Étale cohomology [Lecture notes] by Uwe Jannsen

By Uwe Jannsen

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3 (c)). 1. 11 Let (X , T ) be a site and U an object in X . The functor H i (U, −) := H i (U, T ; −) : Sh(X , T ) → Ab F ⇝ H i (U, F ) is the i-th right derivative of the left exact functor F ⇝ F (U ) =: Γ(U, F ) =: H 0 (U, F ) . H i (U, F ) is called i-th cohomology of F on U (or i-th cohomology group on U with coeﬃcients in F ). By construction, H i (U, F ) = H i (I · (U )), where F → I · is an injective resolution F in Sh(X , T ). 12 If X is a topological space and F a sheaf on X, then H i (X, F ) is the usual cohomology of sheaves on X.

Ii) By definition, H i (Gk , −) is the i-th right derived functor of M → H 0 (Gk , M ) = M Gk . Hence ´etale cohomology of fields is Galois cohomology. 61 9 Henselian rings Henselian rings, and in particular the strictly henselian rings, play the same role in the ´etale topology as the local rings in the Zariski topology. Let A be a local ring with maximal ideal m and factor field k = A/m. 1 Let x be the closed point of X = Spec(A). A is called henselian, if the following equivalent conditions hold.

By construction, H i (U, F ) = H i (I · (U )), where F → I · is an injective resolution F in Sh(X , T ). 12 If X is a topological space and F a sheaf on X, then H i (X, F ) is the usual cohomology of sheaves on X. 13 Let f : (X ′ , T ′ ) → (X , T ) a morphism of sites. Then Ri f∗ is the i-th right derivative of the left exact functor f∗ : Sh(X ′ , T ′ ) → (Sh(X , T ) . Ri f∗ F is called the i-th higher direct image of F under f . Hence Ri f∗ F = Hi (f∗ I · ), where F → I · is an injective resolution in Sh(X ′ , T ′ ).