By Lei Fu

Etale cohomology is a vital department in mathematics geometry. This e-book covers the most fabrics in SGA 1, SGA four, SGA four half and SGA five on etale cohomology conception, together with respectable thought, etale primary teams, Galois cohomology, etale cohomology, derived different types, base switch theorems, duality, and l-adic cohomology. the necessities for examining this ebook are uncomplicated algebraic geometry and complex commutative algebra.

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It follows that fg(s ) is also a local homomorphism. Hence f is a morphism of schemes. 2. Let g : S → S be a quasi-compact faithfully flat morphism, and let S = S ×S S . For any scheme Z, the sequence Hom(S, Z) → Hom(S , Z) ⇒ Hom(S , Z) is exact. Proof. 1. 3. Let g : S → S be a quasi-compact faithfully flat morphism, S = S ×S S , X and Y two S-schemes, X = X ×S S , Y = Y ×S S , X = X ×S S , and Y = Y ×S S . Then the sequence HomS (X, Y ) → HomS (X , Y ) ⇒ HomS (X , Y ) is exact. Proof. 2, the sequences Hom(X, Y ) → Hom(X , Y ) ⇒ Hom(X , Y ) Hom(X, S) → Hom(X , S) ⇒ Hom(X , S) are exact.

I) Let B0 be an A0 -algebra, Bλ = B0 ⊗A0 Aλ , B = B0 ⊗A0 A, (Cλ ) a direct system of A0 -algebras such that we have a morphism of direct systems from (Aλ ) to (Cλ ), and C = limλ Cλ . If B0 is an A0 -algebra of finite type −→ (resp. finite presentation), then the canonical map lim HomAλ (Bλ , Cλ ) → HomA (B, C) −→ λ is injective (resp. bijective). (ii) For any A-algebra B with finite presentation, there exists an Aλ algebra Bλ with finite presentation for a sufficiently large λ such that B ∼ = Bλ ⊗Aλ A.

7. Assume g : Y → Y is a quasi-compact faithfully flat morphism. Then the topology on Y coincides with the quotient topology induced from Y . 8. Let g : S → S be a quasi-compact faithfully flat morphism, S = S ×S S , O(S) (resp. O(S ), resp. O(S )) the set of open subsets of S (resp. S , resp. S ), and F(S) (resp. F(S ), resp. F(S )) the January 5, 2011 11:3 World Scientific Book - 9in x 6in 34 Etale Cohomology Theory set of closed subsets of S (resp. S , resp. S ). Then the following sequences are exact: p−1 1 O(S) → O(S ) ⇒ O(S ), p−1 2 p−1 1 F(S) → F(S ) ⇒ F(S ).