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I apologize if the question is bit trivial for mathoverflow, but I asked on stack exchange a while ago and haven't got any answer.

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 on the category of $k$-schemes?

If $U \rightarrow X$ is an étale cover, I can prove the equalizer sequence $\mathbb{G}_m / \mu_p(X) \rightarrow \mathbb{G}_m / \mu_p (U) \rightarrow \mathbb{G}_m / \mu_p(U\times_X U)$ is exact when X is reduced. In fact, if $s \in \mathcal{O}_X(U)^\times$ and $(\frac{s\otimes 1}{1\otimes s})^p = 1$, since $U\times_X U$ is again reduced we can deduce $s\otimes 1 = 1\otimes s$ hence $s\in \mathcal{O}_X(X)^\times$. 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!

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  • $\begingroup$ Nontrivial colimits of sheaves in the category of presheaves are almost never sheaves before sheafification, so I doubt it. $\endgroup$
    – Harry Gindi
    Commented Jan 20, 2019 at 3:51
  • $\begingroup$ A quotient presheaf $\mathbf{G}_m / \mathbf \mu_p$ is a sheaf if and only if a natural map $\mathrm{H}^1_{et}(X, \mathbf \mu_p) \to \mathrm{H}^1_{et}(X, \mathbf G_m)$ is injective. In particular, this hold when $\mathrm{H}^1_{et}(X, \mu _p)=0$. $\endgroup$
    – gdb
    Commented Jan 20, 2019 at 8:09
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    $\begingroup$ Note that an example in your question satisfy this property simply because an etale sheaf $\mu_p$ is isomorphic to the zero sheaf (in the small etale site) provided that $X$ is an $\mathbf F_p$-scheme. The reason is that any etale $X$-scheme $U$ is reduced. Indeed, a section $s\in \mu_p(U)$ gives you an element $x\in \mathscr O(U)^*$ s.t. $x^p=1$. Hence, $(x-1)^p=0$ (because we are in char. p!). But this means that $x-1$ is a non-trivial nilpotent in $\mathscr O(U)$. Contradiction. $\endgroup$
    – gdb
    Commented Jan 20, 2019 at 8:16
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    $\begingroup$ Thanks for the comment, gdb. Were you assuming $X$ is reduced when you claimed $\mu_p$ is the zero sheaf? $\endgroup$
    – Taisong Jing
    Commented Jan 20, 2019 at 14:09
  • $\begingroup$ Just so I try to follow the question: what do you mean when you say a sheaf $F$ is representable by something via a morphism? $\endgroup$
    – Qfwfq
    Commented Jan 20, 2019 at 14:20

1 Answer 1

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Let $E$ be an elliptic curve with a $p$-torsion point $Q$. Let $X= E \times (\operatorname{Spec} \mathbb F_p[\epsilon]/\epsilon^2) \times \mathbb G_m$. Let $\sigma$ be the automorphism of $X$ that sends $(P,\epsilon,b)$ to $(P+Q, \epsilon, (1+\epsilon)b)$. Then $\sigma$ has order $p$ and no fixed points, so the induced map to the quotient $Y=X/\sigma$ is etale.

The section $b$ of $\mathbb G_m$ on $Y$ is $\sigma$-invariant when viewed as a section of $\mathbb G_m/\mu_p$. Were $\mathbb G_m / \mu_p$ a sheaf on $X$, it would descend to a section of the presheaf $\mathbb G_m/\mu_p$ on $Y$, which would arise from a section of $\mathbb G_m$ on $Y$, which would lift to a $\sigma$-invariant section of $\mathbb G_m$ on $X$.

But no such section exists, as these would simply be polynomials in $b, b^{-1}$ and $\epsilon$ which mod $\epsilon$ use only $p$-divisible powers of $p$, hence restricted to the closed set $\epsilon=0$, where $\mu_p$ has no nontrivial sections, cannot equal $p$.

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