Neron model of multiplicative group scheme Let $R$ be a discrete valuation ring and $K$ be its fraction field. Consider the multiplicative group scheme $\mathbb{G}_m$, as scheme $\mathbb{G}_m=Spec K[T,T^{-1}]$. How can we construct the Néron model of $\mathbb{G}_m$? Is it $Spec R[T,T^{-1}]$?
I have read it in Bosch's book chapter 10 example 5, but don't really understand the general steps. It is said the the Néron model is only of locally finite type and not of finite type.
 A: As described in the comments, taking $\operatorname{Spec}(R[T,T^{-1}])$ will not work. Instead you have to do a glueing process, which is described in the reference you mentioned. Let me recall the construction, which should also address your point about being locally of finite type, but not of finite type (Recall finite type=locally of finite type + quasi-compact).
Start with $\mathbb{Z}$-copies of $\mathbb{G}_{m,R}$, i.e. the scheme
$$ \coprod_{i\in \mathbb{Z}} X_i$$
where each $X_i\cong \mathbb{G}_{m,R}$ (this is $\pi^i\mathbb{G}_{m,R}$ in the notation of Bosch). Now you have to glue these together:
We identify all $X_i\times_R K\cong \mathbb{G}_{m,K}$ with each other via maps $\phi_{ij}:X_i\times_R K\xrightarrow{\cdot \pi^{j-i}} X_j\times_R K.$ This is an isomorphism, since $\pi$ is now invertible over $K$. Thus your Neron model is
$$ X=(\coprod_{i\in \mathbb{Z}} X_i)/\sim,$$
where $x_i\sim x_j$ if $\phi_{ij}(x_i)=x_j$, or equivalently $\phi_{ji}(x_j)=x_i$.
Now we see that $$ X\to \operatorname{Spec}(R)$$
is indeed locally of finite type: $X$ is covered by the $X_i\cong \mathbb{G}_{m,R}$ which are of finite type over $R$.
But it is not quasi-compact: For each $X_i$ we have a closed subscheme $Z_i$ isomorphic to $\mathbb{G}_{m,k}\subset \mathbb{G}_{m,R}$, where $k=R/\pi$ is the residue field, and these are by construction not glued together. As a consequence
$$ Z_i\cap X_j=\emptyset$$
whenever $i\neq j$. So the covering of $X$ by the (image of the) $X_i$ for all $i\in \mathbb{Z}$ doesn't admit a finite subcover: Omitting one $X_i$ means $Z_i$ will not be contained in your cover anymore.
