Let k be a field. Let G be a group scheme over k which is locally of finite type. Then X = [Spec(k)/G] is an algebraic stack over k (see for example Lemma Tag 06PL).

Let x_0 be the obvious 1-morphism Spec(k) —> X. Let’s look at the associated deformation problem, which in the stacks project is a category cofibred in groupoids

F = F_{X, k, x_0} —> C_k = (Artinian local k-algebras with residue field k)

see Section Tag 07T2. OK, so I was thinking about tangent spaces earlier today and it occurred to me that it is already somewhat fun to consider the example above. Namely, what is the tangent space TF in the situation above?

Your initial reaction might be “it is zero”. If you are a characteristic zero person, then you would be right, but before you read on: can you prove it?

Yeah, so the answer is that it is zero if G is a smooth group scheme over k (which is always the case in characteristic zero, see Lemma Tag 047N). Triviality of TF means that for every pair (T, t_0) where T is a G-torsor over Spec(k[ε]) and t_0 ∈ T(k), there exists a t ∈ T(k[ε]) which reduces to t_0 modulo ε. A torsor for a smooth group scheme is smooth. Hence the infinitesimal lifting criterion of smoothness implies that TF = 0.

But what if G isn’t smooth? In that case TF is always nontrivial. Namely, if TF = 0, then Spec(k) —> X is smooth (argument omitted) which isn’t true because Spec(k) x_X Spec(k) = G. I think that in general

dim(TF) + dim(G) = embedding dimension of G

but I haven’t tried to prove it or look it up. As usual, I welcome suggestions, comments, references, etc.