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I am an algebraic topologist and am trying to understand some computations related to p-adic complex K-theory and equivariant K-theory. However this has led me into the world of algebraic geometry over $\mathbb{Z}_p$ and I need some help with some basic computations.

I would like to understand the moduli stack of G-torsors over the scheme underlying the multiplicative group $\mathbb{G}_m$ where we are working with schemes over the p-adic integers $\mathbb{Z}_p$ and $G$ is a finite group (which we also view as a group scheme over $\mathbb{Z}_p$). I would even be happy to know the cases where $G = \mathbb{Z}/p$ and $G= \mathbb{Z}/q$ with $q \neq p$, i.e., for cyclic groups.

To clarify, for a scheme S defined over the p-adic integers $\mathbb{Z}_p$, the S-points of the stack in question form a groupoid, the groupoid whose objects are S-families of G-torsors over the p-adic multiplicative group $\mathbb{G}_m$. In other words the objects are G-torsors over $$S \times_{spec \; \mathbb{Z}_p} \mathbb{G}_m.$$ The morphisms are the obvious ones, and everything is defined over the base ring $\mathbb{Z}_p$, which should probably be viewed as either a topolological ring or a pro-ring/ind-scheme. I am willing to be quite flexible on this point, as well as the particular topology used (Zariski, etale, flat, etc).

Sometimes it is useful to use the heuristic that $\mathbb{G}_m$ is like the circle $S^1 = \mathbb{R}/\mathbb{Z}$. For the circle the stack of G-torsors looks like the quotient stack $[G/G]$ for the conjugation action. Perhaps the $G$-torsors on $\mathbb{G}_m$ hava a similar description? A simple description of this type would be very useful to me, i.e., as a quotient stack or really any presentation at all. I am not very picky, but I need this for explicit calculations and so certainly need the description to be as explicit as possible.

A good start would be computing the groupoid of G-torsors over $\mathbb{G}_m$ (ignoring S-families) or even the set of isomorphism classes of $G$-torsors over $\mathbb{G}_m$. I tried to compute this by finding some nice cover of $\mathbb{G}_m$, but it is getting late here and I haven't found something that works yet.

I have tried googling around and have come up short. Most of the stuff I have found is either about $\mathbb{G}_m$-torsors, or about $G$-torsors for groups related to $\mathbb{G}_m$, or work which only works over a field of some sort or another. To clarify I am interested in $\mathbb{G}_m$ over $\mathbb{Z}_p$, so not over a field, in particular not over a field which has pth roots of unity. However I would be interested in the answer over those other rings in-so-far as they help get me the answer over $\mathbb{Z}_p$.

If this question is too basic, I would be happy even for a reference with some similar examples carefully worked out so I can learn explicitly how to do this kind of calculation.

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  • $\begingroup$ Please clarify: do you want to "describe" (in some way) the moduli stack $BG$ classifying $G$-torsors over varying $S$-schemes with $S = {\rm{Spec}}(k[t,1/t])$ for a field (or ring?) $k$ (just $k = \mathbf{C}$?), or do you just want to describe the set/groupoid of such $G$-torsors over $S$ itself? Your question only speaks of the latter (aside from the appearance of "stack" for the circle example), but the title says "moduli space", so what exactly do you mean by "moduli space"? Exactly what base scheme is your "$\mathbf{G}_m$" living over? A field? Anything? $\endgroup$
    – user36938
    Aug 5, 2013 at 16:47
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    $\begingroup$ If your ${\mathbf{G}}_m$ is over a field containing a primitive $p$th root of unity then you know how to analyze $\mathbf{Z}/(p)$-torsors over the base using Kummer theory, but is that the sort of thing you want to generalize? $\endgroup$
    – user36938
    Aug 5, 2013 at 16:50
  • $\begingroup$ @user36938: I have edited the question, and hopefully this helps to clarify what I am hoping to learn. In particular the base scheme is spec of the p-adic integers. I am not familiar with Kummer theory (I'll start looking it up); does it answer the related question over a field with pth roots of unity? $\endgroup$
    – Chris Schommer-Pries
    Aug 5, 2013 at 21:08
  • $\begingroup$ Chris S-P: Kummer theory does not answer the question when the group scheme $\mu_p$ is not constant over the base. But are you willing to replace $\mathbf{Z}_p$ with its strict henselization or completion thereof (i.e., $W(\overline{\mathbf{F}}_p)$)? If so, then Kummer theory will take care of finite cyclic groups of order prime to $p$. There must be lots of people in your department who can discuss this with you in person. $\endgroup$
    – user36938
    Aug 6, 2013 at 1:30
  • $\begingroup$ Two minor points of nomenclature: Since you aren't using the group structure in $\mathbb{G}_m$, it is probably better to call it the punctured affine line $X = \mathbb{A}^1 \setminus \{0 \}$. The object in question is generally called the Hom stack $\underline{\operatorname{Hom}}_{\mathbb{Z}_p}(X, BG)$, and often abbreviated $Bun_G(X)$. $\endgroup$
    – S. Carnahan ♦
    Aug 6, 2013 at 1:31

1 Answer 1

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I think if you look, you can find the answer in the literature. In the case that $G$ is Abelian and has order prime to $p$, this should follow from Mumford's theory of biextensions, "Cartier duality", Exercise 7.1 of Dolgachev's "Invariant Theory", etc.

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  • $\begingroup$ I looked up exercise 7.1 in Dolgachev's "Lectures on Invariant Theory". It has to do with $\mathbb{G}_m$ bundles over a connected algebraic group G. This is opposite to what I was asking. I want G-bundles not $\mathbb{G}_m$-bundles. Moreover in my case G is a finite group, so not connected. $\endgroup$
    – Chris Schommer-Pries
    Aug 6, 2013 at 9:13
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    $\begingroup$ @Chris: It is not the opposite. The same argument as in the exercise applies to any (tame) multiplicative group $T$. For every smooth, connected, linear algebraic group $G$ (over a field in that exercise, but the same argument works relatively), every $T$-torsor over $G$ is induced from a unique extension of group schemes of $G$ by $T$. Now take $G$ to be $\mathbb{G}_m$ and take $T$ to be a finite Abelian group of order prime to $p$. $\endgroup$
    – Jason Starr
    Aug 6, 2013 at 11:43

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