# Inertia jumps

Let X be an algebraic stack. Let x_1, x_0 be points of X such that x_1 specializes to x_0 (here point means equivalence class of morphisms from spectra of fields). What can we say about the automorphism group schemes G_1 and G_0 of x_1 and x_0?

I wanted to write a bit about this question, and lead up to some related questions on higher algebraic stacks. But now I realize that (a) in the general case there is not a lot I can say, and (b) I haven’t though enough about this. Maybe you can help me out.

Let (U, R, s, t, c) is a groupoid in algebraic spaces such that X = [U/R]. Then there may exist a specialization of points u_1 -> u_0 of U such that u_i maps to x_i, but this is not always the case as examples of algebraic spaces show (for the unsuspecting reader we point out that in the stacks project an algebraic stack/space is defined with no separation conditions whatsoever). If this holds, then we see that the stabilizer group algebraic space G —> U has fibres G_{u_1} and G_{u_0} which are geometrically isomorphic to G_1 and G_0. This implies that dim(G_0) >= dim(G_1) for example.

Can we say anything more  if the generic stabilizer G_1 is trivial? In other words, given G_1 = \{1\} are there some G_0 which are “forbidden”?

Let’s reformulate the question in a slightly different form: Suppose that R is a valuation ring and G is a group algebraic space locally of finite type over R. Does there exist an algebraic stack X and a morphism Spec(R) —> X whose automorphism group scheme is G?

General remark: If G is locally of finite presentation and flat then the answer is yes, since in that case the quotient stack [Spec(R)/G] is algebraic.

Consider the case where R = k[[t]], char(k) = p > 0 and G = Spec(R[x]/(x^p, tx)) with group law given by addition. I.e., G is the group scheme whose special fibre is \alpha_p and whose general fibre is the trivial group scheme at the special fibre. Does G occur? The answer is yes. Namely, let \alpha_p act on affine 2-space over k by letting x act as the matrix (1 x) (0 1)
and let Spec(R) —> A^2 be given by t maps to (1, t). If you compute the automorphism scheme of this you get G.

Does such a construction work for every complete discrete valuation ring R and finite group scheme H over the residue field of R? If R is equicharacteristic p then a similar construction works, but if R has mixed characteristic I’m not so sure how to do this. Namely, if the group scheme has a flat deformation over R, then I think you can make it work, but if not, then I do not know how to construct a suitable algebraic stack. Do you?

There are noncommutative finite group schemes over fields of characteristic p which do not lift to characteristic zero. There are group schemes of order p^2 which do not lift, see paper by Oort and Mumford from 1968. I also think the kernel of frobenius on GL_n if p is not too small relative to n should not lift, but I do not know why I think so… So these may be good examples to try.