In Milne's book Lectures on étale cohomlogy(link), there is the following proposition.
Proposition 13.4. Let X be a connected regular scheme with generic point $\eta$. There is a short exact sequence on $X_{et}$ $$0 \to \mathbb{G}_m \to i_{\eta,\ast} \mathbb{G}_m \to \bigoplus_{\operatorname{codim}(Z)=1} i_{Z,\ast} \mathbb{Z} \to 0,$$ where the direct sum $Z$ runs over the Weil divisors of $X$.
I'm having trouble understanding what this sequence is, and why it is exact.
I am perfectly happy with this exact sequence in the Zariski topology, but I don't understand its construction in the étale topology. The first map is an adjunction; that is fine. To construct the second map, we want to define it on an étale neighborhood $\pi : U \to X$, so what we want a map $$\Gamma(\pi^{-1}(\eta), \mathbb{G}_m) \to \bigoplus_{Z} \Gamma(\pi^{-1}(Z), \mathbb{Z}_Z).$$ Now the left hand side is just the multiplicative group of invertible rational functions on $U$, and the right hand side is, well, a $\mathbb{Z}^{\pi_0(\pi^{-1} Z)}$ for each Weil divisor $Z$.
Given a rational function on $U$, I suspect this map will be something like "evaluating" its order of zero/pole at a connected component of $\pi^{-1} Z$. Because $U$ is regular (being étale over a regular scheme), it looks well-defined. But the issue I run into is that $\pi^{-1} Z$ can have connected components that have multiple irreducible components. For a concrete example, I could take $X = \mathbb{A}^2$, $Z$ a nodal curve in $X$, and create $U$ so that $\pi^{-1} Z$ is finite étale over $Z$ has two irreducible components that are normalizations of $Z$.
We could try and define the order as the sum of the orders of the irreducible components of $\pi^{-1} Z$. But even if we do that, the kernel has more things that what we want. We can have rational functions that have a pole of order $1$ on one irreducible component of $\pi^{-1} Z$ and a zero of order $1$ on the other component, so that they cancel out. This is definitely in the kernel, but not in $\Gamma(U, \mathbb{G}_m)$.
So the question is, is there something wrong with my reasoning? What is the correct definition of the maps, and why is the sequence exact?
On the other hand, the book seems to use this sequence only in the case when $X$ is a curve over an algebraically closed field. In that case, the above issue doesn't arise. This is what the Stacks project does.
What Milne states is that $$ 0\to \mathbb{G}_{m,X}\to i_{\eta*}\mathbb{G}_{m,\eta}\to \bigoplus_{\operatorname{codim}z=1}i_{z*}\mathbb{Z}\to 0 $$ is exact in the étale topology, where $i_z$ is the inclusion map $\{z\}=\operatorname{Spec} k(z)\to X$. For each étale morphism $\pi\colon U\to X$ and codim $1$ point $z\in X$ we have $$ \Gamma(U,i_{z*}\mathbb{Z}) = \bigoplus_{w\in \pi^{-1}(z)}\mathbb{Z}, $$ which is not equal to $\mathbb{Z}^{\pi_0(\pi^{-1}(\overline{\{z\}}))}$. Therefore the problem of "cancellation" does not occur.