Projection formula for a closed immersion and an invertible sheaf

The canonical map. Pullback of modules is left adjoint to pushforward (Pullback of modules is left adjoint to pushforward), so the identity of the ${\mathcal{O}}_Z$-module $i^{\ast }\mathcal{L}$ corresponds to a canonical ${\mathcal{O}}_X$-linear map $\lambda :\mathcal{L}\to i_{\ast }{i}^{\ast }\mathcal{L}$. There is also the canonical map $i^{-1}\mathcal{L}\to i^{\ast }\mathcal{L}={\mathcal{O}}_Z{\otimes }_{i^{-1}{\mathcal{O}}_X}{i}^{-1}\mathcal{L}$, $s\mapsto 1\otimes s$ (Pullback of a module along a morphism of ringed spaces). Define, for every open $U\subseteq X$, the ${\mathcal{O}}_X(U)$-bilinear map $\mathcal{L}(U)\times (i_{\ast }\mathcal{G})(U)⟶(i^{\ast }\mathcal{L}{\otimes }_{{\mathcal{O}}_Z}\mathcal{G})(i^{-1}U),(\mathrm{\ell },s)⟼\lambda (\mathrm{\ell }){\mathrm{\mid }}_{i^{-1}U}\otimes s\mathrm{.}$ These maps are compatible with the restriction maps, so they assemble into a morphism from the tensor presheaf of Tensor product of sheaves of modules to the sheaf $i_{\ast }(i^{\ast }\mathcal{L}\otimes \mathcal{G})$, and hence, by the universal property of sheafification, into a morphism of ${\mathcal{O}}_X$-modules $\mathrm{\Theta }:\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{i}_{\ast }\mathcal{G}⟶i_{\ast }\left(i^{\ast }\mathcal{L}{\otimes }_{{\mathcal{O}}_Z}\mathcal{G}\right)\mathrm{.}$

Stalks off $Z$. Let $x\in X\setminus Z$. Since $Z$ is closed, $X\setminus Z$ is an open neighbourhood of $x$ with $i^{-1}(X\setminus Z)=\mathrm{\varnothing }$, so in the colimit of [F2] the groups $\mathcal{G}(i^{-1}U)$ vanish for all open $U\subseteq X\setminus Z$; as these $U$ are cofinal among the neighbourhoods of $x$, the stalk $(i_{\ast }\mathcal{G}{)}_x$ is $0$. Hence $(\mathcal{L}\otimes i_{\ast }\mathcal{G}{)}_x\cong {\mathcal{L}}_x\otimes (i_{\ast }\mathcal{G}{)}_x=0$ by [F3]. Applying the same argument to the quasi-coherent module $i^{\ast }\mathcal{L}\otimes \mathcal{G}$ in place of $\mathcal{G}$ gives $(i_{\ast }(i^{\ast }\mathcal{L}\otimes \mathcal{G}){)}_x=0$. Thus ${\mathrm{\Theta }}_x$ is a map $0\to 0$, hence bijective.

Stalks on $Z$. Let $z\in Z$. Every open neighbourhood of $z$ in $Z$ has the form $U\cap Z$ with $U$ an open neighbourhood of $z$ in $X$ (Closed immersions of schemes), and these are cofinal in the neighbourhood system of $z$ in $Z$ by [F1]; comparing the colimit description [F2] of the stalk of $i_{\ast }\mathcal{G}$ with the defining colimit of the stalk of $\mathcal{G}$ (The stalk of a presheaf at a point) gives a canonical isomorphism $(i_{\ast }\mathcal{G}{)}_z\cong {\mathcal{G}}_z$, compatible with the ${\mathcal{O}}_{X,z}$-module structure because the action on ${\mathcal{G}}_z$ factors through ${\mathcal{O}}_{X,z}\to {\mathcal{O}}_{Z,z}$. Similarly $(i_{\ast }(i^{\ast }\mathcal{L}\otimes \mathcal{G}){)}_z\cong (i^{\ast }\mathcal{L}\otimes \mathcal{G}{)}_z$.

Put $R={\mathcal{O}}_{X,z}$ and $S={\mathcal{O}}_{Z,z}$. By [F3] the source stalk is ${\mathcal{L}}_z{\otimes }_R{\mathcal{G}}_z$ and the target is $(S{\otimes }_R{\mathcal{L}}_z){\otimes }_S{\mathcal{G}}_z$. The map sends $\mathrm{\ell }\otimes g$ to $(1\otimes \mathrm{\ell })\otimes g$. Its inverse sends $(s\otimes \mathrm{\ell })\otimes g$ to $\mathrm{\ell }\otimes sg$: the $R$-balancing relation in $S{\otimes }_R{\mathcal{L}}_z$ and the $S$-balancing relation of the outer tensor both give the same element, so this formula is well defined. The composites are identities, since $(s\otimes \mathrm{\ell })\otimes g=(1\otimes \mathrm{\ell })\otimes sg$. Thus ${\mathrm{\Theta }}_z$ is an isomorphism.

Conclusion of the isomorphism. By step 1.2 the stalk ${\mathrm{\Theta }}_x$ is bijective for every $x\in X\setminus Z$, and by step 2.1 it is bijective for every $z\in Z$; hence $\mathrm{\Theta }$ is an isomorphism of ${\mathcal{O}}_X$-modules by [F5]. This proves the first clause.

Coherence clause. Assume now that $X$ is locally Noetherian and that $\mathcal{G}$ is coherent. Then $i_{\ast }\mathcal{G}$ is a coherent ${\mathcal{O}}_X$-module by [F7]. The invertible module $\mathcal{L}$ is locally free of rank one (Invertible sheaves), so $X$ is covered by open subschemes $U$ with $\mathcal{L}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U$; over such $U$ the unit isomorphism of [F4] and the stalk computations of [F3] give $(\mathcal{L}\otimes i_{\ast }\mathcal{G}){\mathrm{\mid }}_U\cong (i_{\ast }\mathcal{G}){\mathrm{\mid }}_U$. Since coherence is local on $X$ and $i_{\ast }\mathcal{G}$ is coherent ([F6], [F7]), the sheaf $\mathcal{L}\otimes i_{\ast }\mathcal{G}$ is coherent; its isomorphic image $i_{\ast }(i^{\ast }\mathcal{L}\otimes \mathcal{G})$ under the isomorphism of step 3.1 is coherent as well.

Cohomology and Euler characteristic. With $X$ locally Noetherian and $\mathcal{G}$ coherent, the isomorphism of step 3.1 identifies $H^q(X,\mathcal{L}\otimes i_{\ast }\mathcal{G})$ with $H^q(X,i_{\ast }(i^{\ast }\mathcal{L}\otimes \mathcal{G}))$ for every $q\ge 0$; the quasi-coherent module $i^{\ast }\mathcal{L}\otimes \mathcal{G}$ satisfies $H^q(X,i_{\ast }(i^{\ast }\mathcal{L}\otimes \mathcal{G}))\cong H^q(Z,i^{\ast }\mathcal{L}\otimes \mathcal{G})$ by [F7] and [F6]. Hence $H^q(X,\mathcal{L}\otimes i_{\ast }\mathcal{G})\cong H^q(Z,i^{\ast }\mathcal{L}\otimes \mathcal{G})$ for every $q\ge 0$. When $X$ is proper over a field, the left side is an alternating sum of finite-dimensional vector spaces (Euler characteristic of a coherent sheaf, step 4.1), so the termwise isomorphic right side gives $\chi (X,\mathcal{L}\otimes i_{\ast }\mathcal{G})=\chi (Z,i^{\ast }\mathcal{L}\otimes \mathcal{G})$. The Axiom of Choice enters only through the suppliers named in [F7]; steps 1.1--3.1 make no selection.