TITLE: What interesting/nontrivial results in Algebraic geometry require the existence of universes?
QUESTION [46 upvotes]: Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link).  I'm wondering if there are any results in AG that actually depend on the existence of universes (and what some of the more interesting ones are).
I'm of course aware of the result that as long as we require that the classes of objects and arrows are sets (this is the only valid approach from Bourbaki's perspective),  for every category C, there exists a universe U such that the U-Yoneda lemma holds for U-Psh(C) (this relative approach makes proper classes pointless because every universe allows us to model a higher level of "largeness"), but this is really the only striking application of universes that I know of (and the only result I'm aware of where it's clear that they are necessary for the result).

REPLY [22 votes]: There is a paper of Solomon Feferman from 1969 in which he proposes a conservative extension of ZFC adequate for formulating much of category theory.  Since the need for universes in algebraic geometry seems to arise mainly via the use of category theory, I conjecture that Feferman's approach may be useful in algebraic geometry.  The paper in question is "Set-theoretical foundations of category theory" (in "Reports of the Midwest Category Seminar III", Springer Lecture Notes in Math 106, edited by Saunders Mac Lane, pages 201-247).  Feferman proposes to add to the language of ZFC a new constant symbol $\kappa$ and to add to the axioms the schema saying that $V_\kappa$ is an elementary submodel of the universe $V$ of all sets, with respect to the original language of ZFC.  That is, for each formula $\phi(x_1,\dots,x_n)$ in the original language (i.e., not involving $\kappa$), there is an axiom saying that, for each $x_1,\dots,x_n\in V_\kappa$, the statement $\phi(x_1,\dots,x_n)$ is equivalent to the same statement with all quantified variables restricted to range over $V_\kappa$.  The idea is that this $V_\kappa$ can play the role of a Grothendieck universe, even though it isn't really one.  Furthermore, in situations where one would expect to use two or more universes, one can often get by with one, by a judicious use of the "elementary submodel" axioms.  
Note that one cannot express the notion of "elementary submodel of the universe" as a single formula, since there is no uniform way to express "truth in the universe" (Tarski's theorem).  But one can express it one formula at a time, as an infinite axiom scheme, since there's no problem expressing truth of a single formula $\phi$ (just say $\phi$).  And this is what Feferman does.
Feferman's axiom system is conservative over ZFC.  That is, if you can prove, in Feferman's system, a sentence that doesn't involve $\kappa$, then you can prove the same sentence in ZFC.  (This follows from Levy's reflection principle plus the compactness theorem of first-order logic.)
Many years ago, I taught a category theory course, using Feferman's axioms as the foundation.  For the most part, this foundation worked well, but, if I remember correctly, I ran into a problem proving the existence of Kan extensions.  I don't remember what I did to overcome the problem (nor do I even remember exactly what the problem was).  
I believe Feferman has done additional work improving this foundational system, but I have not yet absorbed that work.