# Flat extension of local rings with a specified extension of residue field [closed]

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 $$S/\mathfrak m_S\cong K$$ ?

## 1 Answer

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

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