Scheme pullback preserves quasi-coherence

Reduction to the affine case: let $U=\mathrm{Spec}B\subseteq X$, $V=\mathrm{Spec}A\subseteq Y$ be affine opens with $f(U)\subseteq V$ and put $g=f{\mathrm{\mid }}_U:U\to V$. By [F1] the restriction of the pullback is $(f^{\ast }\mathcal{F}){\mathrm{\mid }}_U\cong g^{\ast }(\mathcal{F}{\mathrm{\mid }}_V)$, and by [F6] the morphism $g$ is the one induced by the ring map $\phi :A\to B$; fixing the isomorphism $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{M}$, it suffices to construct a canonical isomorphism $g^{\ast }(\tilde{M})\cong \widetilde{(B{\otimes }_{A}M)}$ of ${\mathcal{O}}_U$-modules, since then $f^{\ast }\mathcal{F}{\mathrm{\mid }}_U\cong \widetilde{(B{\otimes }_{A}M)}$ as well.

The comparison morphism: for $b\in B$ let $[m]\in g^{-1}(\tilde{M})(D(b))$ denote the class of $m\in M=\tilde{M}(V)$ under the canonical map from the neighbourhood colimit to its sheafification on the distinguished open $D(b)\subseteq U$, using that $g(D(b))\subseteq V$. Using the universal property of the module tensor product, define a $B_b$-linear map ${\delta }_b:\widetilde{(B{\otimes }_{A}M)}(D(b))=(B{\otimes }_{A}M{)}_b\cong B_b{\otimes }_{A}M⟶g^{\ast }(\tilde{M})(D(b)),\frac{b^{\mathrm{\prime }}}{b^k}\otimes m⟼\frac{b^{\mathrm{\prime }}}{b^k}\cdot [m],$ with the $B_b$-module structure on the target coming from the ring map $g^{-1}{\mathcal{O}}_V(D(b))\to {\mathcal{O}}_U(D(b))=B_b$ that makes $g^{\ast }(\tilde{M})$ an ${\mathcal{O}}_U$-module. For $D(b^{\mathrm{\prime }})\subseteq D(b)$ the square of restriction maps commutes: both composites send $(b^{\mathrm{\prime }\mathrm{\prime }}\otimes m)\mathrm{/}1$ to the restriction of $b^{\mathrm{\prime }\mathrm{\prime }}\cdot [m]$, and the class construction is compatible with restriction. By [F5] the compatible maps ${\delta }_b$ determine a unique morphism $\delta :\widetilde{(B{\otimes }_{A}M)}\to g^{\ast }(\tilde{M})$ of ${\mathcal{O}}_U$-modules.

The morphism is an isomorphism on stalks: fix $\mathfrak{q}\in U$ and put $\mathfrak{p}={\phi }^{-1}(\mathfrak{q})=g(\mathfrak{q})\in V$. By [F2] and [F3], the stalk of the tensor pullback is $(g^{\ast }(\tilde{M}){)}_{\mathfrak{q}}\cong (O_U{)}_{\mathfrak{q}}{\otimes }_{(g^{-1}{\mathcal{O}}_V{)}_{\mathfrak{q}}}(g^{-1}\tilde{M}{)}_{\mathfrak{q}}\cong B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{\tilde{M}}_{\mathfrak{p}}\cong B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{M}_{\mathfrak{p}},$ while by [F2] and [F3] the stalk of the associated sheaf is ${\widetilde{(B{\otimes }_{A}M)}}_{\mathfrak{q}}\cong (B{\otimes }_{A}M{)}_{\mathfrak{q}}\cong B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{M}_{\mathfrak{p}};$ the map ${\delta }_{\mathfrak{q}}$ is $B_{\mathfrak{q}}$-linear and sends the class of $1\otimes m$ to the class of $1\otimes m$ under these identifications, and since the elements $1\otimes m$ generate $B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{M}_{\mathfrak{p}}$ as a $B_{\mathfrak{q}}$-module, ${\delta }_{\mathfrak{q}}$ is the canonical isomorphism between the two copies of $B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{M}_{\mathfrak{p}}$.

The affine isomorphism: a morphism of sheaves of modules is an isomorphism if and only if its stalk maps are isomorphisms, so step 2.1 shows that $\delta$ is an isomorphism of ${\mathcal{O}}_U$-modules; hence $g^{\ast }(\tilde{M})\cong \widetilde{(B{\otimes }_{A}M)}$ and, by step 1.1, $f^{\ast }\mathcal{F}{\mathrm{\mid }}_U\cong \widetilde{(B{\otimes }_{A}M)}$. This proves the affine form (2), and it shows that for every admissible pair $(U,V)$ the restriction of $f^{\ast }\mathcal{F}$ to the affine open $U$ is an associated sheaf, hence quasi-coherent on $U$ by [F4].

Global quasi-coherence: let $x\in X$. Choose an affine open $U_0=\mathrm{Spec}{B}_0\subseteq X$ containing $x$; since $\mathcal{F}$ is quasi-coherent there are an affine open $V=\mathrm{Spec}A\subseteq Y$ containing $f(x)$ and an $A$-module $M$ with $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{M}$. The set $U_0\cap f^{-1}(V)$ is an open neighbourhood of $x$ in $U_0$, so by [F5] it contains a distinguished open $D(b)$ with $x\in D(b)$; the open $D(b)=\mathrm{Spec}(B_0{)}_b$ is affine, maps into $V$, and satisfies $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{M}$. Therefore the family of all affine opens $U\subseteq X$ that admit an affine open $V\subseteq Y$ with $f(U)\subseteq V$ and $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{M}$ covers $X$, and each member has $f^{\ast }\mathcal{F}{\mathrm{\mid }}_U$ associated by step 3.1; by the affine cover criterion [F4], $f^{\ast }\mathcal{F}$ is quasi-coherent. This proves (1).

Choice accounting: the cover used in step 4.1 is the family of all admissible affine opens, which is determined by the data, so no chart, module or isomorphism is selected; the comparison morphism $\delta$ of step 1.2 is built from the canonical class maps of the inverse image colimit, and the identifications of step 2.1 are the canonical stalk and localisation isomorphisms. Hence the only use of the Axiom of Choice is the inherited one recorded in the Statement through [F7].