Quasi-separatedness in pushforward cannot be omitted

The ideal $I$ is the $k$-span of the divisible monomials: every monomial $x^a$ divisible by $g_n$ equals $x^{a-g_n}{g}_n$ and so lies in $I$, and every generator $g_n$ is such a monomial, so the $k$-span of the divisible monomials is contained in $I$ and contains each generator, hence equals $I$; therefore a polynomial lies in $I$ exactly when every monomial in its support is divisible by some $g_n$, by [F1].

$V$ is not quasi-compact: let $F\subseteq \mathbb{N}$ be finite and let $j$ be the least index with $j\mathrm{\notin }F$; the ideal ${\mathfrak{p}}_F=(t-1,z,x_i:i\mathrm{\ne }j)\subseteq A$ is prime, because the substitution $t\mapsto 1$, $z\mapsto 0$, $x_j\mapsto x_j$, $x_i\mapsto 0$ for $i\mathrm{\ne }j$ defines a surjection $A\to k[x_j]$ with kernel ${\mathfrak{p}}_F$, and $k[x_j]$ is a domain by [F2]. Now $x_j\mathrm{\notin }{\mathfrak{p}}_F$, so ${\mathfrak{p}}_F\in D(x_j)\subseteq V$, while $x_i\in {\mathfrak{p}}_F$ for every $i\in F$, so ${\mathfrak{p}}_F\mathrm{\notin }D(x_i)$ for every $i\in F$; hence the finite subfamily indexed by $F$ of the cover $\{D(x_n)\}$ of $V$ does not cover $V$. As $F$ was arbitrary, no finite subfamily covers $V$, and $V$ is not quasi-compact by [F5].

$f$ is quasi-compact: for a quasi-compact open $W\subseteq Y$ the preimage $f^{-1}(W)$ is the gluing of the two copies $W\cap Y_i\cong W$ of $W$ along $W\cap V$ by [F4], hence is covered by the two compact subspaces $W\cap Y_1$ and $W\cap Y_2$, so it is compact by [F5] and $f^{-1}(W)$ is quasi-compact; therefore $f$ is quasi-compact.

$z$ and its multiples are not in $I$: the monomial $t^{m}z$ has $x_n$-exponent $0<n$ for every $n$, so it is divisible by no generator $g_n=t^n{x}_n^{n}z$, hence $t^{m}z\mathrm{\notin }I$ for every $m\ge 0$ by step 1.1; in particular $z\mathrm{\ne }0$ in $A$, and the element $z\mathrm{/}1\in A_t$ is nonzero because $t^{m}z\mathrm{\ne }0$ in $A$ for all $m$ by [F3].

$z$ dies in every $A_{t{x}_n}$: the relation $t^n{x}_n^{n}z=0$ holds in $A$, and in $A_{t{x}_n}$ the images of $t$ and $x_n$ are units by [F3], so $z=(t^n{x}_n^n{)}^{-1}\cdot t^n{x}_n^{n}z=0$ in $A_{t{x}_n}$ for every $n\ge 1$.

The map $A\to {\prod }_{n\ge 1}{A}_{x_n}$ is injective: let $a\in A$ be nonzero and lift it to $H\mathrm{\notin }I$; by step 1.1 some monomial $m$ in the support of $H$ is divisible by no $g_n$, and since the support is finite there is an index $n$ with $n>m(t)$ and $m(x_n)=0$. For every $N\ge 0$ the monomial $x_n^{N}m$ is then divisible by no $g_j$: a generator with $j=n$ would need $m(t)\ge n$, which fails, and a generator with $j\mathrm{\ne }n$ would divide $m$ itself; hence $x_n^{N}H\mathrm{\notin }I$ by step 1.1, so $x_n^{N}a\mathrm{\ne }0$ in $A$ for every $N$, and $a\mathrm{/}1\mathrm{\ne }0$ in $A_{x_n}$ by [F3]. Therefore a nonzero $a$ has nonzero image in ${\prod }_n{A}_{x_n}$, which is the injectivity claimed.

$f$ is not quasi-separated: the subschemes $Y_1,Y_2\subseteq X$ are affine opens lying over the common affine open $Y$ of the base, and $Y_1\cap Y_2=V$ is not quasi-compact by step 1.2, so the criterion of [F5] fails and $f$ is not quasi-separated.

Global sections: by [F4] a global section of ${\mathcal{O}}_X$ is a pair $(s_1,s_2)\in A\times A$ with equal restrictions to $V=Y_1\cap Y_2$; restricting to the cover $\{D(x_n)\}$ of $V$, this means $s_1-s_2$ maps to $0$ in ${\prod }_n{A}_{x_n}$, so $s_1=s_2$ by step 2.3; hence $\mathrm{\Gamma }(Y,f_{\ast }{\mathcal{O}}_X)=\mathrm{\Gamma }(X,{\mathcal{O}}_X)\cong A$, the diagonal embedding.

The section $(z,0)$ over $D(t)$: the preimage $f^{-1}(D(t))$ is the gluing of the two copies $D(t{)}_1,D(t{)}_2$ of $D(t)$ along $W=D(t)\cap V={\bigcup }_{n}D(t{x}_n)$ by [F4] and [F3], and the pair $(z,0)\in A_t\times A_t$ has equal restrictions to $W$: on each member $D(t{x}_n)$ of the cover of $W$ the first entry restricts to $0$ by step 2.2 and the second entry restricts to $0$ as well, and equality on a cover implies equality on $W$. Hence $(z,0)$ is a section in $\mathrm{\Gamma }(D(t),f_{\ast }{\mathcal{O}}_X)$; it is not of the form $(s,s)$, since $z\mathrm{\ne }0$ in $A_t$ by step 2.1.

$f_{\ast }{\mathcal{O}}_X$ is not quasi-coherent: suppose it were; then by [F6], applied to the quasi-coherent sheaf $\mathcal{G}=f_{\ast }{\mathcal{O}}_X$ and the affine base $Y$, the counit $\widetilde{\mathrm{\Gamma }(Y,\mathcal{G})}\to \mathcal{G}$ would be an isomorphism, and since $\mathrm{\Gamma }(Y,\mathcal{G})\cong A$ by step 3.1 its component on $D(t)$ would identify $A_t$ with $\mathrm{\Gamma }(D(t),f_{\ast }{\mathcal{O}}_X)$. That component is the localisation at $t$ of the restriction map $\mathrm{\Gamma }(Y,\mathcal{G})\to \mathrm{\Gamma }(D(t),\mathcal{G})$, and under the identifications of [F4] this restriction map is the pullback $f^{\mathrm{♯}}$, which sends $s\in A_t$ to the diagonal pair $(s,s)$, because $f$ restricts to the identity on each copy $Y_i$; hence its image is contained in the diagonal and does not contain $(z,0)$, which lies in $\mathrm{\Gamma }(D(t),f_{\ast }{\mathcal{O}}_X)$ by step 3.2. So the component is not surjective, contradicting the isomorphism; therefore $f_{\ast }{\mathcal{O}}_X$ is not quasi-coherent.

Conclusion: by step 1.3 the morphism $f$ is quasi-compact, by step 2.4 it is not quasi-separated, and by step 4.1 the pushforward of the quasi-coherent module ${\mathcal{O}}_X$ is not quasi-coherent on $Y$; hence quasi-compactness alone does not imply that pushforwards of quasi-coherent modules are quasi-coherent, the quasi-separatedness hypothesis in [F7] cannot be dropped, and the false claim is refuted.

Choice accounting: the field $k$, the variables $t,z,x_1,x_2,\dots$, the ideal $I$, the ring $A$, the cover $\{D(x_n)\}$ and the gluing data are explicitly displayed; the auxiliary index of step 2.3 is chosen from a finite set and the index $j$ of step 1.2 is the least natural number outside a finite set, so no infinite simultaneous choice occurs, and the only Axiom of Choice is the inherited one recorded in [F8], used through the gluing theorems, the polynomial-ring universal property and the affine criterion of [F6].