nLab descent along a torsor

Let $C$ be a category equipped with Grothendieck topology $\tau$, and $\pi:\mathcal{F}\to C$ a fibered category which is in fact a stack over the Grothendieck site $(C,\tau)$. Let $G$ be an group object in $C$ and $P$ a $G$-torsor in $C$ over $X\cong P/G$. Then the $G$-equivariant fiber over $P$ (i.e. the category $F^G(P)$ of $G$-equivariant objects over $P$ in the fibered category $\mathcal{F}$ over $C$) is equivalent to the usual fiber $\mathcal{F}(U)= \pi^{-1}(id_U)$ over $X$; this statement is referred to as the descent along the $G$-torsor $P$.

In particular, the assignment of the category of quasicoherent sheaves to every scheme gives a pseudofunctor over the categroy of schemes with the faithfully flat topology. In the faithfully flat topology, for an algebraic group scheme $G$ and a $G$-torsor $P\to X$ in faithfully flat topology, the category of equivariant quasicoherent sheaves over $P$ is equivalent to the category of quasicoherent sheaves over $X$.

Sometimes, the theorem on the descent along torsors is quite imprecisely referred to as the equivariant descent theorem.

An analogue in affine noncommutative algebraic geometry is the Schneider's descent theorem relating relative Hopf modules over the Hopf-Galois extension with the usual modules over the base. It has generalizations for other distributive laws, e.g. when the Hopf modules are replaced by entwined modules; also the generalization when the Hopf algebras are replaced by weak Hopf algebras, Hopf algebroids and alike. There are also globalized (non-affine) versions using coaction compatible localizations.

• Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory MR2223406; math.AG/0412512 pp. 1–104 in Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. 2005. x+339 pp. MR2007f:14001
• Tomasz Brzeziński, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arXiv:0912.0213
• Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770 MR2011b:14004