In mathematics, given a ''G''-
torsor
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non ...
''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.
It is a basic example of
descent
Descent may refer to:
As a noun Genealogy and inheritance
* Common descent, concept in evolutionary biology
* Kinship, one of the major concepts of cultural anthropology
**Pedigree chart or family tree
** Ancestry
** Lineal descendant
**Heritag ...
, 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
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of a finite
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
''L''/''K'', for the ''G''-torsor
, this generalizes classical Galois descent (cf.
field of definition
In mathematics, the field of definition of an algebraic variety ''V'' is essentially the smallest field to which the coefficients of the polynomials defining ''V'' can belong. Given polynomials, with coefficients in a field ''K'', it may not be ...
).
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
*
*
External links
Stack of Tannakian categories? Galois descent?
Algebraic geometry
Topology
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