# 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 , 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)