Descent along torsors

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 $\operatorname {Spec} L\to \operatorname {Spec} K$ , 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.

Notes

Vistoli, Theorem 4.46

References

Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (Updated September 2, 2008)

External links

Stack of Tannakian categories? Galois descent?

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[1] Vistoli, Theorem 4.46