descent along a torsor

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

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 GG and a GG-torsor PXP\to X in faithfully flat topology, the category of equivariant quasicoherent sheaves over PP is equivalent to the category of quasicoherent sheaves over XX.

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

Created on November 29, 2012 at 21:06:36. See the history of this page for a list of all contributions to it.