This is the appropriate generalization of “local homeomorphism” from topology or “local isomorphism” from real differential geometry. Equivalently, is étale if and only if any of the following conditions hold:
A morphism of varieties over an algebraically closed field is étale at a point if it induces an isomorphism between the completed local rings and . If and are over an arbitrary field , then the required condition becomes that is a separable algebraic extension of , where , and induces an isomorphism between and .
A morphism of nonsingular varieties over an algebraically closed field is étale if and only if induces an isomorphism on the tangent spaces. In the differentiable category, the implicit function theorem implies that such a function is actually an isomorphism on some small neighborhood. On schemes, of course, the Zariski topology is too coarse for this to be the case. One way to define a finer “topology”, making the scheme into a site, is by using étale maps.
The word étale comes from French, where it can be used to describe a calm or slack sea.
- 1 Jean Dieudonné, A Panorama of Pure Mathematics, Academic Press, 1982.
- 2 Robin Hartshorne, Algebraic Geometry, Springer–Verlag, 1977 (GTM 52).
|Date of creation||2013-03-22 14:08:40|
|Last modified on||2013-03-22 14:08:40|
|Last modified by||mps (409)|