# Moduli stack of principal bundles

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 -algebra *R*,

- the category of principal
*G*-bundles over the relative curve .

In particular, the category of -points of , that is, , is the category of *G*-bundles over *X*.

Similarly, 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 .

## Basic properties

It is known that is a smooth stack of dimension where is the genus of *X*. It is not of finite type but of locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf. the Harder–Narasimhan stratification.) If *G* is a split reductive group, then the set of connected components is in a natural bijection with the fundamental group .[2]

## The Atiyah–Bott formula

## Behrend's trace formula

This is a (conjectural) version of the Lefschetz trace formula for when *X* is over a finite field, introduced by Behrend in 1993.[3] It states:[4] if *G* is a smooth affine group scheme with semisimple connected generic fiber, then

where (see also Behrend's trace formula for the details)

*l*is a prime number that is not*p*and the ring of l-adic integers is viewed as a subring of .- is the geometric Frobenius.
- , the sum running over all isomorphism classes of G-bundles on
*X*and convergent. - for a graded vector space , provided the series on the right absolutely converges.

*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.

## Notes

- http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf
- Heinloth 2010, Proposition 2.1.2
- http://www.math.ubc.ca/~behrend/thesis.pdf
- Lurie 2014, Conjecture 1.3.4.

## References

- J. Heinloth, Lectures on the moduli stack of vector bundles on a curve, 2009 preliminary version
- J. Heinloth, A.H.W. Schmitt, The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves, 2010 preprint, available at http://www.uni-essen.de/~hm0002/.
- Gaitsgory, D; Lurie, J.; Weil's Conjecture for Function Fields. 2014,