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Galois Groups and Fundamental Groups by Tamás Szamuely

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5

Fundamental groups of schemes

Though the theory of the previous chapter is sufficient for many applications, a genuine understanding of the algebraic fundamental group only comes from Grothendieck’s definition of the fundamental group for schemes. His theory encompasses the classification of finite covers of complex algebraic varieties of any dimension, Galois theory for extensions of arbitrary fields and even notions coming from arithmetic such as specialization modulo a prime. Moreover, it is completely parallel to the topological situation and clarifies the role of base points and universal covers – these have been somewhat swept under the carpet in the last chapter. In his original account in [29] Grothendieck adopted an axiomatic viewpoint ...

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