In algebraic geometry, given a smooth algebraic group G, a G-torsor or a principal G-bundle P over a scheme X is a scheme (or even algebraic space) with the action of G that is locally trivial in the given Grothendieck topology in the sense that the base change
Y
×
X
P
along "some" covering map
Y
→
X
is the trivial torsor
Y
×
G
→
Y
(G acts only on the second factor). Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme
G
X
=
X
×
G
(i.e.,
G
X
acts simply transitively on
P
.)
The definition may be formulated in the sheaf-theoretic language: a sheaf P on the category of X-schemes with some Grothendieck topology is a G-torsor if there is a covering
{
U
i
→
X
}
in the topology, called the local trivialization, such that the restriction of P to each
U
i
is a trivial
G
U
i
-torsor.
A line bundle is nothing but a
G
m
-bundle, and, like a line bundle, the two points of views of torsors, geometric and sheaf-theoretic, are used interchangeably (by permitting P to be a stack like an algebraic space if necessary).
Examples and basic properties
Examples
A
G
L
n
-torsor on "X" is a principal
G
L
n
-bundle on "X".
If
L
/
K
is a finite Galois extension, then
Spec
L
→
Spec
K
is a
Gal
(
L
/
K
)
-torsor (roughly because the Galois group acts simply transitively on the roots.) This fact is a basis for Galois descent. See integral extension for a generalization.
Remark: A G-torsor P over X is isomorphic to a trivial torsor if and only if
P
(
X
)
=
Mor
(
X
,
P
)
is nonempty. (Proof: if there is an
s
:
X
→
P
, then
X
×
G
→
P
,
(
x
,
g
)
↦
s
(
x
)
g
is an isomorphism.)
Let P be a G-torsor with a local trivialization
{
U
i
→
X
}
in étale topology. A trivial torsor admits a section: thus, there are elements
s
i
∈
P
(
U
i
)
. Fixing such sections
s
i
, we can write uniquely
s
i
g
i
j
=
s
j
on
U
i
j
with
g
i
j
∈
G
(
U
i
j
)
. Different choices of
s
i
amount to 1-coboundaries in cohomology; that is, the
g
i
j
define a cohomology class in the sheaf cohomology (more precisely Čech cohomology with sheaf coefficient) group
H
1
(
X
,
G
)
. A trivial torsor corresponds to the identity element. Conversely, it is easy to see any class in
H
1
(
X
,
G
)
defines a G-torsor on X, unique up to an isomorphism.
If G is a connected algebraic group over a finite field
F
q
, then any G-bundle over
Spec
F
q
is trivial. (Lang's theorem.)
Most of constructions and terminology regarding principal bundles in algebraic topology carry over in verbatim to G-bundles. For example, if
P
→
X
is a G-bundle and G acts from the left on a scheme F, then one can form the associated bundle
P
×
G
F
→
X
with fiber F. In particular, if H is a closed subgroup of G, then for any H-bundle P,
P
×
H
G
is a G-bundle called the induced bundle.
If P is a G-bundle that is isomorphic to the induced bundle
P
′
×
H
G
for some H-bundle P', then P is said to admit a reduction of structure group from G to H.
Let X be a smooth projective curve over an algebraically closed field k, G a semisimple algebraic group and P a G-bundle on a relative curve
X
R
=
X
×
Spec
k
Spec
R
, R a finitely generated k-algebra. Then a theorem of Drinfeld and Simpson states that, if G is simply connected and split, there is an étale morphism
R
→
R
′
such that
P
×
X
R
X
R
′
admits a reduction of structure group to a Borel subgroup of G.
If P is a parabolic subgroup of a smooth affine group scheme G with connected fibers, then its degree of instability, denoted by
deg
i
(
P
)
, is the degree of its Lie algebra
Lie
(
P
)
as a vector bundle on X. The degree of instability of G is then
deg
i
(
G
)
=
max
{
deg
i
(
P
)
|
P
⊂
G
parabolic subgroups
}
. If G is an algebraic group and E is a G-torsor, then the degree of instability of E is the degree of the inner form
E
G
=
Aut
G
(
E
)
of G induced by E (which is a group scheme over X); i.e.,
deg
i
(
E
)
=
deg
i
(
E
G
)
. E is said to be semi-stable if
deg
i
(
E
)
≤
0
and is stable if
deg
i
(
E
)
<
0
.
