In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Atiyah–Bott formula says
the
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
:
of the
moduli stack of principal bundles In algebraic geometry, given a smooth projective curve ''X'' over a finite field \mathbf_q and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by \operatorname_G(X), is an algebraic stack given b ...
is a
free graded-commutative algebra on certain homogeneous generators. The original work of
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
and
Raoul Bott
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...
concerned the integral cohomology ring of
.
See also
*
Borel's theorem
In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring.
See also
*Atiyah–Bott formula
In algebraic geometry, the Atiyah–Bott formula s ...
, which says that the cohomology ring of a classifying stack is a polynomial ring.
Notes
References
*
*
Theorems in algebraic geometry
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