Let $K$ be as in the title with tilt $K^\flat$. $W = W(K^{\circ\flat})$ satisfies a universal property: it is the unique $p$-adically complete $p$-torsion free $\mathbb{Z}_p$-algebra $A$ with $A / pA \cong K^{\circ\flat}$. It comes equipped with a surjective morphism $\theta: W \to K^\circ$ which mod $p$ induces the morphism $K^{\circ\flat} \to K^{\circ} / (p)$. See Bhatt, Lectures on Perfectoid Spaces, section 6.1.
What can we say about algebraic properties of $W$ beyond what is in Bhatt? I’m particularly wondering if there is a classification of its ideals (either prime ideals or ideals in general). One observation is that for any other untilt $K’$ of $K^\flat$ we obtain an analogous morphism $\theta’: W \to (K’)^\circ$. Can we intrinsically characterize the prime ideals corresponding to such morphisms $\theta’$? I am wondering if the primes of $W$ can “parametrize” untilts of $K^\flat$ in any useful sense (I know the Fargues-Fontaine curve plays a role in parametrizing untilts, though I am unsure if that’s related). I am also curious about the ideal structure of the 2-truncated Witt vectors $W_2(K^{\circ\flat})$ if that is easier (again, not necessarily just prime ideals). Thanks!
