When is the residue field of the completion of a local ring a finite extension?

Question

Let $(R, \mathfrak m)$ be a Noetherian local ring and $(\hat R, \hat {\mathfrak m} )$ be its $\mathfrak m$-adic completion.

My question is: When can we say $\hat R / \hat {\mathfrak m}$ is a finitely generated $R / \mathfrak m$-module ?

In particular can we say it when $R$ is excellent, or if it is the homomorphic image of a regular local ring ?

Answer 1

The natural map $R/\mathfrak{m}\to \hat R / \hat {\mathfrak m}$ is an isomorphism, so $\hat R / \hat {\mathfrak m}$ is trivially finitely generated over $R/\mathfrak{m}$.