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# Atiyah–Bott formula

In algebraic geometry, the Atiyah–Bott formula says[1] the cohomology ring

\( \operatorname{H}^*(\operatorname{Bun}_G(X), \mathbb{Q}_l) \)

of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. Atiyah and Bott's original work concerned integral cohomology ring of Bun_{G}(*X*).

See also

Borel's theorem, which says the cohomology ring of a classifying stack is a polynomial ring.

Notes

Gaitsgory–Lurie, § 6.2.

References

Atiyah, M. F. and R. Bott.; "The Yang-Mills equations over Riemann surfaces." Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505.

Gaitsgory, D; Lurie, J.; "Weil's Conjecture for Function Fields." 2014, [1]

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