Different notions of torsors in algebraic geometry

Question

In what follows $X$ will be a scheme and $G$ a group scheme. In the examples I will take $X=\mathbb{P}^1_k$ and $G=\mathbb{G}_{m}$. When reading about "the torsor..." I found many definitions, not sure which are equivalent and why exactly:

1. (Principle $G$-bundle) A $G$-torsor $P$ over a scheme $X$ is a scheme $P\to X$ with a $G$ action $G\times P\to P$ , which is locally trivial in the sense that there is a covering map $Y\to X$ (in the Zariski,etale,fppf,... topology), s.t. $Y\times _X P\to Y$ is isomorphic to $Y\times G\to Y$.

   Example: $P=X\times G$ or $\mathbb{A}^2_k-\{0\} \to X$ the Hopf bundle

2. (Sheaf version) A $G$-torsor $P$ is a sheaf on $(Sch/X)$ (with the Zariski, etale,fppf,... topology), if there is a covering $\{U_i\to X\}$, s.t. $P|_{U_i}$ is a trivial $G_{U_i}$ torsor (on sheafs a trivial torsor is a sheaf with a $G$ simply transitive action and non empty global sections).

   Example: $Hom(-,X)$ for a principal $G$-bundle as in 1) ?

3. (Small sheaf version) Let $\mathcal{G}$ be a sheaf of abelian groups on the topological space $X$. A $\mathcal{G}$-torsor is a sheaf $P$ on $X$ with a $\mathcal{G}$ action with non-empty stalks on which $G$ acts simply transitive.

4. (Tannakian version) A $G$-torsor is an exact tensor functor $Rep_k(G)\to Bun_X$, where $X$ is a scheme over $k$. (cf this paper 4.1.)

If $G$ happens to be $GL_n$ then we even have two more notions:

5. $P$ is a vector bundle, i.e. a locally free sheaf of rank $n$

6. $P$ is a geometric vector bundle, i.e. $P\to X$ is a morphism which is locally of the form $pr_2:\mathbb{A}_n\times U_i\to U_i$ in a compatible way (cf. Hartshore ex.5.19 for more details)

Now here is what I think is equivalent:

• If we consider the etale topology, 1. and 4. are equivalent
• I think I read that 1. and 2. are equivalent in some good cases, if some affinity condition is fulfilled
• In the $GL_n$ case 3., 5. and 6. are all equivalent
• I am not sure if 1. 2. or 4. are connected to 3. (well $Hom(-,\mathbb{G}_m)=\mathcal{O}_X^*$ when restricted to X seems no coincidence?)

Further questions:

• What do $\check{\mathrm{H}}^1(X,\mathcal{G})$ and $\check{\mathrm{H}}^1(X,G)$ really capture?
• Is the line bundle corresponding to the Hopf bundle in 1. $\mathcal{O}(-1)$?

Answer 1

This is an incomplete answer but too long for a comment.

1. is the standard definition of a torsor in the category of schemes.
2. and 3. are a generalization for groups that are not representable. So if $T$ is a torsor in the sense of 1., then $Hom(.,T)$ are torsors in the sense of 2. and 3. But 2. is more general : if $\tau$ is a topology and $\mathcal{G}$ is a $\tau$-sheaf of group which is not representable, then we can still define $\mathcal{G}$-torsors as a sheaf which is locally for the $\tau$-topology isomorphic to $\mathcal{G}$. I believe (I hope someone can confirm this) that if $\tau$ is subcanonical (such as most of the topology fppf, étale, Zarisky, but not the h-topology) that if $\mathcal{G}$ is representable by a scheme $G\in Sch/X$, then any $\mathcal{G}$-torsor is representable. So if $\mathcal{G}$ is representable, then this is the same thing as definition 1.
3. vs 2. Well obviously, a sheaf $\mathcal{G}$ on $Sch/X$ restricts as a sheaf $\mathcal{G}|_X$ on $X$ for the same topology and a $\mathcal{G}$-torsor on $Sch/X$ will restrict as a $\mathcal{G}|_X$-torsor on $X$. As I said before, torsor in the sense of 1. defines torsor in the sense of 3., but they are not representable anymore, but they are the restriction of a representable sheaf on $Sch/X$. Hence torsor in the sense of 2. provides a useful generalization of both 1. and 3. Unfortunately, this is a bit too wide because a lot of torsors in the sense of 2. "does not look like torsors" (say $X=\operatorname{Spec}k$ and $\tau=Zar$, then a torsor $T$ can be very different on $\operatorname{Spec}k$ and on $\mathbb{P}^1_k$) and because a lot of torsors in the sense of 2. restricts to the same torsor on $X$.
4. I am not comfortable with this definition. But also, I am not sure about what $Bun/X$. Do you mean maps $p:Y\to X$ which are locally a projection $U\times F\to U$ or do you mean all maps $p:Y\to X$ ?

Now for the case of $GL_n$, I disagree that 5. and 6. are definitions of torsors. They are instead associated spaces. From a vector bundle of rank $n$ (a sheafy version or geometric one, this is the same, and by Hilbert 90, every topology is equivalent in this case) defines a $GL_n$ by looking at the space of frames. Conversely, from a space of frames, we can form the associated space. But this is not the same geometrical object. Even in the case $n=1$ : from the vector bundle $p:E\to X$, we have to remove the image of the zero-section. But yes, there is a bijective correspondence between the $GL_n$-torsors and rank $n$ vectors bundle.

As for your final questions :

• $\check H^1(X,\mathcal{G})$ classifies the $\mathcal{G}$-torsor in the sense of 1. if $\mathcal{G}$ is representable in $Sch/X$ and in the sense of 3. otherwise. We cannot get torsors in the sense of 2. because cohomology on the big site coincide with the cohomology on the small site. This agree with what I said before : torsors in 2. are too wide since there is not much connection between a scheme $Y$ and $X$ if $Y$ is not part of a covering of $X$. But definition 2. still provides a useful category in which both definition 1. and definition 3. lives.
• Yes but note again that $\mathcal{O}(-1)$ is not a $\mathbb{G}_m$-torsor=$\mathcal{O}^*$-torsor, but rather an associated space.