Cosimplicial Objects in Algebraic Geometry

Part of the NATO ASI Series book series (ASIC, volume 407)


Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I, X) over X × X, P(w) = (w(0), w(1)), we get a local system of sets (Poincaré groupoid) over X × X. This construction does not have a straightforward generalization to algebraic varieties over any field. Using cosimplicial objects, we propose a generalization for smooth, algebraic varieties over an arbitrary field of characteristic zero. This leads to a definition of an algebraic fundamental group of De Rham type. We partly calculate the Betti lattice in the algebraic fundamental group for the projective line minus three points.


Vector Bundle Fundamental Group Hopf Algebra Spectral Sequence Characteristic Zero 
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Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  1. 1.Sophia Antipolis Laboratoire de Mathématique U.R.A. au C.N.R.S.Université de NiceNice Cedex 2France

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