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We are taught since when we are young that schemes are cool because they take into account "nilpotents". This means also that we can distinguish between schemes which have the same underlying topological space but for example in the sense that they differ by some "thickening".

I would like to get some intuition and motivation about the notion of universal homeomorphism: by definition a morphism of schemes $f\colon X\to Y$ is a $\mathit{universal\,\, homeomorphism}$ if for every morphism of schemes $Z\to Y$ the pull-back morphism induces a homeomorphism $\lvert Z\times_Y X \rvert \to \lvert Z\rvert$.

I haven't been able to find some discussion about that, and in the stacks project (28.43) they almost apologize for including that section because a universal homeo "is really just an integral, universally injective and surjective morphism". I also have the feeling that they are taken into account especially when one works in positive characteristic.

This might be the question: do we need universal homeomorphisms? do we really need universal homeomorphisms when working over $\mathbb{C}$?

Thank you!

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    $\begingroup$ Another proof is EGA IV$_4$ 18.12 (see 18.12.10), where "universally injective" is called "radiciel" (EGA I, 3.5.4) and characterized nicely in EGA I, 3.5.8. The equivalence of etale sites via pullback along $S_{\rm{red}} \hookrightarrow S$ (see EGA IV$_4$ 18.1.2) holds along any universal homeomorphism $f:S'\rightarrow S$ (3.12 in Ch. I of Freitag & Kiehl's book on etale cohomology and SGA4 Exp. VIII section 1 treat finite universal homeomorphisms, and it follows in general via limit arguments; see Rem.1.4 in SGA4 Exp. VIII). Isn't that nice? Normalization of cusp is such an $f$, in any char. $\endgroup$ – nfdc23 Jul 10 '16 at 17:25
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    $\begingroup$ Here's an example to keep in mind: the normalization of the cuspidal cubic $\{y^2=x^3\}$ (given by $t\mapsto(t^2,t^3)$) is a universal homeomorphism (which is not an isomorphism, even in characteristic zero, even though both the source and the target are reduced). $\endgroup$ – Gro-Tsen Jul 10 '16 at 17:49
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    $\begingroup$ Note also that if $X$ and $Y$ are finite type over $\mathbf{C}$, then a $\mathbf{C}$-morphism $f:X \rightarrow Y$ is a universal homeomorphism if and only if the map of topological spaces $X(\mathbf{C}) \rightarrow Y(\mathbf{C})$ is a homeomorphism. Indeed, the latter is proper if and only if $f$ is proper (SGA1, Exp. XII, 3.2(v)), so the latter is a homeomorphism if and only if $f$ is a finite radiciel surjection, which is equivalent to $f$ being a universal homeomorphism. $\endgroup$ – nfdc23 Jul 10 '16 at 17:56
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Universal homeomorphisms of schemes induce equivalences of etale sites. This is useful in the proof of the following fact: if $K/k$ is an extension of separably closed fields, $X$ is a qcqs scheme over $k$, $F$ is a sheaf of torsion abelian groups with torsion orders not divisible by $\mathrm{char}(k)$, then the natural map $H^q_{et}(X, F)\rightarrow H^q_{et}(X_K, F_K)$ is an isomorphism. The latter is useful to avoid confusion when talking about "geometric" etale cohomology groups.

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