only if it is unramified, which is also equivalent to where is the separable closure of the field K and the right hand side is a finite direct sum, all of whose summands are . This characterization of etale K-algebras is a stepping stone in reinterpreting classical Galois theory (see Grothendieck's Galois theory). Properties 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). Inverse function theorem Étale morphisms f: X → Y are the algebraic counterpart of local diffeomorphisms. More precisely, a morphism between smooth varieties is étale at a point iff the differential between the corresponding tangent spaces is an isomorphism. This is in turn precisely the condition needed to ensure that a map between manifolds is a local diffeomorphism, i.e. for any point y ∈ Y, there is an open neighborhood U of x such that the restriction of f to U is a diffeomorphism. This conclusion does not hold in algebraic geometry, because the topology is too coarse. For example, consider the projection f of the parabola y = x2 to the y-axis. This morphism is étale at every point except the origin (0, 0), because the differential is given by 2x, which does not vanish at these points. However, there is no (Zariski-)local inverse of f, just because the square root is not an algebraic map, not being given by polynomials. However, there is a remedy for