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Let $(R, \mathfrak m_R, k)$ be a Noetherian local ring and $K$ be a field containing $k$.

Then is it true that there is a Noetherian local ring $(S, \mathfrak m_S)$ and a flat ring homomorphism $f: R\to S$ such that $f(\mathfrak m_R)S=\mathfrak m_S$ and $S/\mathfrak m_S\cong K$ ?

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Yes. You can see a proof in Bourbaki, Algèbre Commutative, chapitre IX, Appendice, n.2, Corollaire du Théorème 1.

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  • $\begingroup$ Even without noetherian hypothesis. $\endgroup$ – A.G May 29 at 11:56
  • $\begingroup$ Is it flat that gonflement in Bourbaki? $\endgroup$ – user26857 Jun 4 at 18:45
  • $\begingroup$ Yes. It follows easily from the definition. $\endgroup$ – A.G Jun 11 at 21:52

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