Lifting a residue-field map to W(k₀)→𝒪

Let $p$ be a prime, let $k_0$ be a finite field of characteristic $p$, and let $\mathcal{O}$ be a commutative ring which is a domain and a discrete valuation ring, complete for the adic topology of its maximal ideal $\mathfrak{m}_{\mathcal{O}}$ (in the sense of IsAdicComplete for the ideal IsLocalRing.maximalIdeal 𝒪). Assume that the image of $p$ in $\mathcal{O}$ lies in $\mathfrak{m}_{\mathcal{O}}$, and let $f\colon k_0 \to \mathcal{O}/\mathfrak{m}_{\mathcal{O}}$ be a ring homomorphism from $k_0$ to the residue field of $\mathcal{O}$. Then there exists a ring homomorphism $g$ from the ring $W(k_0)$ of $p$-typical Witt vectors of $k_0$ to $\mathcal{O}$ such that $g$ is a local homomorphism, i.e. the preimage under $g$ of the non-units of $\mathcal{O}$ consists of non-units (the IsLocalHom predicate), and such that the composite of $g$ with the residue map $\mathcal{O} \to \mathcal{O}/\mathfrak{m}_{\mathcal{O}}$ equals the composite of the constant-coefficient homomorphism $W(k_0) \to k_0$ with $f$. Only existence is asserted; no uniqueness of $g$ is claimed.

This is the lifting property of the Witt vectors of a finite (hence perfect) field of characteristic $p$: any map of $k_0$ into the residue field of a complete discrete valuation ring of mixed characteristic $(0,p)$ is induced by a local homomorphism out of $W(k_0)$, the coefficient ring of the unramified case of the Cohen structure theorem. It is used to supply coefficient rings: it is cited in the construction of regular local rings prorepresenting stalks on quaternionic moduli problems in the Čerednik–Drinfeld setting, and in the construction of Galois representations attached to cusp forms with prescribed Hecke data.