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ar X iv :1 20 4. 26 28 v1 [ ma th. AG ] 12 A pr 20 12 THE COHERENT COHOMOLOGY RING OF AN ALGEBRAIC GROUP MICHEL BRION Abstract. Let G be a group scheme of finite type over a field, and consider the cohomology ring H?(G) with coefficients in the structure sheaf. We show that H?(G) is a free module of finite rank over its component of degree 0, and is the exterior algebra of its component of degree 1. When G is connected, we determine the Hopf algebra structure of H?(G). 1. Introduction To each scheme X over a field k, one associates the graded-commutative k-algebra H?(X) := ? i≥0H i(X,OX) with multiplication given by the cup product. Any mor- phism of schemes f : X ? X ? induces a pull-back homomorphism of graded algebras f ? : H?(X ?) ? H?(X), and there are Kunneth isomorphisms H?(X) ? H?(Y ) ?=?? H?(X ? Y ). When X is affine, the “coherent cohomology ring” H?(X) is just the algebra O(X) of global sections of OX . Now consider a k-group scheme G with multiplication map µ : G?G ? G, neutral element eG ? G(k), and inverse map ? : G ? G.

- higher direct
- modules ?
- g?x ??
- pull-back u?
- h?
- hopf algebra
- full abelian
- ?? ri
- direct im- age

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Published by | pefav |

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Language | English |

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