By Jean-Pierre Serre, Piotr Achinger, Lukasz Krupa

Translation of: Jean-Pierre Serre, "Faisceaux Algebriques Coherents", The Annals of arithmetic, 2d Ser., Vol. sixty one, No. 2. (Mar., 1955), pp. 197--278

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**Example text**

Ip , F ) = 0 for q > 0. The conditions (a) and (b) of Theorem 1, n◦ 29 are satisfied and the theorem follows. Theorem 5. Let X be an algebraic variety and U = {Ui }i∈I a finite covering of X by open affine subsets. Let 0 → A → B → C → 0 be an exact sequence of sheaves on X, the sheaf A being coherent algebraic. The canonical homomorphism H0q (U, C ) → H q (U, C ) (cf. n◦ 24) is bijective for all q ≥ 0. iq , which follows from Corollary 2 of Theorem 3. Corollary 1. Let X be an algebraic variety and let 0 → A → B → C → 0 be an exact sequence of sheaves on X, the sheaf A being coherent algebraic.

S ) belongs to Γ(U, OU ) since Γ(U, OU ) as a ring; it follows that it is a continuous function on U , thus its zero set is closed, which shows the continuity of φ. If we have x ∈ U and f ∈ Oφ(x),V , we can write f locally in the form f = P/Q, where P and Q are polynomials and 39 §1. Algebraic varieties II Q(φ(x)) = 0. The function f ◦ φ is then equal to P ◦ φ/Q ◦ φ in a neighborhood of x; from what we gave seen, P ◦ φ and Q ◦ φ are regular in a neighborhood of x. d. A composition of two regular maps is regular.

Ip , F ) = 0 for all q > 0. Applying then Proposition 5 of n◦ 29, we see that H q (U, F ) = H q (V, F ), and, as H q (V, F ) = 0 for q > 0 by Theorem 3, the Proposition is proven. Theorem 4. Let X be an algebraic variety, F a coherent algebraic sheaf on X and U = {Ui }i∈I a finite covering of X by open affine subsets. The homomorphism σ(U) : H n (U, F ) → H n (X, F ) is bijective for all n ≥ 0. Consider the family Vα of all finite coverings of X by open affine subsets. By the corollary of Theorem 1, these coverings are arbitrarily fine.

### Coherent Algebraic Sheaves by Jean-Pierre Serre, Piotr Achinger, Lukasz Krupa

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