that . Then there exists a unique Y-morphism such that . Assume that is locally noetherian and f is locally of finite type. For in , let and let be the induced map on completed local rings. Then the following are equivalent: is étale. For every in , the induced map on completed local rings is formally étale for the adic topology. For every in , is a free -module and the fiber is a field which is a finite separable field extension of the residue field . (Here is the maximal ideal of .) f is formally étale for maps of local rings with the following additional properties. The local ring A may be assumed Artinian. If m is the maximal ideal of A, then J may be assumed to satisfy . Finally, the morphism on residue fields may be assumed to be an isomorphism. If in addition all the maps on residue fields are isomorphisms, or if is separably closed, then is étale if and only if for every in , the induced map on completed local rings is an isomorphism. Examples Any open immersion is étale because it is locally an isomorphism. Covering spaces form examples of étale morphisms. For example, if is an integer invertible in the ring then is a degree étale morphism. Any ramified covering has an unramified locus which is étale. Morphisms induced by finite separable field extensions are étale — they form arithmetic covering spaces with group of deck transformations given by . Any ring homomorphism of the form , where all the are polynomials, and where the Jacobian determinant is a unit in , is étale. For example the morphism is etale and corresponds to a degree covering space of with the group of deck transformations. Expanding upon the previous example, suppose that we have a morphism of smooth complex algebraic varieties. Since is given by equations, we can interpret it as a map of complex manifolds. Whenever the Jacobian of is nonzero, is a local isomorphism of complex manifolds by the implicit function theorem. By the previous example, having non-zero Jacobian is the same as being étale. Let be a dominant morphism of finite type with X, Y locally noetherian, irreducible and Y normal. If f is unramified, then it is étale. For a field K, any K-algebra A is necessarily flat. Therefore, A is an etale algebra if and