Quasi-coherence of pushforward for qcqs morphisms

Affine model of one chart: let $g:\mathrm{Spec}C\to \mathrm{Spec}R$ be a morphism of affine schemes with corresponding ring map $\psi :R\to C$, let $\mathcal{G}$ be a quasi-coherent ${\mathcal{O}}_{\mathrm{Spec}C}$-module and put $N=\mathrm{\Gamma }(\mathrm{Spec}C,\mathcal{G})$, regarded as an $R$-module through $\psi$. Then $\mathcal{G}\cong \tilde{N}$ as $C$-modules, and for every $r\in R$ the inverse image $g^{-1}D(r)$ is the distinguished open $D(\psi (r))$, so the sections of the direct image are $\mathrm{\Gamma }(D(r),g_{\ast }\mathcal{G})=\mathcal{G}(g^{-1}D(r))=\mathcal{G}(D(\psi (r)))=N_{\psi (r)}$, the localisation of the $C$-module $N$ at $\psi (r)$.

Covering data inside a fixed affine target chart: fix an affine open $U=\mathrm{Spec}R\subseteq S$; then $U$ is quasi-compact, so $X_U=f^{-1}(U)$ is quasi-compact because $f$ is quasi-compact, and the family of all affine open subschemes of $X_U$, which covers $X_U$, has a finite subcover $U_1,\dots ,U_n$ (with $n=0$ meaning $X_U=\mathrm{\varnothing }$); for each pair $i\le j$ the intersection $U_i\cap U_j$ is quasi-compact by quasi-separatedness, so it has a finite affine cover $U_{ij1},\dots ,U_{ij{m}_{ij}}$, and one may take $U_{ii1}=U_i$. All these are affine opens of $X$ lying over $U$, their corresponding ring maps are ${\psi }_i:R\to \mathrm{\Gamma }(U_i,{\mathcal{O}}_X)$ and ${\psi }_{ijk}:R\to \mathrm{\Gamma }(U_{ijk},{\mathcal{O}}_X)$, and we write $N_i=\mathrm{\Gamma }(U_i,\mathcal{F})$ and $N_{ijk}=\mathrm{\Gamma }(U_{ijk},\mathcal{F})$; only finitely many objects are chosen.

Affine model of the comparison: in the situation of step 1.1 regard each $N_{\psi (r)}$ as an $R_r$-module through the canonical ring map $R_r\to C_{\psi (r)}$; then for every $r\in R$ the map ${\chi }_r:N_r\to N_{\psi (r)}$, $n\mathrm{/}{r}^k\mapsto n\mathrm{/}\psi (r{)}^k$ in fractions, is well defined, $R_r$-linear and bijective: if $n\mathrm{/}{r}^k=n^{\mathrm{\prime }}\mathrm{/}{r}^l$ in $N_r$ then $r^m(r^{l}n-r^k{n}^{\mathrm{\prime }})=0$ for some $m$, and applying $\psi$ gives $\psi (r{)}^m(\psi (r{)}^{l}n-\psi (r{)}^k{n}^{\mathrm{\prime }})=0$, so the images agree in $N_{\psi (r)}$; every element of $N_{\psi (r)}$ is a fraction $n\mathrm{/}\psi (r{)}^k$, so ${\chi }_r$ is surjective; and ${\chi }_r(n\mathrm{/}{r}^k)=0$ means $\psi (r{)}^{m}n=0$ for some $m$, hence $r^{m}n=\psi (r{)}^{m}n=0$ and $n\mathrm{/}{r}^k=0$ in $N_r$, so ${\chi }_r$ is injective; linearity is immediate from the fraction formulas.

The affine model is an associated sheaf: in the situation of steps 1.1 and 2.1 the maps ${\chi }_r$ are compatible with the restriction maps of $\widetilde{N_R}$ and $g_{\ast }\mathcal{G}$, because for $D(s)\subseteq D(r)$ both composites $N_r\to N_{\psi (s)}$ are the canonical localisation maps induced by the ring maps $R_r\to R_s$ and $R_r\to C_{\psi (r)}\to C_{\psi (s)}$; since the distinguished opens form a basis and both sides are sheaves, these compatible isomorphisms on a basis assemble into a unique isomorphism of ${\mathcal{O}}_{\mathrm{Spec}R}$-modules $\widetilde{N_R}\to g_{\ast }\mathcal{G}$ whose component on $D(r)$ is ${\chi }_r$, by restricting a section over an open set to the distinguished opens it contains, mapping each restriction and gluing in the target; the inverses ${\chi }_r^{-1}$ assemble into an inverse in the same way, so $g_{\ast }\mathcal{G}\cong \widetilde{N_R}$ is quasi-coherent.

Application to the covering charts: by step 3.1 applied to $g=f{\mathrm{\mid }}_{U_i}:U_i\to U$ with the quasi-coherent module $\mathcal{F}{\mathrm{\mid }}_{U_i}$ one has $(f{\mathrm{\mid }}_{U_i}{)}_{\ast }(\mathcal{F}{\mathrm{\mid }}_{U_i})\cong \widetilde{(N_i{)}_R}$, a quasi-coherent sheaf on $U$; the same argument applies to each chart $f{\mathrm{\mid }}_{U_{ijk}}:U_{ijk}\to U$ with module $N_{ijk}$, so every summand occurring below is quasi-coherent.

The kernel description: let $E={⨁}_{i=1}^n(f{\mathrm{\mid }}_{U_i}{)}_{\ast }(\mathcal{F}{\mathrm{\mid }}_{U_i})$ and $E^{\mathrm{\prime }}={⨁}_{i\le j}{⨁}_{k=1}^{m_{ij}}(f{\mathrm{\mid }}_{U_{ijk}}{)}_{\ast }(\mathcal{F}{\mathrm{\mid }}_{U_{ijk}})$; restrictions define an ${\mathcal{O}}_U$-linear morphism $d:E\to E^{\mathrm{\prime }}$ whose component on an open $W\subseteq U$ is $d_W((s_i{)}_i{)}_{(i,j,k)}=s_i{\mathrm{\mid }}_{U_{ijk}\cap f^{-1}W}-s_j{\mathrm{\mid }}_{U_{ijk}\cap f^{-1}W}$, and $E,E^{\mathrm{\prime }}$ are quasi-coherent by [F8] and step 4.1. For every open $W\subseteq U$ a family $(s_i)$ lies in $\ker (d_W)$ exactly when $s_i$ and $s_j$ agree on $U_i\cap U_j\cap f^{-1}W$ for all $i,j$, because the opens $U_{ijk}\cap f^{-1}W$ cover that intersection and $\mathcal{F}$ is separated [F6]; such compatible families are exactly the restrictions to the cover $(f^{-1}W\cap U_i{)}_i$ of a section in $\mathcal{F}(f^{-1}W)$, by gluing and locality in the sheaf $\mathcal{F}$ [F6], so there is an objectwise bijection $\ker (d)(W)\cong \mathcal{F}(f^{-1}W)=((f_{\ast }\mathcal{F}){\mathrm{\mid }}_U)(W)$ which is natural in $W$ and $\mathcal{O}(W)$-linear.

Conclusion: by step 5.1 the restriction $(f_{\ast }\mathcal{F}){\mathrm{\mid }}_U\cong \ker d$ is the kernel of a morphism of quasi-coherent ${\mathcal{O}}_U$-modules, hence is quasi-coherent by [F8]; since the affine open $U\subseteq S$ was arbitrary and affine opens cover $S$, locality of quasi-coherence [F3] shows that $f_{\ast }\mathcal{F}$ itself is quasi-coherent.

Choice and edge cases: the only selections are finite subcovers of the fixed covers by affine opens, the finite covers of the quasi-compact intersections, and finitely many distinguished opens, so no infinite simultaneous choice or localisation at infinitely many primes is made, and the only Axiom of Choice is the inherited one recorded in [F10]. The empty and degenerate cases are covered by the same steps: for $X_U=\mathrm{\varnothing }$ one takes $n=0$ and then $E=E^{\mathrm{\prime }}=0$ and $(f_{\ast }\mathcal{F}){\mathrm{\mid }}_U=0$; for $r=0$ both sides of step 2.1 are the zero module; and for $S=\mathrm{\varnothing }$ or $C=0$ the relevant spaces and modules are zero as well.