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
References
- Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (Updated September 2, 2008)
External links
- http://mathoverflow.net/questions/149718/stack-of-tannakian-categories-galois-descent
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