New PDF release: Coherent Algebraic Sheaves

By Jean-Pierre Serre, Piotr Achinger, Lukasz Krupa

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

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Example text

Then the homomorphism ι : A → Γ(V, OV ) is bijective. We apply Corollary 2 above to X = K r , p = 1, F = J (V ), the sheaf of ideals defined by V ; we obtain that every element of Γ(V, OV ) is the restriction of a section of O on X, that is, a polynomial, by Proposition 4 applied to X. 53 §3. Coherent algebraic sheaves on affine varieties 45 II Sections of a coherent algebraic sheaf on an affine variety Theorem 2. Let F be a coherent algebraic sheaf on an affine variety X. For every x ∈ X, the Ox,X –module Fx is generated by elements of Γ(X, F ).

Let G be a coherent algebraic sheaf on V which is zero outside W ; the annihilator of G does not necessarily contain J (W ) (in other words, G not always can be considered as an coherent algebraic sheaf on W ); all we can say is that it contains a power of J (W ). 40 Sheaves of fractional ideals Let V be an irreducible algebraic variety and let K(V ) denote the constant sheaf of rational functions on V (cf. n◦ 36); K(V ) is an algebraic sheaf which is not coherent if dim V > 0. An algebraic subsheaf F of K(V ) can be called a ,,sheaf of fractional ideals” since each Fx is a fractional ideal of Ox,V .

Suppose that there exist charts φ : Xi → Ui and let Tij = φ × φj (Xij ); Tij is the set of (φi (x), φj (x)) for x running over Xi ∩ Xj . The axiom (V AII ) takes therefore the following form: (V AII ) — For each pair (i, j), Tij is closed in Ui × Uj . In this form we recognize Weil’s axiom (A) (cf. [16], p. 167), except that Weil considered only irreducible varieties. Examples of algebraic varieties: Any locally closed subspace U of an affine space, equipped with the induced topology and the sheaf OU defined in n◦ 31 is an algebraic variety.