Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?)

share|improve this question
4  
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 '10 at 0:34
2  
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 '10 at 13:52

1 Answer 1

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.

share|improve this answer
2  
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 '10 at 13:40

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.