TITLE: Who uses radicial morphisms?
QUESTION [24 upvotes]: 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:

Who uses radicial morphisms? 

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

What are radicial morphisms good for?

REPLY [3 votes]: 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.