Sheaf description of $G$-bundles

Question

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, principal $\mathrm{GL}(n,\mathbb{C})$-bundles are given by locally free sheaves of rank $n$.

So...what about other groups? I guess that $\mathrm{SL}(n,\mathbb{C})$ bundles are then locally free sheaves of rank $n$ with top exterior power trivial, but can we phrase everything in terms of the properties of a sheaf and a group?

My guess is that in this context, if we can do it, we'll end up with something that's not quite locally free sheaves of rank n for $\mathrm{GL}(n,\mathbb{C})$, but which will be equivalent.

Note: I'm aware that we could just say something like "the sheaf of local sections of a $G$-bundle" but I'm looking for something intrinsic, a set of properties of the sheaf without reference to the geometric bundle, which can be reconstructed from the sheaf description.

Answer 1

$\newcommand{\O}{\mathcal{O}}$ $\newcommand{\F}{\mathcal{F}}$

The way you get a locally free sheaf of rank $n$ from a $GL(n)$-torsor $P$ is by twisting the trivial rank $n$ bundle $\O^n$ (which has a natural $GL(n)$-action) by the torsor. Explicitly, the locally free sheaf is $\F=\O^n\times^{GL(n)}P$, whose (scheme-theoretic) points are $(v,p)$, where $v$ is a point of the trivial bundle and $p$ is a point of $P$, subject to the relation $(v\cdot g,p)\sim (v,g\cdot p)$. Conversely, given a locally free sheaf $\F$ of rank $n$, the sheaf $Isom(\O^n,\F)$ is a $GL(n)$-torsor, and this procedure is inverse to the $P\mapsto \O^n\times^{GL(n)}P$ procedure above. (Note: I'm identifying spaces over the base $X$ with their sheaves of sections, both for regarding $Isom(\O^n,\F)$ as a torsor and for regarding $\O^n\times^{GL_n}P$ as a locally free sheaf.)

Similarly, if you have a group $G$ and a representation $V$, then you can associate to any $G$-torsor $P$ a locally free sheaf of rank $\dim(V)$, namely $V\times^G P$. But I don't know of a characterization of which locally free sheaves of rank $\dim(V)$ arise in this way.

Operations with the locally free sheaf (like taking top exterior power, or any other operation which is basically defined fiberwise and shown to glue) correspond to doing that operation with the representation $V$, so I think you're right that in the case of $SL(n)$ you get exactly those locally free sheaves whose top exterior power is trivial (since $SL(n)$ has no non-trivial $1$-dimensional representations).

Answer 2

If G is an affine algebraic group, a G-bundle is the same as a monoidal functor from G-reps to coherent sheaves. The map one way is take associated bundle, the other involves reconstructing the structure sheaf of the G-bundle from the associated ones. Roughly, you think of the functions on the group as a ring ind-object in the category of representations, and take the corresponding ring object in quasi-coherent sheaves. The Spec of this sheaf of rings is the G-bundle.

For GL(n), you'e lucky, since its category has a simple description: it's (basically) the free monoidal category with a single generator of dimension n. Other groups are a little more complicated, but not much worse.

Answer 3

Adding it so that it's easily found. The thing I was looking for, which is generally not written out except in the case of vector bundles, is that the sheaf of sections of an F-bundle with fiber F is a sheaf of sets that is locally isomorphic in the etale topology to the sheaf hom(-,F) ranging over small enough open subsets of X.