2 - A very brief introduction to étale homotopy

Summary

Abstract The task of these notes is to supply the reader who has little or no experience of simplicial topology with a phrase-book on étale homotopy, enabling them to proceed directly to [5] and [10]. This text contains no proofs, for which we refer to the foundational book by Artin and Mazur [1] in the hope that our modest introduction will make it more accessible. This is only a rough guide and is no substitute for a rigorous and detailed exposition of simplicial homotopy for which we recommend [4] and [8].

Let X be a Noetherian scheme which is locally unibranch (this means that the integral closure of every local ring of X is again a local ring), e.g., a Noetherian normal scheme (all local rings are integrally closed). All smooth schemes over a field fall into this category. The aim of the Artin–Mazur theory is to attach to X its étale homotopy type Ét(X). This is an object of a certain category pro − H, the pro-category of the homotopy category of CW-complexes. The aim of these notes is to explain this construction.