Etale morphisms

In the last section, we have seen that many familiar concepts involving differentials can be transferred from differentials and analytic geometry to algebraic geometry. But one very important theorem in the differential and analytic situations is false in the algebraic case - the implicit function theorem. This asserts that if we are given k differentiable (resp. analytic) functions /i. / near a point x in Rn+k (resp. Cn+fc) such that 

2nd an arbitrary morphism / X - Y of finite type is etale, if for all x G X, there are open neighbourhoods U C X of x and V C Y of f(x) such that F U) C V and such that /, restricted to U, looks like a morphism of the above type  

This intuitively reasonable definition, like the provisional ones we made for the tangent cone and tangent space, is not really intrinsic. 

One has a right to ask for an equivalent form involving only the local rings of X and Y and not dragging in affine space. There is such a reformulation, but it involves the concept of flatness so we have to put it off until 10. This clumsy form is adequate for the present. 

11 The word apparently refers to the appearance of the sea at high tide under a full moon in certain types of weather. 

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The morphism Ag%s Spec(Z) is surjective and smooth of relative dimension g(g A 1). Proof. Let V -+ Ag,d be a surjective etale morphism and let V = U XAg,d Agj. The schemes U and V are of finite type over Spec(Z) as A3 d and Agj are of finite type over Spec(Z). To prove the first assertion we have to show that the morphism V U is a locally closed immersion. Indeed, if we have this then for any morphism S A3td the morphism... 

We will mainly deal with morphlsms of finite type, in which case the concepts of formally smooth (resp formally etale) morphism and smooth (resp. etale) morphism obviously coincide. A smooth morphism f X— S is also called a smooth family of schemes parametrized bv 5, and X is said to be smooth over 5. 

If we modify the previous definitions asking that the map (1) is bijective (instead of only being surjective) for all C and I as above, we obtain the notions of formally etale and of etale morphism... 

Suppose 5 -> 5 is a finite etale morphism such that c i ZlpZs Y and < 2,5 ... 

One of the key facts about etale morphisms is Hensel s lemma, of which here is a variant ... 

Moreover, I claim that Yi is the maximal open set such that res(/) /-1(Yi) - Yi is etale. To prove this, we need the main result of this section, which is the intrinsic characterization of etale morphisms referred to in 5 ... 

Proposition 1.1. — Let X. be a scheme of finite type over S and U, —+ X a finite family of etale morphisms in Sch/S. The following conditions are equivalent ... 

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