Moduli stack of principal bundles

In algebraic geometry, given a <a href="/facts/Smooth_curve/xBQ0pTBW">smooth</a> <a href="/facts/Projective_curve/aB2XdB7B">projective curve</a> X over a finite field 
 
 
 
 
 
 F
 
 
 q
 
 
 
 
 {\displaystyle \mathbf {F} _{q}}
 
 and a smooth <a href="/facts/Affine_group_scheme/bJW14FE2">affine</a> <a href="/facts/Group_scheme/bJW14FE2">group scheme</a> G over it, the moduli stack of principal bundles over X, denoted by 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)}
 
, is an <a href="/facts/Algebraic_stack/flrtwvwU">algebraic stack</a> given by: for any 
 
 
 
 
 
 F
 
 
 q
 
 
 
 
 {\displaystyle \mathbf {F} _{q}}
 
-algebra R,

Bun
 
 G
 
 
 ⁡
 (
 X
 )
 (
 R
 )
 =
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)(R)=}
 
 the category of <a href="/facts/Torsor/N27maCdL">principal G-bundles</a> over the relative curve 
 
 
 
 X
 
 ×
 
 
 
 F
 
 
 q
 
 
 
 
 Spec
 ⁡
 R
 
 
 {\displaystyle X\times _{\mathbf {F} _{q}}\operatorname {Spec} R}
 
.
In particular, the category of 
 
 
 
 
 
 F
 
 
 q
 
 
 
 
 {\displaystyle \mathbf {F} _{q}}
 
-points of 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)}
 
, that is, 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 (
 
 
 F
 
 
 q
 
 
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)(\mathbf {F} _{q})}
 
, is the category of G-bundles over X.
Similarly, 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)}
 
 can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)}
 
 as the <a href="/facts/Quotient_stack/lwme8cqw">quotient stack</a> of the space of holomorphic connections on X by the <a href="/facts/Gauge_group_(mathematics)/JkeS71NG">gauge group</a>. Replacing the quotient stack (which is not a topological space) by a <a href="/facts/Homotopy_quotient/N6zTy9fn">homotopy quotient</a> (which is a topological space) gives the <a href="/facts/Homotopy_type/1nWrPxdj">homotopy type</a> of 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)}
 
.
In the finite field case, it is not common to define the homotopy type of 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)}
 
. But one can still define a (<a href="/facts/Smooth_topology/EEvRoqEW">smooth</a>) <a href="/facts/Cohomology/J7FJzTNh">cohomology</a> and <a href="/facts/Homology_(mathematics)/VFJmryEW">homology</a> of 
 
 
 
 
 Bun
 
 G
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Bun} _{G}(X)}
 
.

Moduli stack of principal bundles open-in-new

Moduli stack of principal bundles