Universal covering spaces and fundamental groups in algebraic geometry as schemes

Details Universal covering spaces and fundamental groups in algebraic geometry as schemes Universal covering spaces and fundamental groups in algebraic geometry as schemes

2009-02-19

arxiv.org/abs/0902.3464v4

Mathematics

Algebraic Geometry

v2: Important references added. New example by David Harbater: the group scheme structure on the additive group does not lift

Scientific paper

In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the 'etale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.

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