# Do pushouts along universal homeomorphisms exist?

### References and background on universal homeomorphisms

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

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

is a universal homeomorphism.

### Question

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

In more detail. Suppose $f:V\to U$ a universal homeomorphism, and suppose $p:V\to W$ a morphism. Everything here is a scheme; I can assume $W$ quasicompact and quasiseparated, but I have no control over the map $V\to W$. Now of course I can construct the pushout $P$ of $V\to U$ along $V\to W$ 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.]