Is the quotient presheaf $mathbb{G}_m/mu_p$ an étale sheaf?
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Let $k$ be a field of characteristic $p > 0$. Consider the multiplicative group scheme $mathbb{G}_m$ and its subgroup $mu_p$, the $p$-th roots of unity. It is well known that the quotient presheaf $mathbb{G}_m / mu_p$ is not a sheaf in fppf topology, and its sheafification in fppf topology is representable by $mathbb{G}_m$ via the morphism $mathbb{G}_m xrightarrow{cdot mapsto cdot^p} mathbb{G}_m$. However, is the quotient presheaf $mathbb{G}_m / mu_p$ an étale sheaf?
If $U rightarrow X$ is an étale cover, I can prove the equalizer sequence $mathbb{G}_m / mu_p(X) rightarrow mathbb{G}_m / mu_p (U) rightarrow mathbb{G}_m / mu_p(Utimes_X U)$ is exact when X is reduced. In fact, if $s in mathcal{O}_X(U)^times$ and $(frac{sotimes 1}{1otimes s})^p = 1$, since $Utimes_X U$ is again reduced we can deduce $sotimes 1 = 1otimes s$ hence $sin mathcal{O}_X(X)^times$. However I don't know to prove the general case when $X$ is not reduced. I tried several ways to formulate an induction, but couldn't work it out.
Appreciate any hints towards a proof or a counterproof. Thank you!
algebraic-geometry sheaf-theory etale-cohomology
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
$endgroup$
add a comment |
$begingroup$
Let $k$ be a field of characteristic $p > 0$. Consider the multiplicative group scheme $mathbb{G}_m$ and its subgroup $mu_p$, the $p$-th roots of unity. It is well known that the quotient presheaf $mathbb{G}_m / mu_p$ is not a sheaf in fppf topology, and its sheafification in fppf topology is representable by $mathbb{G}_m$ via the morphism $mathbb{G}_m xrightarrow{cdot mapsto cdot^p} mathbb{G}_m$. However, is the quotient presheaf $mathbb{G}_m / mu_p$ an étale sheaf?
If $U rightarrow X$ is an étale cover, I can prove the equalizer sequence $mathbb{G}_m / mu_p(X) rightarrow mathbb{G}_m / mu_p (U) rightarrow mathbb{G}_m / mu_p(Utimes_X U)$ is exact when X is reduced. In fact, if $s in mathcal{O}_X(U)^times$ and $(frac{sotimes 1}{1otimes s})^p = 1$, since $Utimes_X U$ is again reduced we can deduce $sotimes 1 = 1otimes s$ hence $sin mathcal{O}_X(X)^times$. However I don't know to prove the general case when $X$ is not reduced. I tried several ways to formulate an induction, but couldn't work it out.
Appreciate any hints towards a proof or a counterproof. Thank you!
algebraic-geometry sheaf-theory etale-cohomology
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
$endgroup$
$begingroup$
Now posted also on MathOverflow: Is the quotient presheaf $mathbb{G}_m/mu_p$ an étale sheaf? I think that this answer contains very reasonable advice about cross-posting. The most important thing is probably to link both copies to each other. (Of course, you can have a look at other discussions about cross-posting, too.)
$endgroup$
– Martin Sleziak
Jan 23 at 7:52
add a comment |
$begingroup$
Let $k$ be a field of characteristic $p > 0$. Consider the multiplicative group scheme $mathbb{G}_m$ and its subgroup $mu_p$, the $p$-th roots of unity. It is well known that the quotient presheaf $mathbb{G}_m / mu_p$ is not a sheaf in fppf topology, and its sheafification in fppf topology is representable by $mathbb{G}_m$ via the morphism $mathbb{G}_m xrightarrow{cdot mapsto cdot^p} mathbb{G}_m$. However, is the quotient presheaf $mathbb{G}_m / mu_p$ an étale sheaf?
If $U rightarrow X$ is an étale cover, I can prove the equalizer sequence $mathbb{G}_m / mu_p(X) rightarrow mathbb{G}_m / mu_p (U) rightarrow mathbb{G}_m / mu_p(Utimes_X U)$ is exact when X is reduced. In fact, if $s in mathcal{O}_X(U)^times$ and $(frac{sotimes 1}{1otimes s})^p = 1$, since $Utimes_X U$ is again reduced we can deduce $sotimes 1 = 1otimes s$ hence $sin mathcal{O}_X(X)^times$. However I don't know to prove the general case when $X$ is not reduced. I tried several ways to formulate an induction, but couldn't work it out.
Appreciate any hints towards a proof or a counterproof. Thank you!
algebraic-geometry sheaf-theory etale-cohomology
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
$endgroup$
Let $k$ be a field of characteristic $p > 0$. Consider the multiplicative group scheme $mathbb{G}_m$ and its subgroup $mu_p$, the $p$-th roots of unity. It is well known that the quotient presheaf $mathbb{G}_m / mu_p$ is not a sheaf in fppf topology, and its sheafification in fppf topology is representable by $mathbb{G}_m$ via the morphism $mathbb{G}_m xrightarrow{cdot mapsto cdot^p} mathbb{G}_m$. However, is the quotient presheaf $mathbb{G}_m / mu_p$ an étale sheaf?
If $U rightarrow X$ is an étale cover, I can prove the equalizer sequence $mathbb{G}_m / mu_p(X) rightarrow mathbb{G}_m / mu_p (U) rightarrow mathbb{G}_m / mu_p(Utimes_X U)$ is exact when X is reduced. In fact, if $s in mathcal{O}_X(U)^times$ and $(frac{sotimes 1}{1otimes s})^p = 1$, since $Utimes_X U$ is again reduced we can deduce $sotimes 1 = 1otimes s$ hence $sin mathcal{O}_X(X)^times$. However I don't know to prove the general case when $X$ is not reduced. I tried several ways to formulate an induction, but couldn't work it out.
Appreciate any hints towards a proof or a counterproof. Thank you!
algebraic-geometry sheaf-theory etale-cohomology
algebraic-geometry sheaf-theory etale-cohomology
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
asked Jan 10 at 3:40
Taisong JingTaisong Jing
1161
1161
$begingroup$
Now posted also on MathOverflow: Is the quotient presheaf $mathbb{G}_m/mu_p$ an étale sheaf? I think that this answer contains very reasonable advice about cross-posting. The most important thing is probably to link both copies to each other. (Of course, you can have a look at other discussions about cross-posting, too.)
$endgroup$
– Martin Sleziak
Jan 23 at 7:52
add a comment |
$begingroup$
Now posted also on MathOverflow: Is the quotient presheaf $mathbb{G}_m/mu_p$ an étale sheaf? I think that this answer contains very reasonable advice about cross-posting. The most important thing is probably to link both copies to each other. (Of course, you can have a look at other discussions about cross-posting, too.)
$endgroup$
– Martin Sleziak
Jan 23 at 7:52
$begingroup$
Now posted also on MathOverflow: Is the quotient presheaf $mathbb{G}_m/mu_p$ an étale sheaf? I think that this answer contains very reasonable advice about cross-posting. The most important thing is probably to link both copies to each other. (Of course, you can have a look at other discussions about cross-posting, too.)
$endgroup$
– Martin Sleziak
Jan 23 at 7:52
$begingroup$
Now posted also on MathOverflow: Is the quotient presheaf $mathbb{G}_m/mu_p$ an étale sheaf? I think that this answer contains very reasonable advice about cross-posting. The most important thing is probably to link both copies to each other. (Of course, you can have a look at other discussions about cross-posting, too.)
$endgroup$
– Martin Sleziak
Jan 23 at 7:52
add a comment |
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$begingroup$
Now posted also on MathOverflow: Is the quotient presheaf $mathbb{G}_m/mu_p$ an étale sheaf? I think that this answer contains very reasonable advice about cross-posting. The most important thing is probably to link both copies to each other. (Of course, you can have a look at other discussions about cross-posting, too.)
$endgroup$
– Martin Sleziak
Jan 23 at 7:52