In editing the algebraic geometry notes posted here, prompted by Brian Conrad, I am introducing the notion of radicial morphism. This seems to me to not be a notion that absolutely everyone should see in a first serious schemes course, given the volume of definitions a student must digest (even if this idea is wafer-thin). So I would like to make clear to the learner who this notion is for. But the problem is: I don't know. I have two bits of data: I've never used it professionally, and I know some arithmetic geometers (e.g. in my department) have used it. Thus I ask:

In the answers, I expect to hear multiple interesting answers to the implicit question:

What are radicial morphisms good for?

• Your "This seems to me to not be a notion that absolutely everyone should see" had me rereading the phrase fve times until it made sense. Good work :) Dec 20 '10 at 0:09
• I don't know, but I'm sure I'd prefer the term "universally injective" or even "geometrically injective" over radicial anything. Dec 20 '10 at 1:47
• Anyone who uses etale topology or characteristic-$p$ geometry or char-$p$ alg. groups (e.g., ab. varieties or linear algebraic groups) in a serious way will find it useful. For example, "etale + radicial = open immersion" and "radicial integral surjections have no effect on the etale topos" (e.g., passing from sep. closed ground field to an alg. closed ground field when doing an etale cohom. or fundamental gp calculation). As Allen notes, it's often called "universally injective" (never heard of "geom. injective"), though if you do then you need to prove the lemma that justifies the name... Dec 20 '10 at 1:58
• Allow me to add an example to BCnrd's comment (which I learned from Olsson). Let G be a group scheme of finite type over a field k of char. p, but not smooth over k. Let BG be the classifying stack of G. To compute the etale cohom of BG we'd like to use cohom descent, but pt --> BG is not a presentation (not smooth). The max reduced G_{red} is a group scheme, and is more likely to be smooth (and will be if we make a finite extension of k). The morphism BG_{red} --> BG is rep and radicial, so they have the same cohom, and we can apply descent. This works over a general base, using devissage. Dec 20 '10 at 11:15
• Dear shenghao: As tiny correction, for imperfect $k$, $G_{\rm{red}}$ may not be subgp; see Ex. A.8.3 in "Pseudo-reductive groups" for an example over any imperfect field (with $G$ finite). As you say, if we first make a suitable finite purely insep. (!) extn on $k$ and then kill the nilpotents we get a smooth subgp. That's good enough for purposes of etale cohomology, but for other purposes can be a nuisance. Here's a version avoiding change of $k$: the quotient of $G$ by inf'tml kernel of (perhaps not-flat!) $n$-fold rel. Frob. is smooth for suff. large $n$; see SGA3, VII_A, 8.3. Dec 20 '10 at 15:53

Grothendieck's version of the Ax-Grothendieck theorem is that if you have an $S$-scheme $X$ of finite type, then any radicial $S$-endomorphism $f: X\to X$ is surjective: EGA IV_3, Prop. (10.4.11).
Such a wonderfully unexpected theorem certainly justifies the investment in learning what radicial means...

• Thanks for the answer! I checked the Wikipedia page for the Ax-Grothendieck theorem which led to the confirmation of my speculation that polynomial algebras over a commutative ring $R$ are Hopfian (any $R$-endomorphism is an automorphism). Feb 15 '17 at 6:54

A reasonable prerequisite for a course in Algebraic Geometry is a course in Galois Theory including some characteristic $p$ results. Every finite algebraic extension of fields $E|K$ (meaning $K \subset E$) factors trough an intermediate separable extension $E_s|K$ such that $E|E_s$ is purely inseparable. This theorem sheds light on the structure of field extensions.

I consider finite separable extensions of fields as the one important idea behind the concept of etale map. An etale map is a flat family of finite separable extensions. The counterpart of purely inseparable extensions in geometry are radical morphisms. The topological characterization as universally injective morphisms makes them interesting also. I would not discuss etale maps without treating radical morphisms.

Another reason is that the Frobenius maps (relative, absolute...) are the essential tools for understanding the cohomology of varieties in characteristic $p$ and this is probably the main example of radical morphisms. For these two reasons I would say that radical morphisms belong in an introductory course on schemes.

This isn't exactly the same thing, but it's obviously related (and its a special case of radicial), so it might be a place to look. Brian Conrad pointed this relation out to me several months ago (here on mathoverflow). Radicial morphisms are closely related to (but slightly weaker than) the following notion which appears in algebraic geometry and especially in commutative algebra:

An extension of rings $R \subseteq S$ is called weakly subintegral if the induced map on Spec is a bijection (this is where it differs from radicial) and if the extensions of residue fields is inseparable. This is the same thing as being a universal homeomorphism if I recall correctly.

Having no finite bijective (on Spec) birational radicial extensions is called being weakly normal. Weak normality probably has a lot of papers written about it over the years.

Especially in equal-characteristic zero, being weakly normal is also sometimes called being semi-normal (although semi-normality is a distinct notion in general). Both these conditions are used in the study of various moduli spaces of algebraic varieties.

Here is a fun example of a place where radicial morphisms get used: in the valuative criterion for when a map of schemes is a locally closed embedding (i.e. the composition of a closed embedding and an open embedding), due to Mochizuki.

Roughly speaking, this valuative criterion * says that if we are given a map $$X \to Y$$ and a map from the spectrum of a DVR $$T$$ to $$Y$$ such that both points of the DVR map to $$X$$ with the natural diagrams commuting, then we have a map from $$T$$ to $$X$$ making the diagrams commute.

Finite type maps of noetherian schemes satisfy * if and only if they can be expressed as the composition of a radicial map and an open immersion. From this, one deduces that finite type maps of noetherian schemes are locally closed immersions if and only if they are monomorphisms and satisfy *.

The only proof I know for this valuative criterion for locally closed immersions, which seems like a natural thing to want, goes by first proving the statement about radicial maps.

Of course, none of this should go in a first course on schemes, but I just thought it was a fun application of radicial maps.