this situation, using the étale topology. The precise statement is as follows: if is étale and finite, then for any point y lying in Y, there is an étale morphism V → Y containing y in its image (V can be thought of as an étale open neighborhood of y), such that when we base change f to V, then (the first member would be the pre-image of V by f if V were a Zariski open neighborhood) is a finite disjoint union of open subsets isomorphic to V. In other words, étale-locally in Y, the morphism f is a topological finite cover. For a smooth morphism of relative dimension n, étale-locally in X and in Y, f is an open immersion into an affine space . This is the étale analogue version of the structure theorem on submersions. See also Purity (algebraic geometry) References Bibliography J. S. Milne (2008). Lectures on Etale Cohomology Morphisms of schemes