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 .
^Lurie, Jacob (April 3, 2013), Tamagawa Numbers in the Function Field Case (Lecture 2)(PDF), archived from the original (PDF) on 2013-04-11, retrieved 2014-01-30
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