Étale morphism

In algebraic geometry, an étale morphism (etal) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology. The word étale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.

Étale morphisms are preserved under composition and base change. Étale morphisms are local on the source and on the base. In other words, is étale if and only if for each covering of by open subschemes the restriction of to each of the open subschemes of the covering is étale, and also if and only if for each cover of by open subschemes the induced morphisms is étale for each subscheme of the covering. In particular, it is possible to test the property of being étale on open affines The product of a finite family of étale morphisms is étale. Given a finite family of morphisms , the disjoint union is étale if and only if each is étale. Let and , and assume that is unramified and is étale. Then is étale. In particular, if and are étale over , then any morphism between and is étale. Quasi compact étale morphisms are quasi finite. A morphism is an open immersion if and only if it is étale and radicial. If is étale and surjective, then (finite or otherwise).