Lemma 10.20.3. Let $A \to B$ be a local homomorphism of local rings. Assume
$B$ is finite as an $A$-module,
$\mathfrak m_ B$ is a finitely generated ideal,
$A \to B$ induces an isomorphism on residue fields, and
$\mathfrak m_ A/\mathfrak m_ A^2 \to \mathfrak m_ B/\mathfrak m_ B^2$ is surjective.
Then $A \to B$ is surjective.