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Myshkin
Myshkin
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Question: Is the moduli of formal groups smooth?

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-'etale"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.

Question: Is the moduli of formal groups smooth?

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-'etale" 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.

Is the moduli of formal groups smooth?

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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Tomer Schlank
Tomer Schlank
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Question: Is the moduli of formal groups smooth?

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-'etale" 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.