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?<\/p>\n
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.<\/p>\n
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.<\/p>\n
Can we say anything more\u00a0 if the generic stabilizer G_1 is trivial? In other words, given G_1 = \\{1\\} are there some G_0 which are “forbidden”?<\/p>\n
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?<\/p>\n
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.<\/p>\n
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 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?<\/p>\n 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.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading
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\nand let Spec(R) —> A^2 be given by t maps to (1, t). If you compute the automorphism scheme of this you get G.<\/p>\n