# Pullback of sheaf of roots of unity.

Given a closed immersion of integral (and maybe normal) schemes $$i^*:Z\hookrightarrow X$$. Is it true that $$i^*\mu_n$$ is isomorphic to $$\mu_n|_{Z}$$? Here by pullback we consider the sheaf of $$n$$-th root of unity as a sheaf on the small etale site. By the restriction to $$Z$$, I mean the sheaf of $$n$$-th root of unity naturally defined on $$Z$$.

• Thanks for your comment. Is the reason for this being true because $\mu_n$ is locally constant? It becomes constant on some etale cover. Commented Mar 20, 2021 at 15:43
• So by that logic it is true for $\mathbb{G}_m$ too. But let's assume $Z$ is a closed point corresponding to some maximal ideal. I think $i^*\mathbb{G}_m$ is the stalk of $\mathbb{G}_m$, which is given by $\mathbb{G}_m$ of the etale local ring (strict henselization of the loacl ring at the maximal ideal.). This seems to be different than $\mathbb{G}_m$ on a point ($\text{Spec}(k)$ for a field $k$, which is the residue field of the maximal ideal.) they have different global sections, as the local ring has more invertible elements. Commented Mar 20, 2021 at 16:17
• Yes you are right and I was too quick. This is definitively true if $n$ is invertible on the base scheme. I will give a thought if $n$ is not invertible. I'll remove my comments. Commented Mar 20, 2021 at 16:47

This isn't true in general. Let me give a counterexample first, and then discuss two cases in which it works.

Counterexample. Consider $$X=\operatorname{Spec} \mathbb Z_p[\zeta_p]$$, where $$\zeta_p$$ is a $$p$$-th root of unity, and let $$Z=\operatorname{Spec}\mathbb F_p$$, with $$i\colon Z\hookrightarrow X$$ being the inclusion of the special point. All étale $$\mathbb F_p$$-algebras are finite products $$\prod_i\mathbb F_{p^i}$$, hence they don't contain any non-trivial $$p$$-th root of unity. Thus, the stalk of $$\mu_{n,Z}$$ at any geometric point $$\overline{z}$$ of $$Z$$ is $$\{1\}$$. However, any étale $$\mathbb Z_p[\zeta_p]$$-algebra $$B$$ contains the element $$\zeta_p$$, and $$\zeta_p-1$$ is a nonzerodivisor by flatness. Hence also the stalk of $$\mu_{n,X}$$ at $$i(\overline{z})$$ contains the non-trivial root of unity $$\zeta_p$$. Since $$(i^*\mu_{n,X})_{\overline{z}}=(\mu_{n,X})_{i(\overline{z})}$$ (pullback of étale sheaves preserves stalks), this shows that $$i^*\mu_{n,X}$$ cannot coincide with $$\mu_{n,Z}$$.

And now for the two cases in which it works. Neither case needs that $$i$$ is a closed immersion.

Case 1. Assume $$n$$ is invertible on $$X$$ and thus on $$Z$$ too (I believe you already discussed that case in the now deleted comments, but let me spell it out for completeness). Then the canonical map $$i^*\mu_{n,X}\rightarrow \mu_{n,Z}$$ is always an isomorphism, even for $$i$$ not a closed immersion and $$X$$, $$Z$$ not reduced.

This can be seen as follows: $$\mu_{n,X}$$ is always represented by the scheme $$Y=\operatorname{\underline{Spec}}\mathcal O_X[T]/(T^n-1)$$, and similarly $$\mu_{n,Z}$$ is represented by the base change $$i^*Y$$. If $$n$$ is invertible, then $$Y$$ is étale over $$X$$ and (thus) $$i^*Y$$ is étale over $$Z$$. In this case it is completely formal to show that the induced map $$i^*h_Y\rightarrow h_{i^*Y}$$ on representable sheaves is an isomorphism. Indeed, by Yoneda's lemma, it suffices to check that $$\operatorname{Hom}_{\mathrm{Sh}(Z_{\mathrm{\acute{e}t}})}(h_{i^*Y},\mathcal F)\rightarrow \operatorname{Hom}_{\mathrm{Sh}(Z_{\mathrm{\acute{e}t}})}(i^*h_Y,\mathcal F)$$is an isomorphism for all sheaves $$\mathcal F\in \mathrm{Sh}(Z_{\mathrm{\acute{e}t}})$$. This follows from the calculation $$\operatorname{Hom}_{\mathrm{Sh}(Z_{\mathrm{\acute{e}t}})}(h_{i^*Y},\mathcal F)\cong\Gamma(i^*Y,\mathcal F)\cong\Gamma(Y,i_*\mathcal F)\cong\operatorname{Hom}_{\mathrm{Sh}(X_{\mathrm{\acute{e}t}})}(h_{Y},i_*\mathcal F)\cong \operatorname{Hom}_{\mathrm{Sh}(Z_{\mathrm{\acute{e}t}})}(i^*h_{Y},\mathcal F)\,.$$ The first and third isomorphism follow from Yoneda's lemma, the second holds by definition of $$i_*\mathcal F$$, and the fourth by the $$(i^*,i_*)$$-adjunction. In the first and the third isomorphism, we critically used that $$i^*Y$$ and $$Y$$ are étale over $$Z$$ and $$X$$, respectively. This explains why $$i^*\mathbb G_{m,X}\rightarrow \mathbb G_{m,Z}$$ may fail to be an isomorphism, as you note in the comments: $$\mathbb G_{m,X}$$ is representable by an $$X$$-scheme, but not by an étale one.

Case 2. $$X$$ and (thus) $$Z$$ have characteristic $$p$$. We'll only need that $$X$$ and $$Z$$ are reduced and $$i$$ can be any map. Moreover, the question is local, so we may assume $$X=\operatorname{Spec} A$$ and $$Z$$ are affine. Finally, $$\mu_{n,X}$$ is the direct sum of its subsheaves $$\mu_{\ell^{m}}$$, where $$\ell$$ is a prime divisor of $$n$$ and $$\ell^m$$ is the highest power of $$\ell$$ dividing $$n$$. For $$\ell\neq p$$, case 1 does it, hence we may assume $$n=p^m$$.

Note that every $$\mathbb F_p$$-algebra $$B$$ with a non-trivial $$p^m$$-th root of unity $$\zeta$$ is necessarily non-reduced, as $$0=\zeta^{p^m}-1=(\zeta-1)^{p^m}$$. But every étale algebra over the reduced ring $$A$$ is reduced again. For $$A$$ noetherian, this follows from the fact that a noetherian ring is reduced iff it satisfies Serre's conditions $$(R_0)$$ and $$(S_1)$$ (the little brother of Serre's normality criterion) and that the conditions $$(R_k)$$ and $$(S_k)$$ ascend along étale maps. In general, we can write any étale map $$A\rightarrow B$$ as a filtered colimit $$\operatorname{colimit} (A_\alpha\rightarrow B_\alpha)$$ of étale maps between finite type $$\mathbb Z$$-algebras. Replacing all $$A_\alpha$$ by their reductions, the above argument shows that $$B$$ can be written as a filtered colimit of reduced rings $$B_\alpha$$, whence $$B$$ is reduced itself.

Therefore, the étale sheaves $$\mu_{p^m,X}$$ and $$\mu_{p^m,Z}$$ vanish if $$X$$ and $$Z$$ are reduced of characteristic $$p$$, so $$i^*\mu_{p^m,X}\rightarrow \mu_{p^m,Z}$$ is trivially an isomorphism.