(2) A morphism f : X → Y of schemes is called étale if f is locally of finite type, flat and unramified.

Definition 1. .

(1) A locally of finite type morphism f : X → Y of schemes is called unramified at x ∈ X if O X,x

.

m _{f (x)} O X,x is a finite separable field extension of k (f (y)).

(2) A morphism f : X → Y of schemes is called étale if f is locally of finite type, flat and unramified.

Proposition 2. Let A be a finite algebra over a field k. TFAE:

(a) A is separable over k.

(b) A = A ⊗ k k is isomorphic to a finite product of copies of k.

(c) A is isomorphism to a finite product of separable field extension of k.

(d) The discrimiant of any basis of A over k is nonzero.

proof: (a) ⇒(b): A has only finitely many prime ideals and they are all maximal. By the assumption, their intersection is zero. The conclusion follows from Chinese remainder theorem.

(b) ⇒(c): By Chinese remainder theorem, A 

J is isomorphic to a finite product Q

k _i of finite field extension of k, where J is the Jacobson radical of A. Then Hom _k A, k

has P

[k i : k] s elements. Since Hom _k A, k

= Hom _k A, k

, which has [A : k] elements by the assumption, we have [A : k] = [A : k] = X

[k _i : k] _s ≤ X

[k _i : k] =  A  J : k 

≤ [A : k]

The equality holds, which implies k _i are separable.

(c) ⇒(d): If A = Q

k _i with k _i separable field extension of k, then disc (A) = Q

disc (k _i ), which is nonzero since k _i are separable.

(d) ⇒(a): If x ∈ J A

, then xa is nilpotent for any a ∈ A, and hence Tr _A/k (xa) = 0. Note that the discriminant of A and A are the same, therefore x = 0.

Recall the Hensel’s lemma in nunber theory:

Theorem 3. Let A be a complete discrete valuation ring with k the residue field and f ∈ A[T ] monic.

If f = g ₀ h ₀ ∈ k[T ] for some g 0 , h ₀ ∈ k[T ] monic coprime, then f = gh for some g, h ∈ A[T ] monic with g = g ₀ , h = h ₀ .

Definition 4. A local ring A is called Henselian if the conclusion of Hensel’s lemma holds.

Theorem 5. Let A be a local ring and x be the closed point of X = Spec A. TFAE:

(a) A is Henselian.

(b) Any finite A-algebra B is a direct product of local rings B = Q B i .

(c) If f : Y → X is quasi-finite and separable, then Y = Y 0 t . . . t Y n , where x / ∈ f (Y 0 ) and for i ≥ 1, Y _i = Spec B _i is finite over X, where B _i are local rings.

(d) If f : Y → X is étale and there is a point y ∈ Y such that f (y) = x and k (y) = k (x), then f has a section s : X → Y .

(e) Let f ₁ , . . . , f _n ∈ A[T 1 , . . . , T _n ]. If there exists an a = (a ₁ , . . . , a _n ) ∈ k ⁿ such that f _i (a) = 0 and det

 ∂f _i

∂T _j

 (a)



6= 0, then there is a b ∈ A ⁿ such that b = a and f i (b) = 0.

[c.f. Étale Cohomology, Milne, p.32]

Proposition 6. Any complete local ring A is Henselian.

proof: Let B be an étale A-algebra, and suppose that there is a section s ₀ : B → k. Write A r = A /m ^r+1 . It suffices to show that there exist compatible sections s _r : B → A r , then they induce a section s : B → A.

It is clear for r = 0, and for r > 0, the existence of s _r follows from the existence of s _{r −1} and the following fact: Given an X-morphism g ₀ : X ₀ ^′ → Y , there is an X-morphism g : X ^′ → Y such that the diagram commutes

Y X ₀ ^′

X X ^′

f g

g

[c.f. EGA.IV.17][c.f. Milne, p.30]

A ring A is a subring of its completion b A, hence any local ring A is a subring of Henselian ring. We define the Henselization of A to be the Henselian ring A ^h with a local homomorphism i : A → A ^h such that for any other local homomorphism from A to a Henselian local ring factors through i uniquely. It is clear that the Henselization is unique if it exists. To prove the existence of the Henselization, we introduce the étale neighborhood.

Definition 7. An étale neighborhood of a local ring A is a pair (B, q) where B is an étale A-algebra and q is a prime ideal of B lying over m such that the induced map k → k (q) is an isomorphism.

Lemma 8. .

(a) If (B, q) and (B ^′ , q ^′ ) are étale neighborhoods of A with Spec B ^′ connected, then there is at most one

A-homomorphism f : B → B ^′ such that f ⁻¹ (q ^′ ) = q.

(b) Let (B, q) and (B ^′ , q ^′ ) be étale neighborhoods of A. Then there is an étale neighborhood (B ^′′ , q ^′′ ) of A with Spec B ^′′ connected and A-homomorphisms f : B → B ^′′ , f ^′ : B ^′ → B ^′′ such that f ⁻¹ (q ^′′ ) = q, f ^′−1 (q ^′′ ) = q ^′ .

proof: (a) Use the fact: Let f, g : Y ^′ → Y be X-morphisms with Y ^′ connected and Y étale separated over X. If there existsa point y ^′ ∈ Y ^′ such that f (y ^′ ) = g (y ^′ ) = y and the maps k (y) → k (y ^′ ) induced by f, g coincide, then f = g.

(b) Let C = B ⊗ A B ^′ . Then we have a map C → k induced by B → k and B ^′ → k. Let q ^′′ be the kernel.

Take c / ∈ q ^′′ and let B ^′′ = C _c . Then (B ^′′ , q ^′′ B ^′′ ) is as desired.

Corollary 9. For any local ring A, the Henselization A ^h exists.

proof: The étale neighborhoods of A with connected sepctra form a filtered direct system. Define A ^h , m ^h

= lim −→ (B, q). Then A ^h is a local A-algebra with maximal ideal m ^h and A ^h /m ^h = k, and it is indeed a Henselian ring.

Definition 10. Let X be a scheme and let x ∈ X. An étale neighborhood of x is a pair (Y, y) where Y is an étale X-scheme and y ∈ Y is mapped to x such that k (x) = k (y).

Similarly, the connected étale neighborhoods of x form a filtered system and lim −→ Γ (Y, O Y ) = O X,x ^h . Definition 11. A Henselian ring A is strictly Henselian if the residue field of A is separably algebraically closed.

Some of above conclusion can be rewrittent for strictly Henselian rings. The strict Henselization of A is a pair A ^sh , i

, where A ^sh is a strictly Henselian ring and i : A → A ^sh is a local homomorphism such that for any other local homomorphism from A to a strictly Henselization factors through i.

Definition 12. Let X be a scheme and x : Spec k → X a geometric point of X, where k is a separably closed field. An étale neighborhood of x is a commutative diagram

Spec k U

X

x

with U → X being étale.

Similarly O ^sh _X,x = lim −→ Γ (U, O U ) where the limit is taken over all étale neighborhood of x.

Proposition 13. .

(a) A composite of étale morphisms is étale.

(b) An étale morphism X → Y remains étale after an arbitrary base extension Z → Y . (c) Given morphisms X − → Y ^f − → Z. If g ◦ f and g are étale, then so is f. ^g

In the version of rings:

(c ^′ ) Let A → B → C be ring extensions. If B and C are étale over A, then C is étale over B.

For a scheme X, denote Ét (X) to be the category of all étale extensions of X, considered as a full subcategory of all X-schemes. All of morphisms in Ét (X) are étale by (c). Similarly define Ét (A) for a ring A.

Definition 14. A presheaf F on Ét (X) of abelian groups is a contravariant functor F : Ét (X) → (Ab)

A presheaf F on Ét (A) of abelain groups is a covariant functor F : Ét (A) → (Ab) Definition 15. A finite family B = 

U _i −→ U, i ∈ I ^ϕ



of étale morphisms is called an étale covering of a scheme U if U = ∪

i ∈I ϕ _i (U _i ).

Definition 16. A presheaf F is called a sheaf if the sequence F (U) → Y

i

F (U i ) ⇒ Y

i,j

F (U i × U U _j )

is exact for all coverings (U _i → U).

Consider the category of all étale coverings of a fixed object B ∈ Ét (A). A map between two coverings B → B ^′ , where

B = (B → B i , i ∈ I) , B ^′ = B → B _j ^′ , j ∈ J

is given by a map σ : J → I of the index sets and a family of homomorphisms B σ(j) → B j ^′ . For each covering B = (B → B i ), denote F (B) the set of all families s i ∈ F (B i ) with the above compatibility property.

Lemma 17. .

(a) For any two coverings B and B ^′ , there is a covering B ^′′ together with morphisms B → B ^′′ and

B ^′ → B ^′′ .

(b) Two morphisms B ⇒ B ^′ induce the same map F (B) → F (B ^′ ).

So we may consider f F (B) = lim −→ _B F (B). In general, f F is not a sheaf, however, we have the following proposition:

Proposition 18. .

(a) If F (B) → F (B) is injective, then f F is a sheaf.

(b) f F (B) → f F (B) is always injective.

Therefore c F = f F is always a sheaf, called the sheaf generated by the presheaf F . f

The category Ét (X) is small, then the collection of all presheaves with the natural transformations as morphisms forms a category. We consider the category of sheaves as a full subcategory.

Fact: The category of presheaves of abelian groups and the category of sheaves of abelian groups are abelian, and every sheaf is a subsheaf of an injective sheaf. The functor F 7→ F (X) (respectively, F (A)) is left exact.

Hence now we define the étale cohomology as

H _ét ⁱ (X, F ) = R ⁱ ( F 7→ F (X)) , H ét ⁱ (A, F ) = R ⁱ ( F 7→ F (A))

In particular, if X = Spec A, for any sheaf F on X, let F 0 be a sheaf on A defined by F 0 (B) = F (Spec B).

This gives an equivalence of categories, therefore we have

H _ét ⁱ (Spec A, F ) = H ét ⁱ (A, F 0 )

Definition 19. Let F be a presheaf. Define the stalk of F at the geometric point x to be the limit F x = lim −→ F (U )

where U runs through all étale neighborhoods of x.

Definition 20. Let f : X → Y be a morphism of schemes, F a sheaf on X. The direct image is defined to be

(f _∗ F ) (U) = F (X × Y U ) , U → Y étale

which is a sheaf on Y . Note that the functor F 7→ f _∗ F is left exact, hence we can define the higher derived image R ⁱ f _∗ F .

Proposition 21. R ⁱ f _∗ F agrees with the sheaf generated by the presheaf U 7→ H ⁱ (X × Y U, F U )

where we denote by F U the restriction of F w.r.t. the étale map X × Y U → X.