Witt vectors, the cotangent complex, and a solid construction of $B_{dR}^+$ 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With the understanding that mathoverflow may not be the best place to push on unpublished results (feel free to close the question if it crosses the line), I wondered if anyone has a reference for or could offer a fully deformation theoretic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: 
 A: Let me point out that, the original statement in the video of Dustin Clausen's
talk is essentially correct (although it might have been inadvertent,
according to his answer), thanks to Arpon Raksit's paper Hochschild homology and the derived de Rham cohomology revisited (arXiv link).
For sake of simplicity, we fix a base ring $R$. Roughly speaking, there is a
nonconnective version of animated $R$-algebras due to Bhatt–Mathew, which is
named after derived (commutative) $R$-algebras in Raksit's paper (§4). The
point is that, animated $R$-algebras are precisely modules over the monad
$\mathbb{L}{\operatorname{Sym}} : D (R)_{\geq 0} \rightarrow D
(R)_{\geq 0}$ on the $\infty$-category of connective $R$-module spectra (or
connective $R$-modules, by abuse of terminology). Using Goodwillie calculus,
one can extend to the whole $\infty$-category $D (R)$ of $R$-modules,
obtaining a monad $\mathbb{L}{\operatorname{Sym}} : D (R)
\rightarrow D (R)$, and derived $R$-algebras are modules over this monad,
whose $\infty$-category will be denoted by
${\operatorname{DAlg}}_R$.
This formalism applies more generally to derived algebraic contexts (4.2.1) in
place of $D (R)$, and in particular, to the $\infty$-category
$\widehat{{\operatorname{DF}}}^{\geq 0} (R)$ of completely
(nonnegatively decreasingly) filtered $R$-modules, obtaining a monoid, whose
modules are completely filtered derived $R$-algebras, or equivalently,
nonnegative $h_-$-differential graded derived $R$-algebras
${\operatorname{DG}}_-^{\geq 0} {\operatorname{DAlg}}_R$
due to 5.1.5.
Similar to 5.3.2 (and even easier), the functor
$\operatorname{gr}^0\colon{\operatorname{DG}}_-^{\geq 0} {\operatorname{DAlg}}_R
\rightarrow {\operatorname{DAlg}}_R$ admits a left adjoint
${\operatorname{Cpl}} : {\operatorname{DAlg}}_R
\rightarrow {\operatorname{DG}}_-^{\geq 0}
{\operatorname{DAlg}}_R$. There are two ways to understand this functor:

*

*By adjunction, $\operatorname{Cpl}(A)$ is the initial completely filtered derived $R$-algebra $B$ equipped with a map $A\to\operatorname{gr}^0(B)$ of derived $R$-algebras. In some sense, this is closely related to the final object in the "pro-infinitesimal site" of $A/R$ (reminder: the site is opposite to the category of rings);

*If we start with a quotient $R / I$ where
$I \subseteq R$ is an ideal generated by a Koszul-regular sequence, then the
image under this left adjoint is the $I$-completion $R_I^{\wedge}$ along with
the $I$-adic filtration (the readers could convince themselves that this
object is somehow initial), so this functor should be understood as a variant
of adic completion, which justifies the notation.

The proof of 5.3.6 essentially leads to the following:
Proposition. Given a derived
$R$-algebra $A$, the $n$-th associated graded piece
${\operatorname{gr}}^n ({\operatorname{Cpl}} (A))$ is
given by the symmetric product $\mathbb{L}{\operatorname{Sym}}_A^n
({\operatorname{gr}}^1 ({\operatorname{Cpl}} (A)))$ (and
in particular, ${\operatorname{gr}}^0
({\operatorname{Cpl}} (A)) = A$), and we have an equivalence
$\mathbb{L}_{A / R} \simeq
\Sigma {\operatorname{gr}}^1 ({\operatorname{Cpl}} (A))$ of $A$-modules where $\mathbb{L}_{A / R}$ is the
(algebraic) cotangent complex of $R$.
In particular, consider the case that $R =\mathbb{Z}$. If $A =\mathbb{F}_p$,
or more generally, $A$ is given by a perfect $\mathbb{F}_p$-algebra, then the
cotangent complex $\mathbb{L}_{A / R}$ has
${\operatorname{Tor}}$-amplitude in $[1, 1]$, therefore the
$A$-module ${\operatorname{gr}}^1 ({\operatorname{Cpl}}
(A))$, and consequently the $A$-modules ${\operatorname{gr}}^{\ast}
({\operatorname{Cpl}} (A))$, are flat. It follows that the
underlying derived ring ${\operatorname{Fil}}^0
({\operatorname{Cpl}} (A))$ of ${\operatorname{Cpl}}
(A)$ is concentrated in degree $0$, and there is a surjective map
${\operatorname{Fil}}^0 ({\operatorname{Cpl}} (A))
\rightarrow {\operatorname{gr}}^0 ({\operatorname{Cpl}}
(A)) = A$ of rings whose kernel $I$ is quasiregular, and by Quillen's result,
${\operatorname{Fil}}^n ({\operatorname{Cpl}} (A))$ are
simply given by $I^n$.
On the other hand, if we consider a reasonable ($\infty$-)category of
deformations of the $R$-algebra $A$, say the category of surjective maps $B
\twoheadrightarrow A$ of $R$-algebras whose kernel is finitely generated (the
completeness with respect to a non-finitely generated ideal is usually
pathologic), then this ($\infty$-)category should be a full subcategory
${\operatorname{Def}}_R$ of $\{ S \in
{\operatorname{DG}}_-^{\geq 0} {\operatorname{DAlg}}_R |
{\operatorname{gr}}^0 (S) \simeq A \}$. Having this in mind, when
the cotangent complex $\mathbb{L}_{A / R}$ is, as an $A$-module, equivalent to
a shift $P [1]$ where $P$ is a finite projective $A$-module, then
${\operatorname{Cpl}} (A)$ lies in this subcategory
${\operatorname{Def}}_R$ and the universal property follows.
A: 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.)
A: $\newcommand\BdR{B_\text{dR}^+}\newcommand\Ainf{A_\text{inf}}$Thank you for the question! Actually, you've caught me out.  Though I didn't realize it at the time, I was indeed cheating and should have said things more carefully.  Hopefully I can partially atone here.  Below I will describe two reasons why I was cheating, one mathematical and one moral.
Edit: In his answer, Z.M explains that the mathematical reason does not apply, and in Peter's answer he explains that the moral reason does not apply!  So I was overly pessimistic on both counts, and in spite my ignorance my claims from my talk are perfectly substantiated!
But for now let me just give the take-away:
I should not have implied that we give a new construction of $\BdR$.  Rather I should have said that we give a universal property for $\BdR$: it is the universal pro-nilpotent thickening in solid rings.
The two reasons why I was cheating:

*

*I don't think it's actually true that the contangent complex being $F[1]$ implies, for purely formal reasons, a full universal one-parameter formal deformation.  Certainly it does give the first-order deformation.  But that only uses that the $H_1$ is $F$ and the $H_0$ is $0$, not that the others vanish.  Using the vanishing of the cotangent complex in higher degrees, you do get a good deal of knowledge about higher deformations by iteratively using transitivity triangles, and it feels like it should be saying a lot, but as far as I know it's still incomplete.  To get around this problem and actually make a full one-parameter formal deformation, one option is to work relative to an already-existing one.  So in the perfect field case, you can use that $\mathbb Z$ is already a one-parameter deformation of $\mathbb F_p$ and that the cotangent complex of $k$ over $\mathbb F_p$ vanishes; this is the approach discussed in section six of Bhargav's notes, which is where I learned it.  In the perfectoid case, you at least see in the same way that if you construct $\BdR$ by hand for a base perfectoid, then you get it for all perfectoids over that one.  Not as snazzy as what I claimed, but it's also not nothing: for example, $\Ainf$ and $\BdR$ for the $p$-cyclotomic extension are very explicit.


*Even if it were true that you get the construction for formal reasons from the cotangent calculation, the way I know how to get the cotangent calculation uses $\Ainf$.  So it would still be morally, if not mathematically, incorrect to claim this is a new construction of $\BdR$.
