ag.algebraic geometry - Do pushouts along universal homeomorphisms exist?

References and background on universal homeomorphisms

Definition [EGA I (2d ed.) 3.8.1]. A morphism $f:Vto U$ is a universal homeomorphism if for any morphism $U'to U$, the pullback $Vtimes_UU'to U'$ is a homeomorphism.

Theorem [EGA IV 18.12.11]. A morphism is a universal homeomorphism if and only if it is surjective, integral, and radicial.

Theorem ["Topological invariance of the étale topos," SGA I Exp IX, 4.10 and SGA IV Exp. VIII, 1.1] If $f:Vto U$ is a universal homeomorphism, then the induced morphism $f:V_{textrm{ét}}to U_{textrm{ét}}$ of the small étale topoi is an equivalence.

General examples. Any nilimmersion, any purely inseparable field extension (or any base change thereby), the geometric Frobenius of an $mathbf{F}_p$-scheme [SGA V Exp. XIV=XV, § 1, No. 2, Pr. 2(a)].

Theorem. Suppose $X$ a reduced scheme with finitely many irreducible components. Denote by $X'$ its normalization. Then the natural morphism $X'to X$ is a universal homeomorphism if and only if $X$ is geometrically unibranch.

Specific example. Suppose $k$ an algebraically closed field of characteristic $2$. Consider the subring $k[x^2,xy,y]subset k[x,y]$. The induced morphism

$$mathrm{Spec}k[x,y]tomathrm{Spec}k[x^2,xy,y]$$

is a universal homeomorphism.

Question

Do pushouts along universal homeomorphisms exist in the category of schemes?

In more detail. Suppose $f:Vto U$ a universal homeomorphism, and suppose $p:Vto W$ a morphism. Everything here is a scheme; I can assume $W$ quasicompact and quasiseparated, but I have no control over the map $Vto W$. Now of course I can construct the pushout $P$ of $Vto U$ along $Vto W$ as a locally ringed space with no trouble (just take the underlying space of $W$ along with the fiber product $O_W times_{p_{star}O_V}p_{star}O_U$), but I can't show that $P$ is a scheme. Is it?

Thoughts

Of course the key point here is that $f$ is a universal homeomorphism, not just some run-of-the-mill morphism. So one can try to treat the cases where $f$ is schematically dominant or a nilimmersion separately.

Update

If $f$ is a nilimmersion, then I now see how to prove this completely. I still have no idea how to proceed the schematically dominant case.

[EDIT: I removed the additional question.]

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