Let X be a DM stack over a field k. We follow the definition in Laumon and Moret-Bailly's book, so that its diagonal is quasi-compact (and hence diagonal is of finite type). Then is the diagonal necessarily finite?


**Edit:** I meant to ask if the inertia $I\to X$ is finite. Recall that the inertia stack $I$ is defined to be the 2-fiber product of $X$ with $X$ over $X\times X,$ where the two maps $X\to X\times X$ are both the diagonal map. This question is equivalent (I think) to the following. Let $G\to S$ be an etale $S$-group scheme of finite type, where $S$ is a $k$-scheme of finite type. Then $G$ is finite over $S.$ 



Is the inertia stack of a Deligne-Mumford stack always finite?
I'm not sure about the suggested equivalence in the last two sentences of your question, but at least the statement about etale group schemes has a negative answer.


That is, it is possible to have an etale group scheme $G \rightarrow S$,
with $G$ and $S$ both finite type over a field $k$, but $G$ not finite over $S$.


For example, let $H$ be the constant group scheme ${\mathbb Z}/2{\mathbb Z}$ over $S$,
let $s$ be some fixed closed point of $S$, and let $G := H \setminus 1\_s,$
where $1\_s$ is the non-zero element of the fibre $({\mathbb Z}/2{\mathbb Z})\_s$.


Then $G$ is open in $H$, hence etale over $S$. Assuming that $S$ is positive dimensional,
it is certainly not finite (we deleted one
point of one fibre), and it is a subgroup scheme of $H$. (If $T$ is an $S$-scheme,
then $G(T)$ is the subgroup of $H(T)$ consisting of points whose values at points of $T$ lying
over $s$ are trivial.)


