I was looking at Homomorphism of local rings which is about proving a particular theorem on local rings. More precisely, we are given a local homomorphism $f\colon(A,\mathfrak m_A)\to(B,\mathfrak m_B)$ of local Noetherian rings so that i) $A/\mathfrak m_A\cong B/\mathfrak m_B$, ii) $\mathfrak m_A\to\mathfrak m_B/\mathfrak m_B^2$ is surjective and iii) $B$ is a finitely generated $A$-module (all assumptions relative to $f$).
Claim. $f$ is surjective.
The linked post goes on using the Snake Lemma on
$$ \require{AMScd} \begin{CD} 0 @>>> \mathfrak m_A @>>> A @>>> A/\mathfrak m_A @>>> 0 \\ @. @VVV @V{f}VV @VVV \\ 0 @>>> \mathfrak m_B @>>> B @>>> B/\mathfrak m_B @>>> 0 \end{CD} $$
which a commutative diagram of $A$-modules with exact rows.
Given the unanswered question in a comment to the answer I was wondering which of the above assumptions are really necessary to conclude like this. Because, interestingly enough, the usual approach (sketched in the answer to the given post) seems to require more assumptions!
So, for the above diagram to commute we certainly need that $f$ is a local homomorphism (i.e. $f(\mathfrak m_A)\subset\mathfrak m_B$). Morever, the isomorphism $A/\mathfrak m_A\cong B/\mathfrak m_B$ allows us to reduce the question of surjectivity of $f\colon A\to B$ to surjectivity of $f\colon\mathfrak m_A\to\mathfrak m_B$. And here it becomes tricky. We want to conclude that $\mathfrak m_A\twoheadrightarrow\mathfrak m_B$ as $A$-modules.
The best approach for this would be to use Nakayama for $(A,\mathfrak m_A)$. We then have to prove that $\mathfrak m_B=\mathfrak m_A+\mathfrak m_A\mathfrak m_B$ and additionally need that $\mathfrak m_B$ is a finitely generated $A$-module (this can guaranteed by assuming that $A$ is Notherian and that $B$ is a finitely generated $A$-module). However, now the assumption $\mathfrak m_A\twoheadrightarrow\mathfrak m_B/\mathfrak m_B^2$ does not seem to be strong enough. The linked answer glossed over this by claiming that we need $\mathfrak m_B=\mathfrak m_A+\mathfrak m_B^2$ instead, which is not correct as far as I can tell (this would be Nakayama for $B$-modules).
Is it possible to make this approach work? If so, which assumptions are necessary to ensure that the arguments goes through? I know another argument which does work (and also weakens some of the assumptions) but I am interested in this particular one using the Snake Lemma.
Thanks in advance!
