [F3], and these isomorphisms are compatible with the restriction maps, which on both sides are the canonical localisations; sheaves are determined by their sections on the basis of principal opens.

The $A$-module $M$ is finitely generated, because it is finitely generated over the quotient ring $B$ by 1.3; hence $\tilde{M}$ is a coherent ${\mathcal{O}}_U$-module by [F3], since $A$ is Noetherian. As $U$ was an arbitrary affine open of the locally Noetherian scheme $X$, coherence of $i_{\ast }\mathcal{F}$ follows.

Boundary and choice accounting. If $Z=\mathrm{\varnothing }$ then $\mathcal{F}=0$, $i_{\ast }\mathcal{F}=0$ and both sides of the isomorphism are the zero group in every degree; if $X=\mathrm{\varnothing }$ then also $Z=\mathrm{\varnothing }$; if $\mathcal{F}=0$ the isomorphism is $0\cong 0$. Affine $X$ is the case in which the local verification of 1.3-1.5 is already global. The Axiom of Choice is a hypothesis, consumed through the affine edge-map theorem [F2] and the affine equivalence [F3]; the localisation identifications of 1.4 make no further choice.