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Galois Groups and Fundamental Groups by Tamás Szamuely

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6

Tannakian fundamental groups

The theory of the last chapter established an equivalence between the category of finite étale covers of a connected scheme and the category of finite continuous permutation representations of its algebraic fundamental group. We shall now study a linearization of this concept, also due to Grothendieck and developed in detail by Saavedra [81] and Deligne [14]. The origin is a classical theorem from the theory of topological groups due to Tannaka and Krein: they showed that one may recover a compact topological group from the category of its continuous unitary representations. In Grothendieck’s algebraic context the group is a linear algebraic group, or more generally an affine group scheme, and one studies the category ...

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