Question about description of $H^1_{et}(X, \mu_n)$

Question

On this page of Stacks project the pair $(\mathcal{L},\alpha)$ mentioned inside is described as follows:

Fix an integer $n$ and a scheme $X$. We form a pair $(\mathcal{L},\alpha)$ where $\mathcal{L}$ is an invertible sheaf on $X$ and $\alpha: \mathcal{L}^{\otimes n} \rightarrow \mathcal{O}_X$ is a trivialization of the $n$-th tensor power of $\mathcal{L}$. We can show that the collection of isomorphism classes of such pairs form an abelian group.

The main result of that page is to show that the etale cohomology group $H^1(X,\mu_n)$ can be identified with the group described above. What I am confused about is that there must be some restrictions on $\mathcal{L}$ and the definition of $\alpha$. Is $\mathcal{L}$ simply any invertible sheaf on $X$? It seems to be the case from the way it was written. But then, does that mean that for any $n$, we have a trivialization $\mathcal{L}^{\otimes n} \rightarrow \mathcal{O}_X$?

Answer 1

this trivialisation $\mathcal{L}^n \cong \mathcal{O}_X$ simply means that your étale line bundle $\mathcal{L}$ is a so called $n$-torsion line bundle. In fancy language, $\mathcal{L} \in \mathbf{Pic}_{ét}(X)[n]$.

But be aware of the fact that all of this takes place in the category of étale sheaves on $X$! A line bundle $\mathcal{L}$ is not defined on opens of $X$ but on étale covers $X'\rightarrow X$.

The relationship with the group of roots of unity comes from two facts:

1. First $\mathbf{Pic}_{ét}(X)\cong H_{ét}^1(X,\mathbf{G}_m)$ tells you that (étale) cocycles define uniquely your line bundles (or the transition functions thereof, this is basically Cech cohomology). $\mathbf{G}_m$ is the sheaf of units on X.
2. there is an exact (Kummer sequence) $1 \rightarrow \mu_n \rightarrow \mathbf{G}_m \xrightarrow{g\mapsto g^n} \mathbf{G}_m \rightarrow 1$ of étale sheaves on $X$. Surjectivity follows basically from the fact that on stalks the local rings are strictly Henselian. Now apply the LES of cohomology to see that the kernel of raising a line bundle $\mathcal{L} $to its $n$th power $\mathcal{L^n}$ is $1 \rightarrow H_{ét}^1(X,\mu_n) \rightarrow \mathbf{Pic}(X) \xrightarrow{n} \mathbf{Pic}(X)$. This gives your claim.
3. intuition: these line bundles are defined by equations $f_i \in\mu_n$, so if you multiply them by $n$ they become the trivial sheaf $\mathcal{O}_X$.