One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in commutative algebra, and really sounds as though it should be easy (the graded case, for example, is easy), but it's not. The question at least goes as far back as Serre's FAC, but it wasn't proved until 1976, by Quillen **EDIT:** and also independently by Suslin.

I decided that this is the sort of fact that I should know a rough outline of how to prove, but the paper was not very helpful. Usually when someone kills off a famous conjecture in 5 pages, it's because they've developed some fantastic new piece of machinery people didn't have before. And, indeed, Quillen is famous for inventing some fancy and wonderful machinery, and the paper is only 5 pages long, but as far as I can tell, none of that fancy machinery actually appears in the proof.

So, what was it that Quillen saw, that Serre missed?

Algebra. $\endgroup$4more comments