On Local Rings

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Joined: Tue Nov 10, 2015 8:25 pm

On Local Rings

#1

Post by Tsakanikas Nickos »

Let \( \displaystyle f \ \colon A \longrightarrow B \) be a local homomorphism of local noetherian rings \( A \) and \( B \), such that
  1. \( A / \mathfrak{m}_{A} \cong B / \mathfrak{m}_{B} \)
  2. \( \mathfrak{m}_{A} \twoheadrightarrow \mathfrak{m}_{B} / \mathfrak{m}^{2}_{B} \)
  3. \( B \) is a finitely generated \( A \)-module
Then \( f \) is surjective.
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