Nakayama’s lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe around 10 times) in my life.
But for some reason I just cannot get this lemma, i.e. I have tendency to forget it. Last time this happened just a couple of days ago, in the book of Shafarevich (Basic Algebraic geometry in 1.5.3.) This lemma is used to prove that for finite maps between quasiprojective varieties the image of a closed set is closed, and again this lemma sounded as something foreign to me (so again I went through the proof of the lemma)…
Question. Is there a path to get some stable understanding of Nakayama’s lemma and its corollaries? I would be especially happy if there were some geometric intuition underlying this lemma. Or some geometric example. Or maybe there is a nice article of this topic? Some mnemonic rule? (or one just needs to get used to the lemma?)
It’s sort of like the inverse function theorem, and that is why it is so strong. If you have n functions vanishing at the origin of kn and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin. Or equivalently if their differentials generate the cotangent space at the origin. So in a [not necessarily noetherian, thanks Georges!] local ring (O,m), Nakayama’s lemma says you can detect that elements of the maximal ideal generate that ideal, hence act sort of like coordinate functions, just by knowing their differentials, i.e. their residues in the Zariski cotangent space m/m2, generate that linear space.
Those versions of the lemma you linked to are almost unrecognizable forms of this simple statement, but that’s the way abstract math goes as we know. But the idea is the same, you have a hypotheses about a truncated version of your statement, and you get out the fuller version. The Jacobson radical stuff is there to disguise the fact that it doesn’t say much unless you are in a local setting. I.e. in a local ring the Jacobson radical is pretty big and you get a better result. In a polynomial ring with tiny Jacobson radical you get nothing.