In algebraic geometry, given a smooth projective curve X over a finite field
and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by
, is an algebraic stack given by:[1] for any
, is the category of G-bundles over X.
can also be defined when the curve X is over the field of complex numbers.
Roughly, in the complex case, one can define
as the quotient stack of the space of holomorphic connections on X by the gauge group.
Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of
In the finite field case, it is not common to define the homotopy type of
But one can still define a (smooth) cohomology and homology of
is a smooth stack of dimension
It is not of finite type but locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf.
the Harder–Narasimhan stratification), also for parahoric G over curve X see [2] and for G only a flat group scheme of finite type over X see.
[3] If G is a split reductive group, then the set of connected components
is in a natural bijection with the fundamental group
[4] This is a (conjectural) version of the Lefschetz trace formula for
when X is over a finite field, introduced by Behrend in 1993.
[5] It states:[6] if G is a smooth affine group scheme with semisimple connected generic fiber, then where (see also Behrend's trace formula for the details) A priori, neither left nor right side in the formula converges.
Thus, the formula states that the two sides converge to finite numbers and that those numbers coincide.