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## Torsors for finite group schemes

Let $k$ be a field of characteristic $p > 0$ (assume $k$ is perfect if it helps). Let $G$ be a connected finite group scheme of height one over $k$. Then $G$ is determined by its Lie algebra $\mathfrak{g}$, as a restricted Lie algebra: it can be recovered as the spectrum of the Hopf algebra dual to the restricted universal enveloping algebra $U( \mathfrak{g} )$.

Let $R$ be a local Artinian $k$-algebra with residue field $R / \mathfrak{m} \simeq k$. I would like to understand the category of $G$-torsors over $R$ (with respect to the flat topology, say) which are trivialized over $R / \mathfrak{m}$. This category feels very infinitesimal,'' so it seems reasonable to expect that there is a way to describe its objects concretely in terms of linear-algebraic data related to $\mathfrak{g}$ (and without ever mentioning the flat topology). Can this be done? (And if so, how?)

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Jacob, in the "toy" case of commutative $G$ (perhaps even $\alpha_p$) is there an answer you have in mind, or at least desired features of such an answer? – BCnrd Nov 20 2010 at 0:34
I don't know how to answer my question even in that case. Of course, when G is given as the kernel of Frobenius on a smooth group scheme, I can describe G-torsors in terms of the smooth group scheme, as mentioned in the answer below. But that's cheating: I want a description in terms of the Lie algebra. If it helps, it may be sufficient for my application to treat the "toy case" at the other extreme, where g is a free restricted Lie algebra (so G is not a finite group scheme, and some care should be taken with the meaning of "torsor", but I believe the question is still sensible). – Jacob Lurie Nov 20 2010 at 13:52

Recall that if $G\rightarrow S$ is a flat group scheme, then a $G$-torsor is an $S$-scheme $X\rightarrow S$ with a $G$-action $G\times_SX\rightarrow X$ such that $G\times_SX\rightarrow X\times_SX$ given by $(g,x)\mapsto (gx,x)$ is an isomorphism and such that $X\rightarrow S$ is faithfully flat. This already gets rid of the flat topology but in the current case where $S=\mathrm{Spec} R$ a local Artinian ring and we also give ourselves an isomorphism of $G$-schemes $X\times_S\mathrm{k}=G\times_S\mathrm{Spec}k$ then it is enough that $X\rightarrow S$ is a flat non-empty $G$-scheme as the fact that $G\times_SX\rightarrow X\times_SX$ is an isomorphism can be checked upon reduction to $k$.

As then $X$ is affine we are talking about a finite flat (non-zero) $R$-algebra $T=R[X]$ which is a $R[G]$-comodule which is also a comodule algebra (i.e., the product $T\bigotimes_RT\rightarrow T$ as well as the unit $R\rightarrow T$ arecomodule maps) together with a comodule algebra isomorphism $T\bigotimes_Rk\rightarrow k[G]$. If we specialise further to the actual case at hand, the category of $U({\frak g}\bigotimes_kR)^\ast$-comodules is isomorphic (for once this is really an isomorphism) as a tensor category to the category of ${\frak g}\bigotimes_kR$-modules (as $p$-Lie algebra). This gives us a description purely in terms of ${\frak g}\bigotimes_kR$-modules.

Note that there is also a very concrete description of $G$-torsors (for $G\rightarrow S$ finite flat for simplicity over an affine $S=\mathrm{Spec}R$). For an $R$-algebra $R'$ we have a tautological map of group schemes $G(R')\hookrightarrow (R'[G]^\ast)^\times$ (an element $f\in G(R')$ is by definition an $R'$-homomorphism $R'[G]^\ast$, it is easily seen to land in $(R'[G]^\ast)^\times$ and is tautologically a group homomorphism). This gives us an embedding of group flat schemes $G\hookrightarrow (R[G]^\ast)^\times$. This gives us a (half-)long exact sequence of cohomology sets associated to $G\hookrightarrow (R[G]^\ast)^\times\rightarrow (R[G]^\ast)^\times/G$. If $R$ is Artinian all $(R[G]^\ast)^\times$-torsor are trivial as they correspond to locally free rank $1$-modules over $R[G]^\ast$ (right modules to be specific) which are all trivial as $R[G]^\ast$ is also Artinian. Hence $G$-torsors correspond to sections of $(R[G]^\ast)^\times/G$ modulo the action of the section of $(R[G]^\ast)^\times$. However, it is in general difficult to get a concrete description of $(R[G]^\ast)^\times$ and even when you have one the orbits may be difficult to figure out.

Addendum: In the orbit description I forgot to add the condition that everything should map to the identity in $k$. Note also that the orbit description is no doubt the closest you can get in general to usual description of for instance $\alpha_p$ and $\mu_p$-torsors where one extracts a $p$'th root of a function (resp. an invertible function). In some other cases one also gets simpler descriptions. For instance if $H$ is a smooth $R$-group scheme and $G$ is the kernel of the Frobenius map, then every $G$-torsor with a trivialisation over $k$ is obtained from a section of $H^{(p)}$.

Addendum 1: As an answer to Jacob's followup question, tautologically the image of $G$ in $(R[G]^\ast)^\times$ consists of the elements $f$ for which $\Delta(f)=f\otimes f$, where $\Delta$ is the coproduct. This means (I hope) that if we let $V$ be the graph of $\Delta$ as a subspace of $R[G]^\ast\times R[G]^\ast\bigotimes [G]^\ast$, then $G$ is the stabiliser of $V$. From this we can get a $1$-dimensional subspace by takint the $\dim V$'th exterior power. Mind you this gives a $1$-dimensional subspace not a vector. In many cases one does however get the same result when looking at a fixed vector.

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So I guess in these terms what I'd like is a more explicit description of the quotient U(g)^x/G. For example, is there some naturally occuring affine space on which U(g)^x acts, where one of the stabilizers is G? – Jacob Lurie Nov 20 2010 at 13:40