To use Etale methods in proofs or not

Question

My question concerns a statement made by Fong and James in "Geometry of the Simple groups..."(1998); the statement says that they offer a proof "independent of etale methods." Am I to presume that "etale methods" are not as desirable as other methods in writing proofs? (I'm new to etale and hence may be oversensitive to what seems to be a judgement statement.) If etale methods are not desirable, obviously I would like to know why not. What is the down-side of etale methods?

Answer 1

"Étale" refers to methods from algebraic geometry, e.g., Étale morphism of schemes, Étale fundamental group, or Étale cohomology, and other things, used in number theory, algebraic geometry and other fields. This need not be "bad", of course. It just may be more technical than, say, abstract algebra. For this reason it may be desirable to have a proof "independent of these methods".

Answer 2

There are two ways a proof can be simple. It can be easy to remember, because it fits into a pretty story. (For example, the proof that subgroups of free groups are free using algebraic topology and the fact that coverings of graphs are graphs.)

Alternatively, it can have few pre-requisites and be easily accessible to anyone with patience and the ability to understand the statement of the theorem. (For example, the "elementary" proofs of the prime number theorem avoiding complex analysis.)

(These don't have to be in conflict, but they often are.)

In the theory of representations of finite groups of Lie type, there is amazing power to using the Deligne-Lusztig theory. At the same time, someone who is mainly interested in finite groups may prefer to digest a longer proof using only elementary techniques rather than spend a few years absorbing the foundations of algebraic geometry.

Of course, this is all subjective. Certainly, though, more proofs are better. This last principle gives a final reason the authors might want to explicitly point out that their proof is "étale-free": it counts as evidence that their proof is genuinely new.