Moduli stack of principal bundles

In algebraic geometry, given a smooth projective curve X over a finite field F q {\displaystyle \mathbf {F} _{q}} and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by Bun G ⁡ ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} , is an algebraic stack 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 principal G-bundles 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.

Source: Wikipedia — Moduli stack of principal bundles (CC BY-SA 4.0)

Moduli stack of principal bundles

In algebraic geometry, given a smooth projective curve X over a finite field F q {\displaystyle \mathbf {F} _{q}} and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by Bun G ⁡ ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} , is an algebraic stack 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 principal G-bundles 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.

Source: Wikipedia "Moduli stack of principal bundles" · CC BY-SA 4.0

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