Let $F$ be a field. Does the category $C\_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object?


This is, for instance, true if $F=\mathbb{F}\_{p}$ for some prime $p$: If $R$ is a local ring with residue field $\mathbb{F}\_{p}$, then any $x\in\mathbb{Z}\setminus(p)$ must map to something invertible under the morphism $\mathbb{Z}\longrightarrow R$. Hence that morphism factors as $\mathbb{Z}\longrightarrow\mathbb{Z}\_{(p)}\longrightarrow R$; thus $\mathbb{Z}\_{(p)}$ is the initial object.


But what happens in the more general case? I guess it should be true at least if $F$ is of finite type over either $\mathbb{Q}$ or $\mathbb{F}\_{p}$ (where $p$ is a prime), but I have no idea how to prove it.



Does the category of local rings with residue field $F$ have an initial object?
A typical approach is to ask for your universal property among complete DVRs, not among all local rings, because there you have a very nice positive result. Given a perfect field $k$ of characteristic $p$, the Witt ring $W(k)$ is initial among complete DVRs of characteristic $0$ equipped with an isomorphism of residue field with $k$. This is Theorem II.5.4 of Serre's book "Local Fields."


