Skip to main content
MathOverflow

Return to Question

replaced http://mathoverflow.net/ with https://mathoverflow.net/
Source Link
Community Bot
Community Bot
  • 1
  • 2
  • 3

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric PowersSmoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.

EDIT. As abx noted in the comments, my question is based on a false expectation. Give this, let me modify the question: how does one prove that $X^{[n]}$ is $k$-smooth by reducing to the $X = \mathbb{A}^1_k$ case? That is, how to carry out the reduction?

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.

EDIT. As abx noted in the comments, my question is based on a false expectation. Give this, let me modify the question: how does one prove that $X^{[n]}$ is $k$-smooth by reducing to the $X = \mathbb{A}^1_k$ case? That is, how to carry out the reduction?

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.

EDIT. As abx noted in the comments, my question is based on a false expectation. Give this, let me modify the question: how does one prove that $X^{[n]}$ is $k$-smooth by reducing to the $X = \mathbb{A}^1_k$ case? That is, how to carry out the reduction?

Notice removed Improve details by CommunityBot
Bounty Ended with no winning answer by CommunityBot
Notice added Improve details by Lisa S.
Bounty Started worth 50 reputation by Lisa S.
added 258 characters in body
Source Link
Lisa S.
Lisa S.
  • 2.7k
  • 11
  • 26

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.

EDIT. As abx noted in the comments, my question is based on a false expectation. Give this, let me modify the question: how does one prove that $X^{[n]}$ is $k$-smooth by reducing to the $X = \mathbb{A}^1_k$ case? That is, how to carry out the reduction?

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.

EDIT. As abx noted in the comments, my question is based on a false expectation. Give this, let me modify the question: how does one prove that $X^{[n]}$ is $k$-smooth by reducing to the $X = \mathbb{A}^1_k$ case? That is, how to carry out the reduction?

Source Link
Lisa S.
Lisa S.
  • 2.7k
  • 11
  • 26

Symmetric power of an etale map of curves

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.