# 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.]