Witt vectors over ̄ k as a Cohen ring with universal property

Let $q$ be a prime and let $\bar k$ be an algebraically closed field of characteristic $q$. The assertion is that there exists a type $O$ carrying a commutative ring structure which is a domain, a discrete valuation ring and of characteristic zero, together with a $\mathbb{Z}_q$-algebra structure on $O$ for which $O$ is adically complete with respect to the ideal generated by the image of $q \in \mathbb{Z}_q$ and for which that ideal is maximal, a ring isomorphism $e \colon O/\mathfrak{m}_O \xrightarrow{\sim} \bar k$ from the residue field of $O$ onto $\bar k$, and a ring homomorphism $\iota \colon W_q(\mathbb{F}_{q^2}) \to O$ from the ring of Witt vectors of the Galois field of order $q^2$, such that the following holds: for every commutative local Artinian ring $B$ and every surjective ring homomorphism $\rho \colon B \to \bar k$ whose kernel is the maximal ideal of $B$, there is a unique ring homomorphism $f \colon O \to B$ with $\rho \circ f = e \circ (O \to O/\mathfrak{m}_O)$. No compatibility is imposed on $\iota$ beyond its existence; it is part of the data provided for later use, not constrained by the universal property.

This packages $W(\bar k)$ as a complete unramified discrete valuation ring of characteristic zero, with uniformiser $q$ and residue field $\bar k$, satisfying the universal property of a Cohen ring with respect to Artin local rings with residue field $\bar k$: it is the canonical coefficient ring for deformation-theoretic arguments. It is used in the construction of a two-dimensional regular tower with a versal pullback property for fake elliptic curves.