Question

Let $\pi:P\mapsto B$ is a $G$-principal bundle, which means $G$ acts on $P$ freely and $\pi$ is a locally trivial fibration. Here is a well-known theorem:

THeorem: The inverse image functor $\pi^*$ gives an equivalence from $Sh_G(P)$ to $Sh(B)$, and the inverse functor is given by $\pi_* ^G$.

Let me explain some notations. the object in $Sh_G(P)$ is a pair $(\mathcal{F},\alpha)$, where $\alpha:p^*\mathcal{F}\simeq a^* \mathcal{F}$, and satisfy cocycle condition. Here $p:G\times P\mapsto P$ is the projection, and $a:G\times P\mapsto P$ is the action. The functor $\pi_* ^G$ is given by assigning open subset V to $\mathcal{F}(\pi^{-1}(U))^G$.
We need to show that
(1). $\pi^*$ is fully-faithful, i.e., for any two sheaves $\mathcal{F}$ and $\mathcal{G}$, $Hom(\mathcal{F},\mathcal{G})\simeq Hom_G(\pi^*(\mathcal{F}),\pi^*(\mathcal{G}))$.

(2). $\pi^*$ is essentially surjective, i.e., for any $G$-equivariant sheaf $\mathcal{H}$, there exists an isomorphism $\mathcal{H}\simeq \pi^* \pi_*^G(\mathcal{H})$.

(1) follows from $\pi_*^G\circ\pi^*\simeq Id$, which is relatively easy.
For (2), I have checked when $B$ is a point. For general case, I can reduce it to the following isomorphism: $\pi_*(\mathcal{H})_b ^G \simeq$

$ \Gamma(\pi^{-1}(b), \mathcal{H}|_{\pi^{-1}(b)})^G$, where $b\in B$.

I know for nice group $G$, for example when $G$ is compact, this is really an isomorphism, because we have base change. For general group, I have no idea at all.

Who can help to finish this proof? Or give some other idea?

Answer 1

A general setup and general nonsense proof for this kind of theorems for the "descent along torsors" is in

• Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, FGA explained, 1–104

For a clean elementary categorical approach I would also recommend recent article

• Tomasz Brzezinski, On synthetic interpretation of quantum principal bundles, arxiv/0912.0213

and some $n$lab remarks at codomain fibration.

In the algebraic setup, torsors are usually considered in the flat topology. Related issues are J. Milne's book on etale cohomology. Basic theorems on the descent of sheaves along torsors are from Grothendieck and there are also analogues in various cohomologies, and for equivariant K-theory (look also at Thomason). A noncommutative version is possible with Hopf algebras instead of algebraic groups, what in affine case reduces to celebrated Schneider's theorem. In this regard cf. Hopf-Galois extension in $n$lab and for nonaffine generalizations general discussion in my article

• Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, arXiv:0811.4770.

Answer 2

One thing you didn't specify is what topology your sheaves are in; I don't know if this result is even true in the Zariski topology.

In the either the etale topology (for working over arbitrary base fields) or the complex topology (if you're working over the complex numbers), it's easy to finish this proof; any principal bundle is locally trivial (by the inverse function theorem), so your local statement only needs to be proven for the trivial bundle, where it's obvious.