Witt vectors as the unique strict p-ring with residue ring k

Let $p$ be a prime and let $\mathcal O$ be a commutative ring in which the image of $p$ lies in the submonoid of non-zero-divisors, so that multiplication by $p$ on $\mathcal O$ is injective. Let $k$ be a commutative ring of characteristic $p$ which is perfect in the sense that the Frobenius endomorphism $x \mapsto x^p$ of $k$ is bijective, and suppose $k$ is given the structure of an $\mathcal O$-algebra whose structure map $\mathcal O \to k$ is surjective with kernel exactly the ideal generated by $p$; assume further that $\mathcal O$ is complete and separated for the adic topology defined by the ideal generated by $p$. Then there exists a ring isomorphism $e \colon W(k) \to \mathcal O$ from the ring of $p$-typical Witt vectors of $k$ onto $\mathcal O$ such that $e$ followed by the structure map $\mathcal O \to k$ is the zeroth Witt coordinate $x \mapsto x_0$, and $e$ is unique with this property even as a ring homomorphism: any ring homomorphism $g \colon W(k) \to \mathcal O$ whose composite with $\mathcal O \to k$ is the zeroth Witt coordinate coincides with $e$.

This is the classical structure theorem for strict $p$-rings with perfect residue ring: such a ring is determined, up to a unique isomorphism compatible with reduction, by its residue ring, and the Witt vectors realise it. In this development it is used to identify an abstractly given $p$-adically complete base ring with $\mathbf Z_p$ or with $W(k)$, in the two Čerednik–Drinfeld statements that cite it.