TITLE: Is the moduli of formal groups smooth?
QUESTION [21 upvotes]: There is a notion of smooth stack in "homotopical algebraic geometry 2" and a notion of a cotangent complex for certain kinds of stacks (including representable stacks). I have two questions. 

Is the moduli stack of one dimensional commutative formal groups of height less than or equal to $n$ smooth in this sense?
Is the formal moduli stack of deformations of a one dimensional commutative formal group over a perfect field of characteristic $p$ of height equal to $n$ smooth in this sense? Here I mean the quotient stack of deformations by the action of the stabilizer group.

It seems to me that (2) should be true as this is the "ind-étale" quotient of Lubin-Tate. Question (1) should be true because of locality for smoothness, but a reference to the result (or at least the relevant tools in the concrete from) would be nice.

REPLY [4 votes]: Q1: I did not find the definition of a smooth stack in HAG2 (it would be nice if you provide a concrete citation!), but the definition of smoothness I know is: A morphism $X \to Y$ of algebraic stacks is smooth if there is a commutative diagram 
$\require{AMScd}$
\begin{CD}
    U @>f>> V\\
    @V g V V @VV h V\\
    X @>>> Y
    \end{CD}
where $U$ and $V$ are schemes (or algebraic spaces) and $f,h,g$ are smooth and $g$ is surjective. (see http://stacks.math.columbia.edu/tag/075U)
In particular, we can take in our example $X = \mathcal{M}_{FG}^{\leq n}$, $V = Y = Spec \mathbb{Z}_{(p)}$ and $U = Spec \mathbb{Z}_{(p)}[v_1,\dots, v_n, v_n^{-1}]$. So at least in this sense of smooth, the answer is yes: $\mathcal{M}_{FG}^{\leq n}$ is smooth. 
Edit: As explained by Jacob Lurie in the commments, the map $g$ is not smooth and my argument fails. I am sorry for being careless.