Saturday, 3 April 2010

ag.algebraic geometry - 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 GrightarrowS,
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 mathbbZ/2mathbbZ over S,
let s be some fixed closed point of S, and let G:=Hsetminus1s,
where 1s is the non-zero element of the fibre (mathbbZ/2mathbbZ)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.)

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