# Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?

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!

• Nontrivial colimits of sheaves in the category of presheaves are almost never sheaves before sheafification, so I doubt it. Jan 20, 2019 at 3:51
• 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 $\mathrm{H}^1_{et}(X, \mu _p)=0$.
– gdb
Jan 20, 2019 at 8:09
• 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.
– gdb
Jan 20, 2019 at 8:16
• Thanks for the comment, gdb. Were you assuming $X$ is reduced when you claimed $\mu_p$ is the zero sheaf? Jan 20, 2019 at 14:09
• 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? Jan 20, 2019 at 14:20

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 $$p$$.