Descent along torsors (original) (raw)

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In mathematics, given a _G_-torsor X → Y and a stack F, the descent along torsors says there is a canonical equivalence between F(Y), the category of _Y_-points and F(X)G, the category of _G_-equivariant _X_-points.[1] It is a basic example of descent, since it says the "equivariant data" (which is an additional data) allows one to "descend" from X to Y.

When G is the Galois group of a finite Galois extension L/K, for the _G_-torsor Spec ⁡ L → Spec ⁡ K {\displaystyle \operatorname {Spec} L\to \operatorname {Spec} K} {\displaystyle \operatorname {Spec} L\to \operatorname {Spec} K}, this generalizes classical Galois descent (cf. field of definition).

For example, one can take F to be the stack of quasi-coherent sheaves (in an appropriate topology). Then F(X)G consists of equivariant sheaves on X; thus, the descent in this case says that to give an equivariant sheaf on X is to give a sheaf on the quotient X/G.

  1. ^ Vistoli 2008, Theorem 4.46