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.

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