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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.

  1. Is the moduli stack of one dimensional commutative formal groups of height less than or equal to $n$ smooth in this sense?

  2. 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.

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  • $\begingroup$ I'd like to know the answer to this question too. $\endgroup$
    – Nat Stapleton
    Commented Oct 17, 2015 at 14:48
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    $\begingroup$ Q1 can be broken into (a) translate the problem into a question about the structure of $E(k)_*E(k)$ (or $(v_k^{-1}BP)_*(v_k^{-1}BP)$) for $k\leq n$ and (b) answer that question. I will attempt to do (b) if someone else does (a). $\endgroup$
    – Neil Strickland
    Commented Oct 18, 2015 at 18:25

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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.

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    $\begingroup$ The map $g$ is not smooth. I also don't know what the OP has in mind by smoothness, but the usual definition fails for $X = \mathcal{M}^{\leq n}$. If $X$ were smooth over $\mathbb{Z}_{(p)}$ (by the usual definition), then it would have a perfect cotangent complex. But the cotangent complex of $X$ isn't perfect: if you take the fiber at a point corresponding to a formal group of height $m > 0$, then you get an $(m-1)$-dimensional vector space. For a perfect complex, the Euler characteristic of the fibers is locally constant. $\endgroup$
    – Jacob Lurie
    Commented Oct 20, 2015 at 12:52
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    $\begingroup$ @Tomer, I'm not sure what definition you have in mind. The moduli stack admits an pro-etale surjection from the spectrum of a finite field, as you point out. If this qualifies for you as "smooth", then the answer is yes. If "smooth" means "admits a smooth surjection from something smooth" (as in Lennart's answer), then the answer is no. $\endgroup$
    – Jacob Lurie
    Commented Oct 20, 2015 at 15:34
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    $\begingroup$ @JacobLurie, My situation is that I have a square-zero extension of the formal stack MFG^{n} by some suspension of some coherent sheaf on MFG^{n} and (presented as a sheaf of square -zero CDGA's) I want to show that it splits. I found a theory of cotangent complexes for stacks in HAG2 (section 1.4). But it is stated in a very general way, and was hoping for a more direct reference useful for this case. $\endgroup$
    – Tomer Schlank
    Commented Oct 21, 2015 at 8:59
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    $\begingroup$ @Tomer a) The moduli stack of formal groups (no need to mention height) is formally smooth: that is, it satisfies the usual infinitesimal lifting criterion. This follows immediately from the fact that the Lazard ring is polynomial. b) The infinitesimal lifting criterion only applies in the affine case, so it doesn't address your problem. c) The moduli stack of formal groups has a well-defined cotangent complex which will let you rephrase your question in homological terms, provided that by "CDGA" you really mean "simplicial commutative ring"... $\endgroup$
    – Jacob Lurie
    Commented Oct 21, 2015 at 14:24
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    $\begingroup$ (continued) d) The moduli stack of formal groups of height exactly n (regarded as a reduced locally closed substack of the entire moduli stack) has trivial cotangent complex over $\mathbf{F}_p$ (if $n > 0$), so any square-zero extension of that has a unique splitting (provided that the extension lives in simplicial $\mathbf{F}_p$-algebras). $\endgroup$
    – Jacob Lurie
    Commented Oct 21, 2015 at 14:25

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