TITLE: Witt vectors, the cotangent complex, and a solid construction of $B_{dR}^+$
QUESTION [12 upvotes]: In a remarkable lecture delivered on October 29th: New Foundations for functional analysis, Dustin Clausen suggests at the 40 minute mark a remarkable new construction interpretation of Fontaine's ring $B_{dR}^+$ making use of solid modules, the condensed analog of non-Archimedean completeness. This relies on a construction of the solid cotangent complex of a perfectoid field, which happens to be as simple as possible, a suspension of the field, and then proceeds to construct the desired period ring by analogy with the Witt vectors of a perfect ring. It's possible that I have misunderstood this story.
With the understanding that mathoverflow may not be the best place to push on unpublished results (feel free to close the question if it crosses the line), I wondered if anyone has a reference for or could offer a fully deformation theoretic construction of the Witt vectors, to at least have the other side of the analogy.
It seems to me that I can get a decent a priori construction of the Witt vectors by leveraging perfection (for cotangent vanishing and multiplicative lifting via p-adic contracting property of the Frobenius), but it might be difficult to get from here to the formulas, the $F$ and $V$ operators, the fact that when we begin with a field the result is a DVR. Excellent references abound, but I am curious whether there is any which begins and remains with the perspective of the cotangent complex.
The relevant blackboard is:

REPLY [5 votes]: Let me add to Z. M's answer, and note that Dustin has no reason to apologize at all: What he said is literally correct.
Namely, one can directly show that $L_{F/\mathbb Z}^\blacksquare$ is isomorphic to $F[1]$, for any perfectoid field (or even ring) $F$. Here are the steps:

One has $L_{F/\mathbb Z}^\blacksquare = L_{\mathcal O_F/\mathbb Z}^\blacksquare\otimes_{\mathcal O_F} F$. This is in fact true already before solidification, and basically formal.

The solidification $L_{\mathcal O_F/\mathbb Z}^\blacksquare$ is derived $p$-complete. This holds true more generally for any derived $p$-complete solid ring $A$ in place of $\mathcal O_F$. The key input is that solid tensor products preserve derived $p$-complete (connective) complexes of solid modules.

Thus, to show that $L_{\mathcal O_F/\mathbb Z}^\blacksquare\cong \mathcal O_F[1]$, it suffices to show that $L_{(\mathcal O_F/p)/\mathbb F_p}^\blacksquare\cong \mathcal O_F/p[1]$. In fact, the latter holds even before solidification.

Now $\mathcal O_F/p = \mathcal O_F^\flat/t$ for some perfect ring $\mathcal O_F^\flat$ and nonzerodivisor $t$, from which the computation follows easily (using the vanishing of the cotangent complex of perfect rings).


(These types of computations of cotangent complexes, and the avoidance of $A_{\mathrm{inf}}$, are very much as in my thesis. The new thing is really that we can directly characterize $B_{\mathrm{dR}}^+(F)$ as the universal solid pro-infinitesimal thickening of $F$: In classical language the required "completeness" of the thickening was extremely hard to formulate.)