A lemma

Did you know that if R is a ring, M is a finite R-module, and φ : M —> M is a surjective module map, then φ is an isomorphism? Just learned this today. This is Lemma 4.4a in Eisenbud if you want a reference. Or see Lemma Tag 05G8 in the stacks project.

If you know about limit arguments etc, then you immediately see how to prove it for finitely presented modules (reduce to Noetherian case, etc, etc). Thinking about it some more you may come to the conclusion that this is one of those things that is simply not true for finite modules in general. So I enjoy lemmas like this since it feels as if you are getting away with something!

[Edit: Just (6:02 PM) received an alternative proof of this lemma from Thanos D. Papaïoannou which is I would say a more honest and in particular completely standard proof. So now there are two proofs… More anybody?]

3 thoughts on “A lemma”

1. Johan, you must have known this result and forgot, due to the mists of time. It is Theorem 2.4 in Matsumura’s Commutative Ring Theory book. (In the exercises to section 3 is the analogous result for endomorphisms of a noetherian ring.) To me it always feels like getting away with something that Nakayama’s Lemma really works without noetherian hypotheses.

• Ah, but no, because I read the “old” Matsumura, entitled “Commutative Algebra” (Second Edition, 1980). I don’t think it is in there. EGA IV, 8.9.3 has the version of the lemma with the finite presentation hypothesis.