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 [math]\displaystyle{ \operatorname{Spec} L \to \operatorname{Spec} K }[/math], 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 2008, Theorem 4.46
References
- Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory". http://homepage.sns.it/vistoli/descent.pdf.
- Algebraic Geometry I: Schemes. Springer Studium Mathematik - Master. 2020. doi:10.1007/978-3-658-30733-2. ISBN 978-3-658-30732-5. https://books.google.com/books?id=XEiLudn6sq4C&pg=PA458.
External links
 |  |
